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Vincent Selhorst-Jones

Vincent Selhorst-Jones

Completing the Square and the Quadratic Formula

Slide Duration:

Table of Contents

I. Introduction
Introduction to Precalculus

10m 3s

Intro
0:00
Title of the Course
0:06
Different Names for the Course
0:07
Precalculus
0:12
Math Analysis
0:14
Trigonometry
0:16
Algebra III
0:20
Geometry II
0:24
College Algebra
0:30
Same Concepts
0:36
How do the Lessons Work?
0:54
Introducing Concepts
0:56
Apply Concepts
1:04
Go through Examples
1:25
Who is this Course For?
1:38
Those Who Need eExtra Help with Class Work
1:52
Those Working on Material but not in Formal Class at School
1:54
Those Who Want a Refresher
2:00
Try to Watch the Whole Lesson
2:20
Understanding is So Important
3:56
What to Watch First
5:26
Lesson #2: Sets, Elements, and Numbers
5:30
Lesson #7: Idea of a Function
5:33
Lesson #6: Word Problems
6:04
What to Watch First, cont.
6:46
Lesson #2: Sets, Elements and Numbers
6:56
Lesson #3: Variables, Equations, and Algebra
6:58
Lesson #4: Coordinate Systems
7:00
Lesson #5: Midpoint, Distance, the Pythagorean Theorem and Slope
7:02
Lesson #6: Word Problems
7:10
Lesson #7: Idea of a Function
7:12
Lesson #8: Graphs
7:14
Graphing Calculator Appendix
7:40
What to Watch Last
8:46
Let's get Started!
9:48
Sets, Elements, & Numbers

45m 11s

Intro
0:00
Introduction
0:05
Sets and Elements
1:19
Set
1:20
Element
1:23
Name a Set
2:20
Order The Elements Appear In Has No Effect on the Set
2:55
Describing/ Defining Sets
3:28
Directly Say All the Elements
3:36
Clearly Describing All the Members of the Set
3:55
Describing the Quality (or Qualities) Each member Of the Set Has In Common
4:32
Symbols: 'Element of' and 'Subset of'
6:01
Symbol is ∈
6:03
Subset Symbol is ⊂
6:35
Empty Set
8:07
Symbol is ∅
8:20
Since It's Empty, It is a Subset of All Sets
8:44
Union and Intersection
9:54
Union Symbol is ∪
10:08
Intersection Symbol is ∩
10:18
Sets Can Be Weird Stuff
12:26
Can Have Elements in a Set
12:50
We Can Have Infinite Sets
13:09
Example
13:22
Consider a Set Where We Take a Word and Then Repeat It An Ever Increasing Number of Times
14:08
This Set Has Infinitely Many Distinct Elements
14:40
Numbers as Sets
16:03
Natural Numbers ℕ
16:16
Including 0 and the Negatives ℤ
18:13
Rational Numbers ℚ
19:27
Can Express Rational Numbers with Decimal Expansions
22:05
Irrational Numbers
23:37
Real Numbers ℝ: Put the Rational and Irrational Numbers Together
25:15
Interval Notation and the Real Numbers
26:45
Include the End Numbers
27:06
Exclude the End Numbers
27:33
Example
28:28
Interval Notation: Infinity
29:09
Use -∞ or ∞ to Show an Interval Going on Forever in One Direction or the Other
29:14
Always Use Parentheses
29:50
Examples
30:27
Example 1
31:23
Example 2
35:26
Example 3
38:02
Example 4
42:21
Variables, Equations, & Algebra

35m 31s

Intro
0:00
What is a Variable?
0:05
A Variable is a Placeholder for a Number
0:11
Affects the Output of a Function or a Dependent Variable
0:24
Naming Variables
1:51
Useful to Use Symbols
2:21
What is a Constant?
4:14
A Constant is a Fixed, Unchanging Number
4:28
We Might Refer to a Symbol Representing a Number as a Constant
4:51
What is a Coefficient?
5:33
A Coefficient is a Multiplicative Factor on a Variable
5:37
Not All Coefficients are Constants
5:51
Expressions and Equations
6:42
An Expression is a String of Mathematical Symbols That Make Sense Used Together
7:05
An Equation is a Statement That Two Expression Have the Same Value
8:20
The Idea of Algebra
8:51
Equality
8:59
If Two Things Are the Same *Equal), Then We Can Do the Exact Same Operation to Both and the Results Will Be the Same
9:41
Always Do The Exact Same Thing to Both Sides
12:22
Solving Equations
13:23
When You Are Asked to Solve an Equation, You Are Being Asked to Solve for Something
13:33
Look For What Values Makes the Equation True
13:38
Isolate the Variable by Doing Algebra
14:37
Order of Operations
16:02
Why Certain Operations are Grouped
17:01
When You Don't Have to Worry About Order
17:39
Distributive Property
18:15
It Allows Multiplication to Act Over Addition in Parentheses
18:23
We Can Use the Distributive Property in Reverse to Combine Like Terms
19:05
Substitution
20:03
Use Information From One Equation in Another Equation
20:07
Put Your Substitution in Parentheses
20:44
Example 1
23:17
Example 2
25:49
Example 3
28:11
Example 4
30:02
Coordinate Systems

35m 2s

Intro
0:00
Inherent Order in ℝ
0:05
Real Numbers Come with an Inherent Order
0:11
Positive Numbers
0:21
Negative Numbers
0:58
'Less Than' and 'Greater Than'
2:04
Tip To Help You Remember the Signs
2:56
Inequality
4:06
Less Than or Equal and Greater Than or Equal
4:51
One Dimension: The Number Line
5:36
Graphically Represent ℝ on a Number Line
5:43
Note on Infinities
5:57
With the Number Line, We Can Directly See the Order We Put on ℝ
6:35
Ordered Pairs
7:22
Example
7:34
Allows Us to Talk About Two Numbers at the Same Time
9:41
Ordered Pairs of Real Numbers Cannot be Put Into an Order Like we Did with ℝ
10:41
Two Dimensions: The Plane
13:13
We Can Represent Ordered Pairs with the Plane
13:24
Intersection is known as the Origin
14:31
Plotting the Point
14:32
Plane = Coordinate Plane = Cartesian Plane = ℝ²
17:46
The Plane and Quadrants
18:50
Quadrant I
19:04
Quadrant II
19:21
Quadrant III
20:04
Quadrant IV
20:20
Three Dimensions: Space
21:02
Create Ordered Triplets
21:09
Visually Represent This
21:19
Three-Dimension = Space = ℝ³
21:47
Higher Dimensions
22:24
If We Have n Dimensions, We Call It n-Dimensional Space or ℝ to the nth Power
22:31
We Can Represent Places In This n-Dimensional Space As Ordered Groupings of n Numbers
22:41
Hard to Visualize Higher Dimensional Spaces
23:18
Example 1
25:07
Example 2
26:10
Example 3
28:58
Example 4
31:05
Midpoints, Distance, the Pythagorean Theorem, & Slope

48m 43s

Intro
0:00
Introduction
0:07
Midpoint: One Dimension
2:09
Example of Something More Complex
2:31
Use the Idea of a Middle
3:28
Find the Midpoint of Arbitrary Values a and b
4:17
How They're Equivalent
5:05
Official Midpoint Formula
5:46
Midpoint: Two Dimensions
6:19
The Midpoint Must Occur at the Horizontal Middle and the Vertical Middle
6:38
Arbitrary Pair of Points Example
7:25
Distance: One Dimension
9:26
Absolute Value
10:54
Idea of Forcing Positive
11:06
Distance: One Dimension, Formula
11:47
Distance Between Arbitrary a and b
11:48
Absolute Value Helps When the Distance is Negative
12:41
Distance Formula
12:58
The Pythagorean Theorem
13:24
a²+b²=c²
13:50
Distance: Two Dimensions
14:59
Break Into Horizontal and Vertical Parts and then Use the Pythagorean Theorem
15:16
Distance Between Arbitrary Points (x₁,y₁) and (x₂,y₂)
16:21
Slope
19:30
Slope is the Rate of Change
19:41
m = rise over run
21:27
Slope Between Arbitrary Points (x₁,y₁) and (x₂,y₂)
22:31
Interpreting Slope
24:12
Positive Slope and Negative Slope
25:40
m=1, m=0, m=-1
26:48
Example 1
28:25
Example 2
31:42
Example 3
36:40
Example 4
42:48
Word Problems

56m 31s

Intro
0:00
Introduction
0:05
What is a Word Problem?
0:45
Describes Any Problem That Primarily Gets Its Ideas Across With Words Instead of Math Symbols
0:48
Requires Us to Think
1:32
Why Are They So Hard?
2:11
Reason 1: No Simple Formula to Solve Them
2:16
Reason 2: Harder to Teach Word Problems
2:47
You Can Learn How to Do Them!
3:51
Grades
7:57
'But I'm Never Going to Use This In Real Life'
9:46
Solving Word Problems
12:58
First: Understand the Problem
13:37
Second: What Are You Looking For?
14:33
Third: Set Up Relationships
16:21
Fourth: Solve It!
17:48
Summary of Method
19:04
Examples on Things Other Than Math
20:21
Math-Specific Method: What You Need Now
25:30
Understand What the Problem is Talking About
25:37
Set Up and Name Any Variables You Need to Know
25:56
Set Up Equations Connecting Those Variables to the Information in the Problem Statement
26:02
Use the Equations to Solve for an Answer
26:14
Tip
26:58
Draw Pictures
27:22
Breaking Into Pieces
28:28
Try Out Hypothetical Numbers
29:52
Student Logic
31:27
Jump In!
32:40
Example 1
34:03
Example 2
39:15
Example 3
44:22
Example 4
50:24
II. Functions
Idea of a Function

39m 54s

Intro
0:00
Introduction
0:04
What is a Function?
1:06
A Visual Example and Non-Example
1:30
Function Notation
3:47
f(x)
4:05
Express What Sets the Function Acts On
5:45
Metaphors for a Function
6:17
Transformation
6:28
Map
7:17
Machine
8:56
Same Input Always Gives Same Output
10:01
If We Put the Same Input Into a Function, It Will Always Produce the Same Output
10:11
Example of Something That is Not a Function
11:10
A Non-Numerical Example
12:10
The Functions We Will Use
15:05
Unless Told Otherwise, We Will Assume Every Function Takes in Real Numbers and Outputs Real Numbers
15:11
Usually Told the Rule of a Given Function
15:27
How To Use a Function
16:18
Apply the Rule to Whatever Our Input Value Is
16:28
Make Sure to Wrap Your Substitutions in Parentheses
17:09
Functions and Tables
17:36
Table of Values, Sometimes Called a T-Table
17:46
Example
17:56
Domain: What Goes In
18:55
The Domain is the Set of all Inputs That the Function Can Accept
18:56
Example
19:40
Range: What Comes Out
21:27
The Range is the Set of All Possible Outputs a Function Can Assign
21:34
Example
21:49
Another Example Would Be Our Initial Function From Earlier in This Lesson
22:29
Example 1
23:45
Example 2
25:22
Example 3
27:27
Example 4
29:23
Example 5
33:33
Graphs

58m 26s

Intro
0:00
Introduction
0:04
How to Interpret Graphs
1:17
Input / Independent Variable
1:47
Output / Dependent Variable
2:00
Graph as Input ⇒ Output
2:23
One Way to Think of a Graph: See What Happened to Various Inputs
2:25
Example
2:47
Graph as Location of Solution
4:20
A Way to See Solutions
4:36
Example
5:20
Which Way Should We Interpret?
7:13
Easiest to Think In Terms of How Inputs Are Mapped to Outputs
7:20
Sometimes It's Easier to Think In Terms of Solutions
8:39
Pay Attention to Axes
9:50
Axes Tell Where the Graph Is and What Scale It Has
10:09
Often, The Axes Will Be Square
10:14
Example
12:06
Arrows or No Arrows?
16:07
Will Not Use Arrows at the End of Our Graphs
17:13
Graph Stops Because It Hits the Edge of the Graphing Axes, Not Because the Function Stops
17:18
How to Graph
19:47
Plot Points
20:07
Connect with Curves
21:09
If You Connect with Straight Lines
21:44
Graphs of Functions are Smooth
22:21
More Points ⇒ More Accurate
23:38
Vertical Line Test
27:44
If a Vertical Line Could Intersect More Than One Point On a Graph, It Can Not Be the Graph of a Function
28:41
Every Point on a Graph Tells Us Where the x-Value Below is Mapped
30:07
Domain in Graphs
31:37
The Domain is the Set of All Inputs That a Function Can Accept
31:44
Be Aware That Our Function Probably Continues Past the Edge of Our 'Viewing Window'
33:19
Range in Graphs
33:53
Graphing Calculators: Check the Appendix!
36:55
Example 1
38:37
Example 2
45:19
Example 3
50:41
Example 4
53:28
Example 5
55:50
Properties of Functions

48m 49s

Intro
0:00
Introduction
0:05
Increasing Decreasing Constant
0:43
Looking at a Specific Graph
1:15
Increasing Interval
2:39
Constant Function
4:15
Decreasing Interval
5:10
Find Intervals by Looking at the Graph
5:32
Intervals Show x-values; Write in Parentheses
6:39
Maximum and Minimums
8:48
Relative (Local) Max/Min
10:20
Formal Definition of Relative Maximum
12:44
Formal Definition of Relative Minimum
13:05
Max/Min, More Terms
14:18
Definition of Extrema
15:01
Average Rate of Change
16:11
Drawing a Line for the Average Rate
16:48
Using the Slope of the Secant Line
17:36
Slope in Function Notation
18:45
Zeros/Roots/x-intercepts
19:45
What Zeros in a Function Mean
20:25
Even Functions
22:30
Odd Functions
24:36
Even/Odd Functions and Graphs
26:28
Example of an Even Function
27:12
Example of an Odd Function
28:03
Example 1
29:35
Example 2
33:07
Example 3
40:32
Example 4
42:34
Function Petting Zoo

29m 20s

Intro
0:00
Introduction
0:04
Don't Forget that Axes Matter!
1:44
The Constant Function
2:40
The Identity Function
3:44
The Square Function
4:40
The Cube Function
5:44
The Square Root Function
6:51
The Reciprocal Function
8:11
The Absolute Value Function
10:19
The Trigonometric Functions
11:56
f(x)=sin(x)
12:12
f(x)=cos(x)
12:24
Alternate Axes
12:40
The Exponential and Logarithmic Functions
13:35
Exponential Functions
13:44
Logarithmic Functions
14:24
Alternating Axes
15:17
Transformations and Compositions
16:08
Example 1
17:52
Example 2
18:33
Example 3
20:24
Example 4
26:07
Transformation of Functions

48m 35s

Intro
0:00
Introduction
0:04
Vertical Shift
1:12
Graphical Example
1:21
A Further Explanation
2:16
Vertical Stretch/Shrink
3:34
Graph Shrinks
3:46
Graph Stretches
3:51
A Further Explanation
5:07
Horizontal Shift
6:49
Moving the Graph to the Right
7:28
Moving the Graph to the Left
8:12
A Further Explanation
8:19
Understanding Movement on the x-axis
8:38
Horizontal Stretch/Shrink
12:59
Shrinking the Graph
13:40
Stretching the Graph
13:48
A Further Explanation
13:55
Understanding Stretches from the x-axis
14:12
Vertical Flip (aka Mirror)
16:55
Example Graph
17:07
Multiplying the Vertical Component by -1
17:18
Horizontal Flip (aka Mirror)
18:43
Example Graph
19:01
Multiplying the Horizontal Component by -1
19:54
Summary of Transformations
22:11
Stacking Transformations
24:46
Order Matters
25:20
Transformation Example
25:52
Example 1
29:21
Example 2
34:44
Example 3
38:10
Example 4
43:46
Composite Functions

33m 24s

Intro
0:00
Introduction
0:04
Arithmetic Combinations
0:40
Basic Operations
1:20
Definition of the Four Arithmetic Combinations
1:40
Composite Functions
2:53
The Function as a Machine
3:32
Function Compositions as Multiple Machines
3:59
Notation for Composite Functions
4:46
Two Formats
6:02
Another Visual Interpretation
7:17
How to Use Composite Functions
8:21
Example of on Function acting on Another
9:17
Example 1
11:03
Example 2
15:27
Example 3
21:11
Example 4
27:06
Piecewise Functions

51m 42s

Intro
0:00
Introduction
0:04
Analogies to a Piecewise Function
1:16
Different Potatoes
1:41
Factory Production
2:27
Notations for Piecewise Functions
3:39
Notation Examples from Analogies
6:11
Example of a Piecewise (with Table)
7:24
Example of a Non-Numerical Piecewise
11:35
Graphing Piecewise Functions
14:15
Graphing Piecewise Functions, Example
16:26
Continuous Functions
16:57
Statements of Continuity
19:30
Example of Continuous and Non-Continuous Graphs
20:05
Interesting Functions: the Step Function
22:00
Notation for the Step Function
22:40
How the Step Function Works
22:56
Graph of the Step Function
25:30
Example 1
26:22
Example 2
28:49
Example 3
36:50
Example 4
46:11
Inverse Functions

49m 37s

Intro
0:00
Introduction
0:04
Analogy by picture
1:10
How to Denote the inverse
1:40
What Comes out of the Inverse
1:52
Requirement for Reversing
2:02
The Basketball Factory
2:12
The Importance of Information
2:45
One-to-One
4:04
Requirement for Reversibility
4:21
When a Function has an Inverse
4:43
One-to-One
5:13
Not One-to-One
5:50
Not a Function
6:19
Horizontal Line Test
7:01
How to the test Works
7:12
One-to-One
8:12
Not One-to-One
8:45
Definition: Inverse Function
9:12
Formal Definition
9:21
Caution to Students
10:02
Domain and Range
11:12
Finding the Range of the Function Inverse
11:56
Finding the Domain of the Function Inverse
12:11
Inverse of an Inverse
13:09
Its just x!
13:26
Proof
14:03
Graphical Interpretation
17:07
Horizontal Line Test
17:20
Graph of the Inverse
18:04
Swapping Inputs and Outputs to Draw Inverses
19:02
How to Find the Inverse
21:03
What We Are Looking For
21:21
Reversing the Function
21:38
A Method to Find Inverses
22:33
Check Function is One-to-One
23:04
Swap f(x) for y
23:25
Interchange x and y
23:41
Solve for y
24:12
Replace y with the inverse
24:40
Some Comments
25:01
Keeping Step 2 and 3 Straight
25:44
Switching to Inverse
26:12
Checking Inverses
28:52
How to Check an Inverse
29:06
Quick Example of How to Check
29:56
Example 1
31:48
Example 2
34:56
Example 3
39:29
Example 4
46:19
Variation Direct and Inverse

28m 49s

Intro
0:00
Introduction
0:06
Direct Variation
1:14
Same Direction
1:21
Common Example: Groceries
1:56
Different Ways to Say that Two Things Vary Directly
2:28
Basic Equation for Direct Variation
2:55
Inverse Variation
3:40
Opposite Direction
3:50
Common Example: Gravity
4:53
Different Ways to Say that Two Things Vary Indirectly
5:48
Basic Equation for Indirect Variation
6:33
Joint Variation
7:27
Equation for Joint Variation
7:53
Explanation of the Constant
8:48
Combined Variation
9:35
Gas Law as a Combination
9:44
Single Constant
10:33
Example 1
10:49
Example 2
13:34
Example 3
15:39
Example 4
19:48
III. Polynomials
Intro to Polynomials

38m 41s

Intro
0:00
Introduction
0:04
Definition of a Polynomial
1:04
Starting Integer
2:06
Structure of a Polynomial
2:49
The a Constants
3:34
Polynomial Function
5:13
Polynomial Equation
5:23
Polynomials with Different Variables
5:36
Degree
6:23
Informal Definition
6:31
Find the Largest Exponent Variable
6:44
Quick Examples
7:36
Special Names for Polynomials
8:59
Based on the Degree
9:23
Based on the Number of Terms
10:12
Distributive Property (aka 'FOIL')
11:37
Basic Distributive Property
12:21
Distributing Two Binomials
12:55
Longer Parentheses
15:12
Reverse: Factoring
17:26
Long-Term Behavior of Polynomials
17:48
Examples
18:13
Controlling Term--Term with the Largest Exponent
19:33
Positive and Negative Coefficients on the Controlling Term
20:21
Leading Coefficient Test
22:07
Even Degree, Positive Coefficient
22:13
Even Degree, Negative Coefficient
22:39
Odd Degree, Positive Coefficient
23:09
Odd Degree, Negative Coefficient
23:27
Example 1
25:11
Example 2
27:16
Example 3
31:16
Example 4
34:41
Roots (Zeros) of Polynomials

41m 7s

Intro
0:00
Introduction
0:05
Roots in Graphs
1:17
The x-intercepts
1:33
How to Remember What 'Roots' Are
1:50
Naïve Attempts
2:31
Isolating Variables
2:45
Failures of Isolating Variables
3:30
Missing Solutions
4:59
Factoring: How to Find Roots
6:28
How Factoring Works
6:36
Why Factoring Works
7:20
Steps to Finding Polynomial Roots
9:21
Factoring: How to Find Roots CAUTION
10:08
Factoring is Not Easy
11:32
Factoring Quadratics
13:08
Quadratic Trinomials
13:21
Form of Factored Binomials
13:38
Factoring Examples
14:40
Factoring Quadratics, Check Your Work
16:58
Factoring Higher Degree Polynomials
18:19
Factoring a Cubic
18:32
Factoring a Quadratic
19:04
Factoring: Roots Imply Factors
19:54
Where a Root is, A Factor Is
20:01
How to Use Known Roots to Make Factoring Easier
20:35
Not all Polynomials Can be Factored
22:30
Irreducible Polynomials
23:27
Complex Numbers Help
23:55
Max Number of Roots/Factors
24:57
Limit to Number of Roots Equal to the Degree
25:18
Why there is a Limit
25:25
Max Number of Peaks/Valleys
26:39
Shape Information from Degree
26:46
Example Graph
26:54
Max, But Not Required
28:00
Example 1
28:37
Example 2
31:21
Example 3
36:12
Example 4
38:40
Completing the Square and the Quadratic Formula

39m 43s

Intro
0:00
Introduction
0:05
Square Roots and Equations
0:51
Taking the Square Root to Find the Value of x
0:55
Getting the Positive and Negative Answers
1:05
Completing the Square: Motivation
2:04
Polynomials that are Easy to Solve
2:20
Making Complex Polynomials Easy to Solve
3:03
Steps to Completing the Square
4:30
Completing the Square: Method
7:22
Move C over
7:35
Divide by A
7:44
Find r
7:59
Add to Both Sides to Complete the Square
8:49
Solving Quadratics with Ease
9:56
The Quadratic Formula
11:38
Derivation
11:43
Final Form
12:23
Follow Format to Use Formula
13:38
How Many Roots?
14:53
The Discriminant
15:47
What the Discriminant Tells Us: How Many Roots
15:58
How the Discriminant Works
16:30
Example 1: Complete the Square
18:24
Example 2: Solve the Quadratic
22:00
Example 3: Solve for Zeroes
25:28
Example 4: Using the Quadratic Formula
30:52
Properties of Quadratic Functions

45m 34s

Intro
0:00
Introduction
0:05
Parabolas
0:35
Examples of Different Parabolas
1:06
Axis of Symmetry and Vertex
1:28
Drawing an Axis of Symmetry
1:51
Placing the Vertex
2:28
Looking at the Axis of Symmetry and Vertex for other Parabolas
3:09
Transformations
4:18
Reviewing Transformation Rules
6:28
Note the Different Horizontal Shift Form
7:45
An Alternate Form to Quadratics
8:54
The Constants: k, h, a
9:05
Transformations Formed
10:01
Analyzing Different Parabolas
10:10
Switching Forms by Completing the Square
11:43
Vertex of a Parabola
16:30
Vertex at (h, k)
16:47
Vertex in Terms of a, b, and c Coefficients
17:28
Minimum/Maximum at Vertex
18:19
When a is Positive
18:25
When a is Negative
18:52
Axis of Symmetry
19:54
Incredibly Minor Note on Grammar
20:52
Example 1
21:48
Example 2
26:35
Example 3
28:55
Example 4
31:40
Intermediate Value Theorem and Polynomial Division

46m 8s

Intro
0:00
Introduction
0:05
Reminder: Roots Imply Factors
1:32
The Intermediate Value Theorem
3:41
The Basis: U between a and b
4:11
U is on the Function
4:52
Intermediate Value Theorem, Proof Sketch
5:51
If Not True, the Graph Would Have to Jump
5:58
But Graph is Defined as Continuous
6:43
Finding Roots with the Intermediate Value Theorem
7:01
Picking a and b to be of Different Signs
7:10
Must Be at Least One Root
7:46
Dividing a Polynomial
8:16
Using Roots and Division to Factor
8:38
Long Division Refresher
9:08
The Division Algorithm
12:18
How It Works to Divide Polynomials
12:37
The Parts of the Equation
13:24
Rewriting the Equation
14:47
Polynomial Long Division
16:20
Polynomial Long Division In Action
16:29
One Step at a Time
20:51
Synthetic Division
22:46
Setup
23:11
Synthetic Division, Example
24:44
Which Method Should We Use
26:39
Advantages of Synthetic Method
26:49
Advantages of Long Division
27:13
Example 1
29:24
Example 2
31:27
Example 3
36:22
Example 4
40:55
Complex Numbers

45m 36s

Intro
0:00
Introduction
0:04
A Wacky Idea
1:02
The Definition of the Imaginary Number
1:22
How it Helps Solve Equations
2:20
Square Roots and Imaginary Numbers
3:15
Complex Numbers
5:00
Real Part and Imaginary Part
5:20
When Two Complex Numbers are Equal
6:10
Addition and Subtraction
6:40
Deal with Real and Imaginary Parts Separately
7:36
Two Quick Examples
7:54
Multiplication
9:07
FOIL Expansion
9:14
Note What Happens to the Square of the Imaginary Number
9:41
Two Quick Examples
10:22
Division
11:27
Complex Conjugates
13:37
Getting Rid of i
14:08
How to Denote the Conjugate
14:48
Division through Complex Conjugates
16:11
Multiply by the Conjugate of the Denominator
16:28
Example
17:46
Factoring So-Called 'Irreducible' Quadratics
19:24
Revisiting the Quadratic Formula
20:12
Conjugate Pairs
20:37
But Are the Complex Numbers 'Real'?
21:27
What Makes a Number Legitimate
25:38
Where Complex Numbers are Used
27:20
Still, We Won't See Much of C
29:05
Example 1
30:30
Example 2
33:15
Example 3
38:12
Example 4
42:07
Fundamental Theorem of Algebra

19m 9s

Intro
0:00
Introduction
0:05
Idea: Hidden Roots
1:16
Roots in Complex Form
1:42
All Polynomials Have Roots
2:08
Fundamental Theorem of Algebra
2:21
Where Are All the Imaginary Roots, Then?
3:17
All Roots are Complex
3:45
Real Numbers are a Subset of Complex Numbers
3:59
The n Roots Theorem
5:01
For Any Polynomial, Its Degree is Equal to the Number of Roots
5:11
Equivalent Statement
5:24
Comments: Multiplicity
6:29
Non-Distinct Roots
6:59
Denoting Multiplicity
7:20
Comments: Complex Numbers Necessary
7:41
Comments: Complex Coefficients Allowed
8:55
Comments: Existence Theorem
9:59
Proof Sketch of n Roots Theorem
10:45
First Root
11:36
Second Root
13:23
Continuation to Find all Roots
16:00
IV. Rational Functions
Rational Functions and Vertical Asymptotes

33m 22s

Intro
0:00
Introduction
0:05
Definition of a Rational Function
1:20
Examples of Rational Functions
2:30
Why They are Called 'Rational'
2:47
Domain of a Rational Function
3:15
Undefined at Denominator Zeros
3:25
Otherwise all Reals
4:16
Investigating a Fundamental Function
4:50
The Domain of the Function
5:04
What Occurs at the Zeroes of the Denominator
5:20
Idea of a Vertical Asymptote
6:23
What's Going On?
6:58
Approaching x=0 from the left
7:32
Approaching x=0 from the right
8:34
Dividing by Very Small Numbers Results in Very Large Numbers
9:31
Definition of a Vertical Asymptote
10:05
Vertical Asymptotes and Graphs
11:15
Drawing Asymptotes by Using a Dashed Line
11:27
The Graph Can Never Touch Its Undefined Point
12:00
Not All Zeros Give Asymptotes
13:02
Special Cases: When Numerator and Denominator Go to Zero at the Same Time
14:58
Cancel out Common Factors
15:49
How to Find Vertical Asymptotes
16:10
Figure out What Values Are Not in the Domain of x
16:24
Determine if the Numerator and Denominator Share Common Factors and Cancel
16:45
Find Denominator Roots
17:33
Note if Asymptote Approaches Negative or Positive Infinity
18:06
Example 1
18:57
Example 2
21:26
Example 3
23:04
Example 4
30:01
Horizontal Asymptotes

34m 16s

Intro
0:00
Introduction
0:05
Investigating a Fundamental Function
0:53
What Happens as x Grows Large
1:00
Different View
1:12
Idea of a Horizontal Asymptote
1:36
What's Going On?
2:24
What Happens as x Grows to a Large Negative Number
2:49
What Happens as x Grows to a Large Number
3:30
Dividing by Very Large Numbers Results in Very Small Numbers
3:52
Example Function
4:41
Definition of a Vertical Asymptote
8:09
Expanding the Idea
9:03
What's Going On?
9:48
What Happens to the Function in the Long Run?
9:51
Rewriting the Function
10:13
Definition of a Slant Asymptote
12:09
Symbolical Definition
12:30
Informal Definition
12:45
Beyond Slant Asymptotes
13:03
Not Going Beyond Slant Asymptotes
14:39
Horizontal/Slant Asymptotes and Graphs
15:43
How to Find Horizontal and Slant Asymptotes
16:52
How to Find Horizontal Asymptotes
17:12
Expand the Given Polynomials
17:18
Compare the Degrees of the Numerator and Denominator
17:40
How to Find Slant Asymptotes
20:05
Slant Asymptotes Exist When n+m=1
20:08
Use Polynomial Division
20:24
Example 1
24:32
Example 2
25:53
Example 3
26:55
Example 4
29:22
Graphing Asymptotes in a Nutshell

49m 7s

Intro
0:00
Introduction
0:05
A Process for Graphing
1:22
1. Factor Numerator and Denominator
1:50
2. Find Domain
2:53
3. Simplifying the Function
3:59
4. Find Vertical Asymptotes
4:59
5. Find Horizontal/Slant Asymptotes
5:24
6. Find Intercepts
7:35
7. Draw Graph (Find Points as Necessary)
9:21
Draw Graph Example
11:21
Vertical Asymptote
11:41
Horizontal Asymptote
11:50
Other Graphing
12:16
Test Intervals
15:08
Example 1
17:57
Example 2
23:01
Example 3
29:02
Example 4
33:37
Partial Fractions

44m 56s

Intro
0:00
Introduction: Idea
0:04
Introduction: Prerequisites and Uses
1:57
Proper vs. Improper Polynomial Fractions
3:11
Possible Things in the Denominator
4:38
Linear Factors
6:16
Example of Linear Factors
7:03
Multiple Linear Factors
7:48
Irreducible Quadratic Factors
8:25
Example of Quadratic Factors
9:26
Multiple Quadratic Factors
9:49
Mixing Factor Types
10:28
Figuring Out the Numerator
11:10
How to Solve for the Constants
11:30
Quick Example
11:40
Example 1
14:29
Example 2
18:35
Example 3
20:33
Example 4
28:51
V. Exponential & Logarithmic Functions
Understanding Exponents

35m 17s

Intro
0:00
Introduction
0:05
Fundamental Idea
1:46
Expanding the Idea
2:28
Multiplication of the Same Base
2:40
Exponents acting on Exponents
3:45
Different Bases with the Same Exponent
4:31
To the Zero
5:35
To the First
5:45
Fundamental Rule with the Zero Power
6:35
To the Negative
7:45
Any Number to a Negative Power
8:14
A Fraction to a Negative Power
9:58
Division with Exponential Terms
10:41
To the Fraction
11:33
Square Root
11:58
Any Root
12:59
Summary of Rules
14:38
To the Irrational
17:21
Example 1
20:34
Example 2
23:42
Example 3
27:44
Example 4
31:44
Example 5
33:15
Exponential Functions

47m 4s

Intro
0:00
Introduction
0:05
Definition of an Exponential Function
0:48
Definition of the Base
1:02
Restrictions on the Base
1:16
Computing Exponential Functions
2:29
Harder Computations
3:10
When to Use a Calculator
3:21
Graphing Exponential Functions: a>1
6:02
Three Examples
6:13
What to Notice on the Graph
7:44
A Story
8:27
Story Diagram
9:15
Increasing Exponentials
11:29
Story Morals
14:40
Application: Compound Interest
15:15
Compounding Year after Year
16:01
Function for Compounding Interest
16:51
A Special Number: e
20:55
Expression for e
21:28
Where e stabilizes
21:55
Application: Continuously Compounded Interest
24:07
Equation for Continuous Compounding
24:22
Exponential Decay 0<a<1
25:50
Three Examples
26:11
Why they 'lose' value
26:54
Example 1
27:47
Example 2
33:11
Example 3
36:34
Example 4
41:28
Introduction to Logarithms

40m 31s

Intro
0:00
Introduction
0:04
Definition of a Logarithm, Base 2
0:51
Log 2 Defined
0:55
Examples
2:28
Definition of a Logarithm, General
3:23
Examples of Logarithms
5:15
Problems with Unusual Bases
7:38
Shorthand Notation: ln and log
9:44
base e as ln
10:01
base 10 as log
10:34
Calculating Logarithms
11:01
using a calculator
11:34
issues with other bases
11:58
Graphs of Logarithms
13:21
Three Examples
13:29
Slow Growth
15:19
Logarithms as Inverse of Exponentiation
16:02
Using Base 2
16:05
General Case
17:10
Looking More Closely at Logarithm Graphs
19:16
The Domain of Logarithms
20:41
Thinking about Logs like Inverses
21:08
The Alternate
24:00
Example 1
25:59
Example 2
30:03
Example 3
32:49
Example 4
37:34
Properties of Logarithms

42m 33s

Intro
0:00
Introduction
0:04
Basic Properties
1:12
Inverse--log(exp)
1:43
A Key Idea
2:44
What We Get through Exponentiation
3:18
B Always Exists
4:50
Inverse--exp(log)
5:53
Logarithm of a Power
7:44
Logarithm of a Product
10:07
Logarithm of a Quotient
13:48
Caution! There Is No Rule for loga(M+N)
16:12
Summary of Properties
17:42
Change of Base--Motivation
20:17
No Calculator Button
20:59
A Specific Example
21:45
Simplifying
23:45
Change of Base--Formula
24:14
Example 1
25:47
Example 2
29:08
Example 3
31:14
Example 4
34:13
Solving Exponential and Logarithmic Equations

34m 10s

Intro
0:00
Introduction
0:05
One to One Property
1:09
Exponential
1:26
Logarithmic
1:44
Specific Considerations
2:02
One-to-One Property
3:30
Solving by One-to-One
4:11
Inverse Property
6:09
Solving by Inverses
7:25
Dealing with Equations
7:50
Example of Taking an Exponent or Logarithm of an Equation
9:07
A Useful Property
11:57
Bring Down Exponents
12:01
Try to Simplify
13:20
Extraneous Solutions
13:45
Example 1
16:37
Example 2
19:39
Example 3
21:37
Example 4
26:45
Example 5
29:37
Application of Exponential and Logarithmic Functions

48m 46s

Intro
0:00
Introduction
0:06
Applications of Exponential Functions
1:07
A Secret!
2:17
Natural Exponential Growth Model
3:07
Figure out r
3:34
A Secret!--Why Does It Work?
4:44
e to the r Morphs
4:57
Example
5:06
Applications of Logarithmic Functions
8:32
Examples
8:43
What Logarithms are Useful For
9:53
Example 1
11:29
Example 2
15:30
Example 3
26:22
Example 4
32:05
Example 5
39:19
VI. Trigonometric Functions
Angles

39m 5s

Intro
0:00
Degrees
0:22
Circle is 360 Degrees
0:48
Splitting a Circle
1:13
Radians
2:08
Circle is 2 Pi Radians
2:31
One Radian
2:52
Half-Circle and Right Angle
4:00
Converting Between Degrees and Radians
6:24
Formulas for Degrees and Radians
6:52
Coterminal, Complementary, Supplementary Angles
7:23
Coterminal Angles
7:30
Complementary Angles
9:40
Supplementary Angles
10:08
Example 1: Dividing a Circle
10:38
Example 2: Converting Between Degrees and Radians
11:56
Example 3: Quadrants and Coterminal Angles
14:18
Extra Example 1: Common Angle Conversions
-1
Extra Example 2: Quadrants and Coterminal Angles
-2
Sine and Cosine Functions

43m 16s

Intro
0:00
Sine and Cosine
0:15
Unit Circle
0:22
Coordinates on Unit Circle
1:03
Right Triangles
1:52
Adjacent, Opposite, Hypotenuse
2:25
Master Right Triangle Formula: SOHCAHTOA
2:48
Odd Functions, Even Functions
4:40
Example: Odd Function
4:56
Example: Even Function
7:30
Example 1: Sine and Cosine
10:27
Example 2: Graphing Sine and Cosine Functions
14:39
Example 3: Right Triangle
21:40
Example 4: Odd, Even, or Neither
26:01
Extra Example 1: Right Triangle
-1
Extra Example 2: Graphing Sine and Cosine Functions
-2
Sine and Cosine Values of Special Angles

33m 5s

Intro
0:00
45-45-90 Triangle and 30-60-90 Triangle
0:08
45-45-90 Triangle
0:21
30-60-90 Triangle
2:06
Mnemonic: All Students Take Calculus (ASTC)
5:21
Using the Unit Circle
5:59
New Angles
6:21
Other Quadrants
9:43
Mnemonic: All Students Take Calculus
10:13
Example 1: Convert, Quadrant, Sine/Cosine
13:11
Example 2: Convert, Quadrant, Sine/Cosine
16:48
Example 3: All Angles and Quadrants
20:21
Extra Example 1: Convert, Quadrant, Sine/Cosine
-1
Extra Example 2: All Angles and Quadrants
-2
Modified Sine Waves: Asin(Bx+C)+D and Acos(Bx+C)+D

52m 3s

Intro
0:00
Amplitude and Period of a Sine Wave
0:38
Sine Wave Graph
0:58
Amplitude: Distance from Middle to Peak
1:18
Peak: Distance from Peak to Peak
2:41
Phase Shift and Vertical Shift
4:13
Phase Shift: Distance Shifted Horizontally
4:16
Vertical Shift: Distance Shifted Vertically
6:48
Example 1: Amplitude/Period/Phase and Vertical Shift
8:04
Example 2: Amplitude/Period/Phase and Vertical Shift
17:39
Example 3: Find Sine Wave Given Attributes
25:23
Extra Example 1: Amplitude/Period/Phase and Vertical Shift
-1
Extra Example 2: Find Cosine Wave Given Attributes
-2
Tangent and Cotangent Functions

36m 4s

Intro
0:00
Tangent and Cotangent Definitions
0:21
Tangent Definition
0:25
Cotangent Definition
0:47
Master Formula: SOHCAHTOA
1:01
Mnemonic
1:16
Tangent and Cotangent Values
2:29
Remember Common Values of Sine and Cosine
2:46
90 Degrees Undefined
4:36
Slope and Menmonic: ASTC
5:47
Uses of Tangent
5:54
Example: Tangent of Angle is Slope
6:09
Sign of Tangent in Quadrants
7:49
Example 1: Graph Tangent and Cotangent Functions
10:42
Example 2: Tangent and Cotangent of Angles
16:09
Example 3: Odd, Even, or Neither
18:56
Extra Example 1: Tangent and Cotangent of Angles
-1
Extra Example 2: Tangent and Cotangent of Angles
-2
Secant and Cosecant Functions

27m 18s

Intro
0:00
Secant and Cosecant Definitions
0:17
Secant Definition
0:18
Cosecant Definition
0:33
Example 1: Graph Secant Function
0:48
Example 2: Values of Secant and Cosecant
6:49
Example 3: Odd, Even, or Neither
12:49
Extra Example 1: Graph of Cosecant Function
-1
Extra Example 2: Values of Secant and Cosecant
-2
Inverse Trigonometric Functions

32m 58s

Intro
0:00
Arcsine Function
0:24
Restrictions between -1 and 1
0:43
Arcsine Notation
1:26
Arccosine Function
3:07
Restrictions between -1 and 1
3:36
Cosine Notation
3:53
Arctangent Function
4:30
Between -Pi/2 and Pi/2
4:44
Tangent Notation
5:02
Example 1: Domain/Range/Graph of Arcsine
5:45
Example 2: Arcsin/Arccos/Arctan Values
10:46
Example 3: Domain/Range/Graph of Arctangent
17:14
Extra Example 1: Domain/Range/Graph of Arccosine
-1
Extra Example 2: Arcsin/Arccos/Arctan Values
-2
Computations of Inverse Trigonometric Functions

31m 8s

Intro
0:00
Inverse Trigonometric Function Domains and Ranges
0:31
Arcsine
0:41
Arccosine
1:14
Arctangent
1:41
Example 1: Arcsines of Common Values
2:44
Example 2: Odd, Even, or Neither
5:57
Example 3: Arccosines of Common Values
12:24
Extra Example 1: Arctangents of Common Values
-1
Extra Example 2: Arcsin/Arccos/Arctan Values
-2
VII. Trigonometric Identities
Pythagorean Identity

19m 11s

Intro
0:00
Pythagorean Identity
0:17
Pythagorean Triangle
0:27
Pythagorean Identity
0:45
Example 1: Use Pythagorean Theorem to Prove Pythagorean Identity
1:14
Example 2: Find Angle Given Cosine and Quadrant
4:18
Example 3: Verify Trigonometric Identity
8:00
Extra Example 1: Use Pythagorean Identity to Prove Pythagorean Theorem
-1
Extra Example 2: Find Angle Given Cosine and Quadrant
-2
Identity Tan(squared)x+1=Sec(squared)x

23m 16s

Intro
0:00
Main Formulas
0:19
Companion to Pythagorean Identity
0:27
For Cotangents and Cosecants
0:52
How to Remember
0:58
Example 1: Prove the Identity
1:40
Example 2: Given Tan Find Sec
3:42
Example 3: Prove the Identity
7:45
Extra Example 1: Prove the Identity
-1
Extra Example 2: Given Sec Find Tan
-2
Addition and Subtraction Formulas

52m 52s

Intro
0:00
Addition and Subtraction Formulas
0:09
How to Remember
0:48
Cofunction Identities
1:31
How to Remember Graphically
1:44
Where to Use Cofunction Identities
2:52
Example 1: Derive the Formula for cos(A-B)
3:08
Example 2: Use Addition and Subtraction Formulas
16:03
Example 3: Use Addition and Subtraction Formulas to Prove Identity
25:11
Extra Example 1: Use cos(A-B) and Cofunction Identities
-1
Extra Example 2: Convert to Radians and use Formulas
-2
Double Angle Formulas

29m 5s

Intro
0:00
Main Formula
0:07
How to Remember from Addition Formula
0:18
Two Other Forms
1:35
Example 1: Find Sine and Cosine of Angle using Double Angle
3:16
Example 2: Prove Trigonometric Identity using Double Angle
9:37
Example 3: Use Addition and Subtraction Formulas
12:38
Extra Example 1: Find Sine and Cosine of Angle using Double Angle
-1
Extra Example 2: Prove Trigonometric Identity using Double Angle
-2
Half-Angle Formulas

43m 55s

Intro
0:00
Main Formulas
0:09
Confusing Part
0:34
Example 1: Find Sine and Cosine of Angle using Half-Angle
0:54
Example 2: Prove Trigonometric Identity using Half-Angle
11:51
Example 3: Prove the Half-Angle Formula for Tangents
18:39
Extra Example 1: Find Sine and Cosine of Angle using Half-Angle
-1
Extra Example 2: Prove Trigonometric Identity using Half-Angle
-2
VIII. Applications of Trigonometry
Trigonometry in Right Angles

25m 43s

Intro
0:00
Master Formula for Right Angles
0:11
SOHCAHTOA
0:15
Only for Right Triangles
1:26
Example 1: Find All Angles in a Triangle
2:19
Example 2: Find Lengths of All Sides of Triangle
7:39
Example 3: Find All Angles in a Triangle
11:00
Extra Example 1: Find All Angles in a Triangle
-1
Extra Example 2: Find Lengths of All Sides of Triangle
-2
Law of Sines

56m 40s

Intro
0:00
Law of Sines Formula
0:18
SOHCAHTOA
0:27
Any Triangle
0:59
Graphical Representation
1:25
Solving Triangle Completely
2:37
When to Use Law of Sines
2:55
ASA, SAA, SSA, AAA
2:59
SAS, SSS for Law of Cosines
7:11
Example 1: How Many Triangles Satisfy Conditions, Solve Completely
8:44
Example 2: How Many Triangles Satisfy Conditions, Solve Completely
15:30
Example 3: How Many Triangles Satisfy Conditions, Solve Completely
28:32
Extra Example 1: How Many Triangles Satisfy Conditions, Solve Completely
-1
Extra Example 2: How Many Triangles Satisfy Conditions, Solve Completely
-2
Law of Cosines

49m 5s

Intro
0:00
Law of Cosines Formula
0:23
Graphical Representation
0:34
Relates Sides to Angles
1:00
Any Triangle
1:20
Generalization of Pythagorean Theorem
1:32
When to Use Law of Cosines
2:26
SAS, SSS
2:30
Heron's Formula
4:49
Semiperimeter S
5:11
Example 1: How Many Triangles Satisfy Conditions, Solve Completely
5:53
Example 2: How Many Triangles Satisfy Conditions, Solve Completely
15:19
Example 3: Find Area of a Triangle Given All Side Lengths
26:33
Extra Example 1: How Many Triangles Satisfy Conditions, Solve Completely
-1
Extra Example 2: Length of Third Side and Area of Triangle
-2
Finding the Area of a Triangle

27m 37s

Intro
0:00
Master Right Triangle Formula and Law of Cosines
0:19
SOHCAHTOA
0:27
Law of Cosines
1:23
Heron's Formula
2:22
Semiperimeter S
2:37
Example 1: Area of Triangle with Two Sides and One Angle
3:12
Example 2: Area of Triangle with Three Sides
6:11
Example 3: Area of Triangle with Three Sides, No Heron's Formula
8:50
Extra Example 1: Area of Triangle with Two Sides and One Angle
-1
Extra Example 2: Area of Triangle with Two Sides and One Angle
-2
Word Problems and Applications of Trigonometry

34m 25s

Intro
0:00
Formulas to Remember
0:11
SOHCAHTOA
0:15
Law of Sines
0:55
Law of Cosines
1:48
Heron's Formula
2:46
Example 1: Telephone Pole Height
4:01
Example 2: Bridge Length
7:48
Example 3: Area of Triangular Field
14:20
Extra Example 1: Kite Height
-1
Extra Example 2: Roads to a Town
-2
IX. Systems of Equations and Inequalities
Systems of Linear Equations

55m 40s

Intro
0:00
Introduction
0:04
Graphs as Location of 'True'
1:49
All Locations that Make the Function True
2:25
Understand the Relationship Between Solutions and the Graph
3:43
Systems as Graphs
4:07
Equations as Lines
4:20
Intersection Point
5:19
Three Possibilities for Solutions
6:17
Independent
6:24
Inconsistent
6:36
Dependent
7:06
Solving by Substitution
8:37
Solve for One Variable
9:07
Substitute into the Second Equation
9:34
Solve for Both Variables
10:12
What If a System is Inconsistent or Dependent?
11:08
No Solutions
11:25
Infinite Solutions
12:30
Solving by Elimination
13:56
Example
14:22
Determining the Number of Solutions
16:30
Why Elimination Makes Sense
17:25
Solving by Graphing Calculator
19:59
Systems with More than Two Variables
23:22
Example 1
25:49
Example 2
30:22
Example 3
34:11
Example 4
38:55
Example 5
46:01
(Non-) Example 6
53:37
Systems of Linear Inequalities

1h 13s

Intro
0:00
Introduction
0:04
Inequality Refresher-Solutions
0:46
Equation Solutions vs. Inequality Solutions
1:02
Essentially a Wide Variety of Answers
1:35
Refresher--Negative Multiplication Flips
1:43
Refresher--Negative Flips: Why?
3:19
Multiplication by a Negative
3:43
The Relationship Flips
3:55
Refresher--Stick to Basic Operations
4:34
Linear Equations in Two Variables
6:50
Graphing Linear Inequalities
8:28
Why It Includes a Whole Section
8:43
How to Show The Difference Between Strict and Not Strict Inequalities
10:08
Dashed Line--Not Solutions
11:10
Solid Line--Are Solutions
11:24
Test Points for Shading
11:42
Example of Using a Point
12:41
Drawing Shading from the Point
13:14
Graphing a System
14:53
Set of Solutions is the Overlap
15:17
Example
15:22
Solutions are Best Found Through Graphing
18:05
Linear Programming-Idea
19:52
Use a Linear Objective Function
20:15
Variables in Objective Function have Constraints
21:24
Linear Programming-Method
22:09
Rearrange Equations
22:21
Graph
22:49
Critical Solution is at the Vertex of the Overlap
23:40
Try Each Vertice
24:35
Example 1
24:58
Example 2
28:57
Example 3
33:48
Example 4
43:10
Nonlinear Systems

41m 1s

Intro
0:00
Introduction
0:06
Substitution
1:12
Example
1:22
Elimination
3:46
Example
3:56
Elimination is Less Useful for Nonlinear Systems
4:56
Graphing
5:56
Using a Graphing Calculator
6:44
Number of Solutions
8:44
Systems of Nonlinear Inequalities
10:02
Graph Each Inequality
10:06
Dashed and/or Solid
10:18
Shade Appropriately
11:14
Example 1
13:24
Example 2
15:50
Example 3
22:02
Example 4
29:06
Example 4, cont.
33:40
X. Vectors and Matrices
Vectors

1h 9m 31s

Intro
0:00
Introduction
0:10
Magnitude of the Force
0:22
Direction of the Force
0:48
Vector
0:52
Idea of a Vector
1:30
How Vectors are Denoted
2:00
Component Form
3:20
Angle Brackets and Parentheses
3:50
Magnitude/Length
4:26
Denoting the Magnitude of a Vector
5:16
Direction/Angle
7:52
Always Draw a Picture
8:50
Component Form from Magnitude & Angle
10:10
Scaling by Scalars
14:06
Unit Vectors
16:26
Combining Vectors - Algebraically
18:10
Combining Vectors - Geometrically
19:54
Resultant Vector
20:46
Alternate Component Form: i, j
21:16
The Zero Vector
23:18
Properties of Vectors
24:20
No Multiplication (Between Vectors)
28:30
Dot Product
29:40
Motion in a Medium
30:10
Fish in an Aquarium Example
31:38
More Than Two Dimensions
33:12
More Than Two Dimensions - Magnitude
34:18
Example 1
35:26
Example 2
38:10
Example 3
45:48
Example 4
50:40
Example 4, cont.
56:07
Example 5
1:01:32
Dot Product & Cross Product

35m 20s

Intro
0:00
Introduction
0:08
Dot Product - Definition
0:42
Dot Product Results in a Scalar, Not a Vector
2:10
Example in Two Dimensions
2:34
Angle and the Dot Product
2:58
The Dot Product of Two Vectors is Deeply Related to the Angle Between the Two Vectors
2:59
Proof of Dot Product Formula
4:14
Won't Directly Help Us Better Understand Vectors
4:18
Dot Product - Geometric Interpretation
4:58
We Can Interpret the Dot Product as a Measure of How Long and How Parallel Two Vectors Are
7:26
Dot Product - Perpendicular Vectors
8:24
If the Dot Product of Two Vectors is 0, We Know They are Perpendicular to Each Other
8:54
Cross Product - Definition
11:08
Cross Product Only Works in Three Dimensions
11:09
Cross Product - A Mnemonic
12:16
The Determinant of a 3 x 3 Matrix and Standard Unit Vectors
12:17
Cross Product - Geometric Interpretations
14:30
The Right-Hand Rule
15:17
Cross Product - Geometric Interpretations Cont.
17:00
Example 1
18:40
Example 2
22:50
Example 3
24:04
Example 4
26:20
Bonus Round
29:18
Proof: Dot Product Formula
29:24
Proof: Dot Product Formula, cont.
30:38
Matrices

54m 7s

Intro
0:00
Introduction
0:08
Definition of a Matrix
3:02
Size or Dimension
3:58
Square Matrix
4:42
Denoted by Capital Letters
4:56
When are Two Matrices Equal?
5:04
Examples of Matrices
6:44
Rows x Columns
6:46
Talking About Specific Entries
7:48
We Use Capitals to Denote a Matrix and Lower Case to Denotes Its Entries
8:32
Using Entries to Talk About Matrices
10:08
Scalar Multiplication
11:26
Scalar = Real Number
11:34
Example
12:36
Matrix Addition
13:08
Example
14:22
Matrix Multiplication
15:00
Example
18:52
Matrix Multiplication, cont.
19:58
Matrix Multiplication and Order (Size)
25:26
Make Sure Their Orders are Compatible
25:27
Matrix Multiplication is NOT Commutative
28:20
Example
30:08
Special Matrices - Zero Matrix (0)
32:48
Zero Matrix Has 0 for All of its Entries
32:49
Special Matrices - Identity Matrix (I)
34:14
Identity Matrix is a Square Matrix That Has 1 for All Its Entries on the Main Diagonal and 0 for All Other Entries
34:15
Example 1
36:16
Example 2
40:00
Example 3
44:54
Example 4
50:08
Determinants & Inverses of Matrices

47m 12s

Intro
0:00
Introduction
0:06
Not All Matrices Are Invertible
1:30
What Must a Matrix Have to Be Invertible?
2:08
Determinant
2:32
The Determinant is a Real Number Associated With a Square Matrix
2:38
If the Determinant of a Matrix is Nonzero, the Matrix is Invertible
3:40
Determinant of a 2 x 2 Matrix
4:34
Think in Terms of Diagonals
5:12
Minors and Cofactors - Minors
6:24
Example
6:46
Minors and Cofactors - Cofactors
8:00
Cofactor is Closely Based on the Minor
8:01
Alternating Sign Pattern
9:04
Determinant of Larger Matrices
10:56
Example
13:00
Alternative Method for 3x3 Matrices
16:46
Not Recommended
16:48
Inverse of a 2 x 2 Matrix
19:02
Inverse of Larger Matrices
20:00
Using Inverse Matrices
21:06
When Multiplied Together, They Create the Identity Matrix
21:24
Example 1
23:45
Example 2
27:21
Example 3
32:49
Example 4
36:27
Finding the Inverse of Larger Matrices
41:59
General Inverse Method - Step 1
43:25
General Inverse Method - Step 2
43:27
General Inverse Method - Step 2, cont.
43:27
General Inverse Method - Step 3
45:15
Using Matrices to Solve Systems of Linear Equations

58m 34s

Intro
0:00
Introduction
0:12
Augmented Matrix
1:44
We Can Represent the Entire Linear System With an Augmented Matrix
1:50
Row Operations
3:22
Interchange the Locations of Two Rows
3:50
Multiply (or Divide) a Row by a Nonzero Number
3:58
Add (or Subtract) a Multiple of One Row to Another
4:12
Row Operations - Keep Notes!
5:50
Suggested Symbols
7:08
Gauss-Jordan Elimination - Idea
8:04
Gauss-Jordan Elimination - Idea, cont.
9:16
Reduced Row-Echelon Form
9:18
Gauss-Jordan Elimination - Method
11:36
Begin by Writing the System As An Augmented Matrix
11:38
Gauss-Jordan Elimination - Method, cont.
13:48
Cramer's Rule - 2 x 2 Matrices
17:08
Cramer's Rule - n x n Matrices
19:24
Solving with Inverse Matrices
21:10
Solving Inverse Matrices, cont.
25:28
The Mighty (Graphing) Calculator
26:38
Example 1
29:56
Example 2
33:56
Example 3
37:00
Example 3, cont.
45:04
Example 4
51:28
XI. Alternate Ways to Graph
Parametric Equations

53m 33s

Intro
0:00
Introduction
0:06
Definition
1:10
Plane Curve
1:24
The Key Idea
2:00
Graphing with Parametric Equations
2:52
Same Graph, Different Equations
5:04
How Is That Possible?
5:36
Same Graph, Different Equations, cont.
5:42
Here's Another to Consider
7:56
Same Plane Curve, But Still Different
8:10
A Metaphor for Parametric Equations
9:36
Think of Parametric Equations As a Way to Describe the Motion of An Object
9:38
Graph Shows Where It Went, But Not Speed
10:32
Eliminating Parameters
12:14
Rectangular Equation
12:16
Caution
13:52
Creating Parametric Equations
14:30
Interesting Graphs
16:38
Graphing Calculators, Yay!
19:18
Example 1
22:36
Example 2
28:26
Example 3
37:36
Example 4
41:00
Projectile Motion
44:26
Example 5
47:00
Polar Coordinates

48m 7s

Intro
0:00
Introduction
0:04
Polar Coordinates Give Us a Way To Describe the Location of a Point
0:26
Polar Equations and Functions
0:50
Plotting Points with Polar Coordinates
1:06
The Distance of the Point from the Origin
1:09
The Angle of the Point
1:33
Give Points as the Ordered Pair (r,θ)
2:03
Visualizing Plotting in Polar Coordinates
2:32
First Way We Can Plot
2:39
Second Way We Can Plot
2:50
First, We'll Look at Visualizing r, Then θ
3:09
Rotate the Length Counter-Clockwise by θ
3:38
Alternatively, We Can Visualize θ, Then r
4:06
'Polar Graph Paper'
6:17
Horizontal and Vertical Tick Marks Are Not Useful for Polar
6:42
Use Concentric Circles to Helps Up See Distance From the Pole
7:08
Can Use Arc Sectors to See Angles
7:57
Multiple Ways to Name a Point
9:17
Examples
9:30
For Any Angle θ, We Can Make an Equivalent Angle
10:44
Negative Values for r
11:58
If r Is Negative, We Go In The Direction Opposite the One That The Angle θ Points Out
12:22
Another Way to Name the Same Point: Add π to θ and Make r Negative
13:44
Converting Between Rectangular and Polar
14:37
Rectangular Way to Name
14:43
Polar Way to Name
14:52
The Rectangular System Must Have a Right Angle Because It's Based on a Rectangle
15:08
Connect Both Systems Through Basic Trigonometry
15:38
Equation to Convert From Polar to Rectangular Coordinate Systems
16:55
Equation to Convert From Rectangular to Polar Coordinate Systems
17:13
Converting to Rectangular is Easy
17:20
Converting to Polar is a Bit Trickier
17:21
Draw Pictures
18:55
Example 1
19:50
Example 2
25:17
Example 3
31:05
Example 4
35:56
Example 5
41:49
Polar Equations & Functions

38m 16s

Intro
0:00
Introduction
0:04
Equations and Functions
1:16
Independent Variable
1:21
Dependent Variable
1:30
Examples
1:46
Always Assume That θ Is In Radians
2:44
Graphing in Polar Coordinates
3:29
Graph is the Same Way We Graph 'Normal' Stuff
3:32
Example
3:52
Graphing in Polar - Example, Cont.
6:45
Tips for Graphing
9:23
Notice Patterns
10:19
Repetition
13:39
Graphing Equations of One Variable
14:39
Converting Coordinate Types
16:16
Use the Same Conversion Formulas From the Previous Lesson
16:23
Interesting Graphs
17:48
Example 1
18:03
Example 2
18:34
Graphing Calculators, Yay!
19:07
Plot Random Things, Alter Equations You Understand, Get a Sense for How Polar Stuff Works
19:11
Check Out the Appendix
19:26
Example 1
21:36
Example 2
28:13
Example 3
34:24
Example 4
35:52
XII. Complex Numbers and Polar Coordinates
Polar Form of Complex Numbers

40m 43s

Intro
0:00
Polar Coordinates
0:49
Rectangular Form
0:52
Polar Form
1:25
R and Theta
1:51
Polar Form Conversion
2:27
R and Theta
2:35
Optimal Values
4:05
Euler's Formula
4:25
Multiplying Two Complex Numbers in Polar Form
6:10
Multiply r's Together and Add Exponents
6:32
Example 1: Convert Rectangular to Polar Form
7:17
Example 2: Convert Polar to Rectangular Form
13:49
Example 3: Multiply Two Complex Numbers
17:28
Extra Example 1: Convert Between Rectangular and Polar Forms
-1
Extra Example 2: Simplify Expression to Polar Form
-2
DeMoivre's Theorem

57m 37s

Intro
0:00
Introduction to DeMoivre's Theorem
0:10
n nth Roots
3:06
DeMoivre's Theorem: Finding nth Roots
3:52
Relation to Unit Circle
6:29
One nth Root for Each Value of k
7:11
Example 1: Convert to Polar Form and Use DeMoivre's Theorem
8:24
Example 2: Find Complex Eighth Roots
15:27
Example 3: Find Complex Roots
27:49
Extra Example 1: Convert to Polar Form and Use DeMoivre's Theorem
-1
Extra Example 2: Find Complex Fourth Roots
-2
XIII. Counting & Probability
Counting

31m 36s

Intro
0:00
Introduction
0:08
Combinatorics
0:56
Definition: Event
1:24
Example
1:50
Visualizing an Event
3:02
Branching line diagram
3:06
Addition Principle
3:40
Example
4:18
Multiplication Principle
5:42
Example
6:24
Pigeonhole Principle
8:06
Example
10:26
Draw Pictures
11:06
Example 1
12:02
Example 2
14:16
Example 3
17:34
Example 4
21:26
Example 5
25:14
Permutations & Combinations

44m 3s

Intro
0:00
Introduction
0:08
Permutation
0:42
Combination
1:10
Towards a Permutation Formula
2:38
How Many Ways Can We Arrange the Letters A, B, C, D, and E?
3:02
Towards a Permutation Formula, cont.
3:34
Factorial Notation
6:56
Symbol Is '!'
6:58
Examples
7:32
Permutation of n Objects
8:44
Permutation of r Objects out of n
9:04
What If We Have More Objects Than We Have Slots to Fit Them Into?
9:46
Permutation of r Objects Out of n, cont.
10:28
Distinguishable Permutations
14:46
What If Not All Of the Objects We're Permuting Are Distinguishable From Each Other?
14:48
Distinguishable Permutations, cont.
17:04
Combinations
19:04
Combinations, cont.
20:56
Example 1
23:10
Example 2
26:16
Example 3
28:28
Example 4
31:52
Example 5
33:58
Example 6
36:34
Probability

36m 58s

Intro
0:00
Introduction
0:06
Definition: Sample Space
1:18
Event = Something Happening
1:20
Sample Space
1:36
Probability of an Event
2:12
Let E Be An Event and S Be The Corresponding Sample Space
2:14
'Equally Likely' Is Important
3:52
Fair and Random
5:26
Interpreting Probability
6:34
How Can We Interpret This Value?
7:24
We Can Represent Probability As a Fraction, a Decimal, Or a Percentage
8:04
One of Multiple Events Occurring
9:52
Mutually Exclusive Events
10:38
What If The Events Are Not Mutually Exclusive?
12:20
Taking the Possibility of Overlap Into Account
13:24
An Event Not Occurring
17:14
Complement of E
17:22
Independent Events
19:36
Independent
19:48
Conditional Events
21:28
What Is The Events Are Not Independent Though?
21:30
Conditional Probability
22:16
Conditional Events, cont.
23:51
Example 1
25:27
Example 2
27:09
Example 3
28:57
Example 4
30:51
Example 5
34:15
XIV. Conic Sections
Parabolas

41m 27s

Intro
0:00
What is a Parabola?
0:20
Definition of a Parabola
0:29
Focus
0:59
Directrix
1:15
Axis of Symmetry
3:08
Vertex
3:33
Minimum or Maximum
3:44
Standard Form
4:59
Horizontal Parabolas
5:08
Vertex Form
5:19
Upward or Downward
5:41
Example: Standard Form
6:06
Graphing Parabolas
8:31
Shifting
8:51
Example: Completing the Square
9:22
Symmetry and Translation
12:18
Example: Graph Parabola
12:40
Latus Rectum
17:13
Length
18:15
Example: Latus Rectum
18:35
Horizontal Parabolas
18:57
Not Functions
20:08
Example: Horizontal Parabola
21:21
Focus and Directrix
24:11
Horizontal
24:48
Example 1: Parabola Standard Form
25:12
Example 2: Graph Parabola
30:00
Example 3: Graph Parabola
33:13
Example 4: Parabola Equation
37:28
Circles

21m 3s

Intro
0:00
What are Circles?
0:08
Example: Equidistant
0:17
Radius
0:32
Equation of a Circle
0:44
Example: Standard Form
1:11
Graphing Circles
1:47
Example: Circle
1:56
Center Not at Origin
3:07
Example: Completing the Square
3:51
Example 1: Equation of Circle
6:44
Example 2: Center and Radius
11:51
Example 3: Radius
15:08
Example 4: Equation of Circle
16:57
Ellipses

46m 51s

Intro
0:00
What Are Ellipses?
0:11
Foci
0:23
Properties of Ellipses
1:43
Major Axis, Minor Axis
1:47
Center
1:54
Length of Major Axis and Minor Axis
3:21
Standard Form
5:33
Example: Standard Form of Ellipse
6:09
Vertical Major Axis
9:14
Example: Vertical Major Axis
9:46
Graphing Ellipses
12:51
Complete the Square and Symmetry
13:00
Example: Graphing Ellipse
13:16
Equation with Center at (h, k)
19:57
Horizontal and Vertical
20:14
Difference
20:27
Example: Center at (h, k)
20:55
Example 1: Equation of Ellipse
24:05
Example 2: Equation of Ellipse
27:57
Example 3: Equation of Ellipse
32:32
Example 4: Graph Ellipse
38:27
Hyperbolas

38m 15s

Intro
0:00
What are Hyperbolas?
0:12
Two Branches
0:18
Foci
0:38
Properties
2:00
Transverse Axis and Conjugate Axis
2:06
Vertices
2:46
Length of Transverse Axis
3:14
Distance Between Foci
3:31
Length of Conjugate Axis
3:38
Standard Form
5:45
Vertex Location
6:36
Known Points
6:52
Vertical Transverse Axis
7:26
Vertex Location
7:50
Asymptotes
8:36
Vertex Location
8:56
Rectangle
9:28
Diagonals
10:29
Graphing Hyperbolas
12:58
Example: Hyperbola
13:16
Equation with Center at (h, k)
16:32
Example: Center at (h, k)
17:21
Example 1: Equation of Hyperbola
19:20
Example 2: Equation of Hyperbola
22:48
Example 3: Graph Hyperbola
26:05
Example 4: Equation of Hyperbola
36:29
Conic Sections

18m 43s

Intro
0:00
Conic Sections
0:16
Double Cone Sections
0:24
Standard Form
1:27
General Form
1:37
Identify Conic Sections
2:16
B = 0
2:50
X and Y
3:22
Identify Conic Sections, Cont.
4:46
Parabola
5:17
Circle
5:51
Ellipse
6:31
Hyperbola
7:10
Example 1: Identify Conic Section
8:01
Example 2: Identify Conic Section
11:03
Example 3: Identify Conic Section
11:38
Example 4: Identify Conic Section
14:50
XV. Sequences, Series, & Induction
Introduction to Sequences

57m 45s

Intro
0:00
Introduction
0:06
Definition: Sequence
0:28
Infinite Sequence
2:08
Finite Sequence
2:22
Length
2:58
Formula for the nth Term
3:22
Defining a Sequence Recursively
5:54
Initial Term
7:58
Sequences and Patterns
10:40
First, Identify a Pattern
12:52
How to Get From One Term to the Next
17:38
Tips for Finding Patterns
19:52
More Tips for Finding Patterns
24:14
Even More Tips
26:50
Example 1
30:32
Example 2
34:54
Fibonacci Sequence
34:55
Example 3
38:40
Example 4
45:02
Example 5
49:26
Example 6
51:54
Introduction to Series

40m 27s

Intro
0:00
Introduction
0:06
Definition: Series
1:20
Why We Need Notation
2:48
Simga Notation (AKA Summation Notation)
4:44
Thing Being Summed
5:42
Index of Summation
6:21
Lower Limit of Summation
7:09
Upper Limit of Summation
7:23
Sigma Notation, Example
7:36
Sigma Notation for Infinite Series
9:08
How to Reindex
10:58
How to Reindex, Expanding
12:56
How to Reindex, Substitution
16:46
Properties of Sums
19:42
Example 1
23:46
Example 2
25:34
Example 3
27:12
Example 4
29:54
Example 5
32:06
Example 6
37:16
Arithmetic Sequences & Series

31m 36s

Intro
0:00
Introduction
0:05
Definition: Arithmetic Sequence
0:47
Common Difference
1:13
Two Examples
1:19
Form for the nth Term
2:14
Recursive Relation
2:33
Towards an Arithmetic Series Formula
5:12
Creating a General Formula
10:09
General Formula for Arithmetic Series
14:23
Example 1
15:46
Example 2
17:37
Example 3
22:21
Example 4
24:09
Example 5
27:14
Geometric Sequences & Series

39m 27s

Intro
0:00
Introduction
0:06
Definition
0:48
Form for the nth Term
2:42
Formula for Geometric Series
5:16
Infinite Geometric Series
11:48
Diverges
13:04
Converges
14:48
Formula for Infinite Geometric Series
16:32
Example 1
20:32
Example 2
22:02
Example 3
26:00
Example 4
30:48
Example 5
34:28
Mathematical Induction

49m 53s

Intro
0:00
Introduction
0:06
Belief Vs. Proof
1:22
A Metaphor for Induction
6:14
The Principle of Mathematical Induction
11:38
Base Case
13:24
Inductive Step
13:30
Inductive Hypothesis
13:52
A Remark on Statements
14:18
Using Mathematical Induction
16:58
Working Example
19:58
Finding Patterns
28:46
Example 1
30:17
Example 2
37:50
Example 3
42:38
The Binomial Theorem

1h 13m 13s

Intro
0:00
Introduction
0:06
We've Learned That a Binomial Is An Expression That Has Two Terms
0:07
Understanding Binomial Coefficients
1:20
Things We Notice
2:24
What Goes In the Blanks?
5:52
Each Blank is Called a Binomial Coefficient
6:18
The Binomial Theorem
6:38
Example
8:10
The Binomial Theorem, cont.
10:46
We Can Also Write This Expression Compactly Using Sigma Notation
12:06
Proof of the Binomial Theorem
13:22
Proving the Binomial Theorem Is Within Our Reach
13:24
Pascal's Triangle
15:12
Pascal's Triangle, cont.
16:12
Diagonal Addition of Terms
16:24
Zeroth Row
18:04
First Row
18:12
Why Do We Care About Pascal's Triangle?
18:50
Pascal's Triangle, Example
19:26
Example 1
21:26
Example 2
24:34
Example 3
28:34
Example 4
32:28
Example 5
37:12
Time for the Fireworks!
43:38
Proof of the Binomial Theorem
43:44
We'll Prove This By Induction
44:04
Proof (By Induction)
46:36
Proof, Base Case
47:00
Proof, Inductive Step - Notation Discussion
49:22
Induction Step
49:24
Proof, Inductive Step - Setting Up
52:26
Induction Hypothesis
52:34
What We What To Show
52:44
Proof, Inductive Step - Start
54:18
Proof, Inductive Step - Middle
55:38
Expand Sigma Notations
55:48
Proof, Inductive Step - Middle, cont.
58:40
Proof, Inductive Step - Checking In
1:01:08
Let's Check In With Our Original Goal
1:01:12
Want to Show
1:01:18
Lemma - A Mini Theorem
1:02:18
Proof, Inductive Step - Lemma
1:02:52
Proof of Lemma: Let's Investigate the Left Side
1:03:08
Proof, Inductive Step - Nearly There
1:07:54
Proof, Inductive Step - End!
1:09:18
Proof, Inductive Step - End!, cont.
1:11:01
XVI. Preview of Calculus
Idea of a Limit

40m 22s

Intro
0:00
Introduction
0:05
Motivating Example
1:26
Fuzzy Notion of a Limit
3:38
Limit is the Vertical Location a Function is Headed Towards
3:44
Limit is What the Function Output is Going to Be
4:15
Limit Notation
4:33
Exploring Limits - 'Ordinary' Function
5:26
Test Out
5:27
Graphing, We See The Answer Is What We Would Expect
5:44
Exploring Limits - Piecewise Function
6:45
If We Modify the Function a Bit
6:49
Exploring Limits - A Visual Conception
10:08
Definition of a Limit
12:07
If f(x) Becomes Arbitrarily Close to Some Number L as x Approaches Some Number c, Then the Limit of f(x) As a Approaches c is L.
12:09
We Are Not Concerned with f(x) at x=c
12:49
We Are Considering x Approaching From All Directions, Not Just One Side
13:10
Limits Do Not Always Exist
15:47
Finding Limits
19:49
Graphs
19:52
Tables
21:48
Precise Methods
24:53
Example 1
26:06
Example 2
27:39
Example 3
30:51
Example 4
33:11
Example 5
37:07
Formal Definition of a Limit

57m 11s

Intro
0:00
Introduction
0:06
New Greek Letters
2:42
Delta
3:14
Epsilon
3:46
Sometimes Called the Epsilon-Delta Definition of a Limit
3:56
Formal Definition of a Limit
4:22
What does it MEAN!?!?
5:00
The Groundwork
5:38
Set Up the Limit
5:39
The Function is Defined Over Some Portion of the Reals
5:58
The Horizontal Location is the Value the Limit Will Approach
6:28
The Vertical Location L is Where the Limit Goes To
7:00
The Epsilon-Delta Part
7:26
The Hard Part is the Second Part of the Definition
7:30
Second Half of Definition
10:04
Restrictions on the Allowed x Values
10:28
The Epsilon-Delta Part, cont.
13:34
Sherlock Holmes and Dr. Watson
15:08
The Adventure of the Delta-Epsilon Limit
15:16
Setting
15:18
We Begin By Setting Up the Game As Follows
15:52
The Adventure of the Delta-Epsilon, cont.
17:24
This Game is About Limits
17:46
What If I Try Larger?
19:39
Technically, You Haven't Proven the Limit
20:53
Here is the Method
21:18
What We Should Concern Ourselves With
22:20
Investigate the Left Sides of the Expressions
25:24
We Can Create the Following Inequalities
28:08
Finally…
28:50
Nothing Like a Good Proof to Develop the Appetite
30:42
Example 1
31:02
Example 1, cont.
36:26
Example 2
41:46
Example 2, cont.
47:50
Finding Limits

32m 40s

Intro
0:00
Introduction
0:08
Method - 'Normal' Functions
2:04
The Easiest Limits to Find
2:06
It Does Not 'Break'
2:18
It Is Not Piecewise
2:26
Method - 'Normal' Functions, Example
3:38
Method - 'Normal' Functions, cont.
4:54
The Functions We're Used to Working With Go Where We Expect Them To Go
5:22
A Limit is About Figuring Out Where a Function is 'Headed'
5:42
Method - Canceling Factors
7:18
One Weird Thing That Often Happens is Dividing By 0
7:26
Method - Canceling Factors, cont.
8:16
Notice That The Two Functions Are Identical With the Exception of x=0
8:20
Method - Canceling Factors, cont.
10:00
Example
10:52
Method - Rationalization
12:04
Rationalizing a Portion of Some Fraction
12:05
Conjugate
12:26
Method - Rationalization, cont.
13:14
Example
13:50
Method - Piecewise
16:28
The Limits of Piecewise Functions
16:30
Example 1
17:42
Example 2
18:44
Example 3
20:20
Example 4
22:24
Example 5
24:24
Example 6
27:12
Continuity & One-Sided Limits

32m 43s

Intro
0:00
Introduction
0:06
Motivating Example
0:56
Continuity - Idea
2:14
Continuous Function
2:18
All Parts of Function Are Connected
2:28
Function's Graph Can Be Drawn Without Lifting Pencil
2:36
There Are No Breaks or Holes in Graph
2:56
Continuity - Idea, cont.
3:38
We Can Interpret the Break in the Continuity of f(x) as an Issue With the Function 'Jumping'
3:52
Continuity - Definition
5:16
A Break in Continuity is Caused By the Limit Not Matching Up With What the Function Does
5:18
Discontinuous
6:02
Discontinuity
6:10
Continuity and 'Normal' Functions
6:48
Return of the Motivating Example
8:14
One-Sided Limit
8:48
One-Sided Limit - Definition
9:16
Only Considers One Side
9:20
Be Careful to Keep Track of Which Symbol Goes With Which Side
10:06
One-Sided Limit - Example
10:50
There Does Not Necessarily Need to Be a Connection Between Left or Right Side Limits
11:16
Normal Limits and One-Sided Limits
12:08
Limits of Piecewise Functions
14:12
'Breakover' Points
14:22
We Find the Limit of a Piecewise Function By Checking If the Left and Right Side Limits Agree With Each Other
15:34
Example 1
16:40
Example 2
18:54
Example 3
22:00
Example 4
26:36
Limits at Infinity & Limits of Sequences

32m 49s

Intro
0:00
Introduction
0:06
Definition: Limit of a Function at Infinity
1:44
A Limit at Infinity Works Very Similarly to How a Normal Limit Works
2:38
Evaluating Limits at Infinity
4:08
Rational Functions
4:17
Examples
4:30
For a Rational Function, the Question Boils Down to Comparing the Long Term Growth Rates of the Numerator and Denominator
5:22
There are Three Possibilities
6:36
Evaluating Limits at Infinity, cont.
8:08
Does the Function Grow Without Bound? Will It 'Settle Down' Over Time?
10:06
Two Good Ways to Think About This
10:26
Limit of a Sequence
12:20
What Value Does the Sequence Tend to Do in the Long-Run?
12:41
The Limit of a Sequence is Very Similar to the Limit of a Function at Infinity
12:52
Numerical Evaluation
14:16
Numerically: Plug in Numbers and See What Comes Out
14:24
Example 1
16:42
Example 2
21:00
Example 3
22:08
Example 4
26:14
Example 5
28:10
Example 6
31:06
Instantaneous Slope & Tangents (Derivatives)

51m 13s

Intro
0:00
Introduction
0:08
The Derivative of a Function Gives Us a Way to Talk About 'How Fast' the Function If Changing
0:16
Instantaneous Slop
0:22
Instantaneous Rate of Change
0:28
Slope
1:24
The Vertical Change Divided by the Horizontal
1:40
Idea of Instantaneous Slope
2:10
What If We Wanted to Apply the Idea of Slope to a Non-Line?
2:14
Tangent to a Circle
3:52
What is the Tangent Line for a Circle?
4:42
Tangent to a Curve
5:20
Towards a Derivative - Average Slope
6:36
Towards a Derivative - Average Slope, cont.
8:20
An Approximation
11:24
Towards a Derivative - General Form
13:18
Towards a Derivative - General Form, cont.
16:46
An h Grows Smaller, Our Slope Approximation Becomes Better
18:44
Towards a Derivative - Limits!
20:04
Towards a Derivative - Limits!, cont.
22:08
We Want to Show the Slope at x=1
22:34
Towards a Derivative - Checking Our Slope
23:12
Definition of the Derivative
23:54
Derivative: A Way to Find the Instantaneous Slope of a Function at Any Point
23:58
Differentiation
24:54
Notation for the Derivative
25:58
The Derivative is a Very Important Idea In Calculus
26:04
The Important Idea
27:34
Why Did We Learn the Formal Definition to Find a Derivative?
28:18
Example 1
30:50
Example 2
36:06
Example 3
40:24
The Power Rule
44:16
Makes It Easier to Find the Derivative of a Function
44:24
Examples
45:04
n Is Any Constant Number
45:46
Example 4
46:26
Area Under a Curve (Integrals)

45m 26s

Intro
0:00
Introduction
0:06
Integral
0:12
Idea of Area Under a Curve
1:18
Approximation by Rectangles
2:12
The Easiest Way to Find Area is With a Rectangle
2:18
Various Methods for Choosing Rectangles
4:30
Rectangle Method - Left-Most Point
5:12
The Left-Most Point
5:16
Rectangle Method - Right-Most Point
5:58
The Right-Most Point
6:00
Rectangle Method - Mid-Point
6:42
Horizontal Mid-Point
6:48
Rectangle Method - Maximum (Upper Sum)
7:34
Maximum Height
7:40
Rectangle Method - Minimum
8:54
Minimum Height
9:02
Evaluating the Area Approximation
10:08
Split the Interval Into n Sub-Intervals
10:30
More Rectangles, Better Approximation
12:14
The More We Us , the Better Our Approximation Becomes
12:16
Our Approximation Becomes More Accurate as the Number of Rectangles n Goes Off to Infinity
12:44
Finding Area with a Limit
13:08
If This Limit Exists, It Is Called the Integral From a to b
14:08
The Process of Finding Integrals is Called Integration
14:22
The Big Reveal
14:40
The Integral is Based on the Antiderivative
14:46
The Big Reveal - Wait, Why?
16:28
The Rate of Change for the Area is Based on the Height of the Function
16:50
Height is the Derivative of Area, So Area is Based on the Antiderivative of Height
17:50
Example 1
19:06
Example 2
22:48
Example 3
29:06
Example 3, cont.
35:14
Example 4
40:14
XVII. Appendix: Graphing Calculators
Buying a Graphing Calculator

10m 41s

Intro
0:00
Should You Buy?
0:06
Should I Get a Graphing Utility?
0:20
Free Graphing Utilities - Web Based
0:38
Personal Favorite: Desmos
0:58
Free Graphing Utilities - Offline Programs
1:18
GeoGebra
1:31
Microsoft Mathematics
1:50
Grapher
2:18
Other Graphing Utilities - Tablet/Phone
2:48
Should You Buy a Graphing Calculator?
3:22
The Only Real Downside
4:10
Deciding on Buying
4:20
If You Plan on Continuing in Math and/or Science
4:26
If Money is Not Particularly Tight for You
4:32
If You Don't Plan to Continue in Math and Science
5:02
If You Do Plan to Continue and Money Is Tight
5:28
Which to Buy
5:44
Which Graphing Calculator is Best?
5:46
Too Many Factors
5:54
Do Your Research
6:12
The Old Standby
7:10
TI-83 (Plus)
7:16
TI-84 (Plus)
7:18
Tips for Purchasing
9:17
Buy Online
9:19
Buy Used
9:35
Ask Around
10:09
Graphing Calculator Basics

10m 51s

Intro
0:00
Read the Manual
0:06
Skim It
0:20
Play Around and Experiment
0:34
Syntax
0:40
Definition of Syntax in English and Math
0:46
Pay Careful Attention to Your Syntax When Working With a Calculator
2:08
Make Sure You Use Parentheses to Indicate the Proper Order of Operations
2:16
Think About the Results
3:54
Settings
4:58
You'll Almost Never Need to Change the Settings on Your Calculator
5:00
Tell Calculator In Settings Whether the Angles Are In Radians or Degrees
5:26
Graphing Mode
6:32
Error Messages
7:10
Don't Panic
7:11
Internet Search
7:32
So Many Things
8:14
More Powerful Than You Realize
8:18
Other Things Your Graphing Calculator Can Do
8:24
Playing Around
9:16
Graphing Functions, Window Settings, & Table of Values

10m 38s

Intro
0:00
Graphing Functions
0:18
Graphing Calculator Expects the Variable to Be x
0:28
Syntax
0:58
The Syntax We Choose Will Affect How the Function Graphs
1:00
Use Parentheses
1:26
The Viewing Window
2:00
One of the Most Important Ideas When Graphing Is To Think About The Viewing Window
2:01
For Example
2:30
The Viewing Window, cont.
2:36
Window Settings
3:24
Manually Choose Window Settings
4:20
x Min
4:40
x Max
4:42
y Min
4:44
y Max
4:46
Changing the x Scale or y Scale
5:08
Window Settings, cont.
5:44
Table of Values
7:38
Allows You to Quickly Churn Out Values for Various Inputs
7:42
For example
7:44
Changing the Independent Variable From 'Automatic' to 'Ask'
8:50
Finding Points of Interest

9m 45s

Intro
0:00
Points of Interest
0:06
Interesting Points on the Graph
0:11
Roots/Zeros (Zero)
0:18
Relative Minimums (Min)
0:26
Relative Maximums (Max)
0:32
Intersections (Intersection)
0:38
Finding Points of Interest - Process
1:48
Graph the Function
1:49
Adjust Viewing Window
2:12
Choose Point of Interest Type
2:54
Identify Where Search Should Occur
3:04
Give a Guess
3:36
Get Result
4:06
Advanced Technique: Arbitrary Solving
5:10
Find Out What Input Value Causes a Certain Output
5:12
For Example
5:24
Advanced Technique: Calculus
7:18
Derivative
7:22
Integral
7:30
But How Do You Show Work?
8:20
Parametric & Polar Graphs

7m 8s

Intro
0:00
Change Graph Type
0:08
Located in General 'Settings'
0:16
Graphing in Parametric
1:06
Set Up Both Horizontal Function and Vertical Function
1:08
For Example
2:04
Graphing in Polar
4:00
For Example
4:28
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Lecture Comments (3)

2 answers

Last reply by: Professor Selhorst-Jones
Sun Nov 8, 2015 4:27 PM

Post by Peter Ke on November 7, 2015

Hello, I am just curious cause I think I forgot. When square rooting why do you need both the positive and the negative?

For example, x^2 = 1 which is:

x = 1
x = -1

and not just x = 1 only.

Completing the Square and the Quadratic Formula

  • In this lesson, we will be working just with quadratic polynomials: polynomials with degree 2. Thus, quadratics are of the form
    ax2 + bx + c,
    where a, b, c are constant real numbers and a ≠ 0.
  • Whenever we take the square roots on both sides of an equation when doing algebra, we put a ± on one side. Example:  x2 = 4   ⇒   x = ±2.
  • To complete the square for any quadratic, we want to put it in the form
    ()2 − k.
    Once in this form, we can easily set it to 0 and solve.
  • We convert as follows:
    ax2 + bx + c = 0     ⇔    
    x+b

    2a



     
    =b2 − 4ac

    4a2


     
    Don't try to memorize the formula above, instead, watch the video and learn the general method behind it. While the formula will work, it's very difficult to remember. It's much easier to learn the step-by-step method to produce it.
  • From this conversion, we can create the quadratic formula: a formula that gives an easy way to solve for the roots of any quadratic polynomial.
    x =
    −b ±

     

    b2−4ac
     

     

    2a


     
  • To use the quadratic formula above, the polynomial must be set up in the format ax2 + bx + c = 0. The quadratic must be put into that format before you can use the formula.
  • There are three possible numbers of roots for a quadratic to have: 2, 1, or 0. We determine this number from the discriminant (contained in the quadratic formula): b2−4ac. This value tells us how many roots the polynomial has:
    • b2 − 4ac > 0  ⇒   2 roots;
    • b2 − 4ac = 0  ⇒   1 root;
    • b2 − 4ac < 0  ⇒   0 roots.

Completing the Square and the Quadratic Formula

Solve the below equation for all possible values of x:
2x2 + 16 = 50
  • We could solve this problem by factoring like we learned in the last lesson, but it would be difficult to figure out the appropriate factors. We could also use the quadratic formula, but there's an easier way. Because the problem only has x2's and no x's, we can solve it simply by getting a number on one side of the equation and x2 on the other side. Then take the square root.
  • Working through this, we find
    2x2 + 16 = 50     ⇒     2x2 = 34     ⇒     x2 = 17
  • Once we have it in this form, we can take the square root of both sides. ALWAYS REMEMBER, when you take the square root of both sides, you MUST put a ± sign (plus-minus, saying that there are two versions at once: a positive version and a negative version).
    x = ±

     

    17
     
  • Is it possible to simplify √{17} any more? No, because it has no factors other than itself, so we can't pull anything out of the square root.
x = ±√{17}, or, equivalently, x = √{17} and x = − √{17}
Complete the square to find the solutions to
x2 − 10x = −9.
  • The goal of completing the square is to create something of the form (x +  )2. We do this by adding the appropriate number to each side so we factor it in such a way on the left side.
  • Consider how (x +  )2 expands. For any number in that blank (let's use r so we can explore it), we would have
    (x+r)2 = x2 + 2r·x + r2.
    Looking at this, we see to create something in the form (x +  )2, we need to take half of the number multiplying the x in our equation, then square it.
  • Simply put, to complete the square, halve the number on the x, square it, then add it to each side of the equation. In this case, we have −10x, so we will want to add

    −10

    2

    2

     
    = (−5)2 = 25
    to each side of the equation.
  • Doing this, we have x2 − 10x +25 = 16, which we can factor:
    (x−5)2 = 16
  • Now we can take the square root of both sides to obtain our answers. Don't forget to put a ± sign when you take the square root of both sides!
    x−5 = ±4
x = 5 ±4 or, equivalently, x=9 and x=1
Complete the square on the below polynomial to obtain an equivalent expression:

−4x2 − 24x + 36
  • The goal of completing the square is to create something of the form (x +  )2. In this case, we're not working with an equation, so we can't just add things to both sides. That's alright, we'll see a way to deal with this later on.
  • To complete the square, we first need to have there be no coefficient on the x2, and we need to group the x with the x2. We can do this by pulling out the coefficient in front of x2 from it and the x:
    −4x2 − 24 x + 36     ⇒     −4(x2 + 6x) + 36
  • To complete the square of x2 + 6x, we need to add half of 6 squared: ([1/2] ·6)2 = 9. However, we have an issue: This is not an equation, so we can't just add a number to both sides. We can get around this though! Notice that nothing happens if we add 0, so we can add something that is equivalent to 0: 9 −9. As long as we can separate the 9 and the −9, we can use the 9 to complete the square.
  • Add the 9−9 inside of the parentheses, next to the x2+6x:
    −4(x2 + 6x+9−9) + 36.
    The only thing we want is x2+6x+9, though. The −9 just gets in the way. Take it out of the parentheses by distributing the −4 on to it:
    −4(x2 + 6x+9)+ 36 + 36.
  • At this point, we can now complete the square:
    −4(x+3)2+ 72
  • It can be easy to make a mistake, so it's a good idea to check your work. To be sure the answer is right, verify that
    −4(x+3)2+ 72 = −4x2 − 24x + 36,
    which turns out to be true.
−4(x+3)2+ 72
To use the quadratic formula, a polynomial equation must be set up in the format ax2 + bx+c = 0. Identify a, b, and c for the polynomial equation

p2−20 = 0.
  • We need to see our equation in the same shape as ax2+bx+c=0. It's already pretty close, but currently we're missing some stuff. We need to fill in this blank:
    p2 +  p − 20 = 0
  • Since there are no p's in the original equation, we have 0p. We now have p2 + 0p −20=0, which is closer to the format we want, but still not quite the same. To finish it up, we need to realize that p2 is the same thing as 1·p2 and that minus 20 can be written as plus −20.
  • With those final things in mind, we have
    1p2 + 0p + (−20) = 0,
    and it is easy to identify a, b, and c. [Note: When you get used to using the quadratic formula, you don't have to take so many steps to figure out a, b, and c. Once you're comfortable, you can just glance at the equation and immediately figure them out.]
a=1, b=0, c=−20
Using the quadratic formula, solve x2+8x−153=0.
  • We're solving for all the x values that make the polynomial equation true. While we could do this with factoring, we're doing it with the quadratic formula. The quadratic formula says for ax2+bx+c=0, we have
    x =
    −b ±


    b2−4ac

    2a
    .
  • For this problem, we have a=1, b=8, and c=−153. Plugging them in, we have
    x =
    −8 ±


    82−4·1·(−153)

    2·1
    .
  • Simplifying, we have
    x =
    −8 ±


    64+612

    2
        =    
    −8 ±


    676

    2
        =    −8 ±26

    2
        =     −4 ±13
x=−4 ±13 or, equivalently, x=−17 and x=9
Using the quadratic formula, solve −2t2 + 2t + 24=0.
  • We're solving for all the t values that make the polynomial equation true. While we could do this with factoring, we're doing it with the quadratic formula. The quadratic formula says for equations of the form ax2+bx+c=0, we have
    x =
    −b ±


    b2−4ac

    2a
    .
  • For this problem, we have a=−2, b=2, and c=24 and we're using t instead of x. Setting this up, we have
    t =
    −2 ±


    22−4·(−2)·24

    2·(−2)
    .
  • Simplifying, we have
    t =
    −2 ±


    4+192

    −4
        =    
    −2 ±


    196

    −4
        =    −2 ±14

    −4
        =     1

    2
    ± 7

    2
    [Notice that ± and ± effectively mean the exact same thing here, because they are just saying that there are both plus and minus versions.]
t=[1/2] ±[7/2] or, equivalently, t=−3 and t=4
Using the quadratic formula, solve 3(n2+7) − 5n = 3n − n2 + 44.
  • We're solving for all the n values that make the polynomial equation true. While we could do this with factoring (although it would be very, very difficult), we're doing it with the quadratic formula. The quadratic formula says for equations of the form ax2+bx+c=0, we have
    x =
    −b ±


    b2−4ac

    2a
    .
  • Before we can apply the quadratic formula, we first need to get it into that format: ax2+bx+c=0. Right now, our equation looks totally different. To use the quadratic formula, we must expand, simplify, and get everything on one side so we have a 0 on the other side:
    3n2+21 − 5n = 3n − n2 + 44     ⇒     4n2 − 8n − 23 = 0
  • We now have a=4, b=−8, and c=−23 and we're using n instead of x. Setting this up, we have
    n =
    −(−8) ±


    (−8)2−4·4·(−23)

    2·4
    .
  • Simplifying, we have
    n =
    8 ±


    64+368

    8
        =    
    8 ±


    432

    8
        =     8 ±12 √3

    8
        =     1 ± 3√3

    2
n=1 ±[(3√3)/2] or, equivalently, n=1 + [(3√3)/2] ≈ 3.598 and n=1 − [(3√3)/2] ≈ −1.598
Using the discriminant, determine how many solutions exist to the polynomial equation
4x2 − 20x +25=0.
  • The discriminant tells you how many roots/zeros a polynomial has. The quadratic formula is
    x =
    −b ±


    b2−4ac

    2a
    ,
    but the discriminant is just the part underneath the square root: b2 − 4ac.
  • If the discriminant is greater than 0, there are 2 roots. If it equals 0, there is 1 root, and if it is less than 0, there are no roots at all.
  • From the polynomial, we have a=4, b=−20, and c=25. Plugging them into the discriminant we have
    (−20)2 − 4 ·4 ·25     ⇒     400 − 400     ⇒     0
    Thus, the discriminant equals 0, so there is precisely 1 root.
  • Because we're looking for how many ways the polynomial can equal 0, and we know there is only 1 root for the polynomial, then there is only one solution to the polynomial equation.
One solution
Using the discriminant, determine how many roots exist for the function
f(w) = 2w2 +7w +43.
  • The discriminant tells you how many roots/zeros a polynomial has. The quadratic formula is
    x =
    −b ±


    b2−4ac

    2a
    ,
    but the discriminant is just the part underneath the square root: b2 − 4ac.
  • If the discriminant is greater than 0, there are 2 roots. If it equals 0, there is 1 root, and if it is less than 0, there are no roots at all.
  • From the polynomial, we have a=2, b=7, and c=43. Plugging them into the discriminant we have
    72 − 4 ·2 ·43     ⇒     49 − 344     ⇒     −295
    Thus, the discriminant equals −295, so there are no roots at all.
The function has no roots
A ball is thrown upward out of window at a speed of 7 m/s. The window is at a height of 11 m. The height of the ball above the ground can be given by

h(t) = −4.9t2 + 7t + 11,
where t is the number of seconds after the ball has been thrown.
How long does it take for the ball to hit the ground?
  • We must begin by understanding what the problem is asking. We have a function h(t) that tells us the height of a ball, and we've been asked to find out when the ball will hit the ground. To do this, we must realize that because h(t) is the height of the ball above the ground, when h(t) = 0, the ball will be touching the ground. This means we want to solve for what t will make h(t)=0.
  • Plugging in h(t)=0 gives us an equation we can solve:
    0 = −4.9t2 + 7t + 11
    To solve this equation, we'll want to use the quadratic formula. The quadratic formula says for equations of the form ax2+bx+c=0, we have
    x =
    −b ±


    b2−4ac

    2a
    .
  • We have a=−4.9, b=7, and c=11. Setting this up, we have
    t =
    −7 ±


    72 − 4 ·(−4.9) ·11

    2·(−4.9)
    .
  • Simplifying, we get
    t =
    −7 ±


    49 +215.6

    −9.8
        =    
    −7 ±


    264.6

    −9.8
  • This means the quadratic formula gives us two possibilities for t:
    t =
    −7 +


    264.6

    −9.8
    ≈ −0.9456        and        t =
    −7 +


    264.6

    −9.8
    ≈ 2.3741
  • HOWEVER! It makes no sense for the ball to hit the ground at a negative time t. The function only applies after the ball is thrown, that is, in positive time t. Thus, the negative answer is extraneous, and we get rid of it, leaving us with only the positive answer.
The ball hits the ground after approximately 2.3741 s

*These practice questions are only helpful when you work on them offline on a piece of paper and then use the solution steps function to check your answer.

Answer

Completing the Square and the Quadratic Formula

Lecture Slides are screen-captured images of important points in the lecture. Students can download and print out these lecture slide images to do practice problems as well as take notes while watching the lecture.

  • Intro 0:00
  • Introduction 0:05
  • Square Roots and Equations 0:51
    • Taking the Square Root to Find the Value of x
    • Getting the Positive and Negative Answers
  • Completing the Square: Motivation 2:04
    • Polynomials that are Easy to Solve
    • Making Complex Polynomials Easy to Solve
    • Steps to Completing the Square
  • Completing the Square: Method 7:22
    • Move C over
    • Divide by A
    • Find r
    • Add to Both Sides to Complete the Square
  • Solving Quadratics with Ease 9:56
  • The Quadratic Formula 11:38
    • Derivation
    • Final Form
  • Follow Format to Use Formula 13:38
  • How Many Roots? 14:53
  • The Discriminant 15:47
    • What the Discriminant Tells Us: How Many Roots
    • How the Discriminant Works
  • Example 1: Complete the Square 18:24
  • Example 2: Solve the Quadratic 22:00
  • Example 3: Solve for Zeroes 25:28
  • Example 4: Using the Quadratic Formula 30:52

Transcription: Completing the Square and the Quadratic Formula

Hi--welcome back to Educator.com.0000

Today, we are going to talk about completing the square and the quadratic formula.0002

In this lesson, we will be working just with quadratic polynomials--that is, polynomials that have degree 2.0006

Quadratics are of the form ax2 + bx + c, degree 2; a, b, and c are constant real numbers, and a is not equal to 0.0011

Otherwise it wouldn't be a quadratic anymore, because we would have knocked out that x2.0021

In the previous lesson, we talked all about finding the roots of polynomials.0025

Since quadratics appear often in nature, we have to find their roots a lot.0028

However, also from the previous lesson, we saw that finding roots and factors...it is not always an easy business to factor a polynomial.0031

So, wouldn't it be nice if there was an easier way to find the roots of a quadratic than having to figure out the exact factors and all that?0039

And it turns out that there is; this lesson is going to explore that method, which will make it easier for us.0045

First, let's remind ourselves of something we learned long ago in algebra: consider solving x2 = 1.0052

Now, our first automatic response would be to take the square root of both sides, and we might get x = 1.0057

But we have to remember that that is only have of the answer.0062

Hopefully we remember that, when we take the square root of both sides, we have to also introduce a plus and a negative version.0065

Remember, since (-1)2 = 1, and 12 = 1, we actually have two answers for this, x = 1 and x = -1.0071

When you square a negative, it loses its negative-ness and becomes a positive number.0081

So, if we wanted, we could express this as x = ± 1, using this symbol right here, which we call the "plus/minus symbol."0085

The previous idea is given by this important rule--we have to always remember this any time we end up taking the square root.0094

Otherwise we will introduce mistakes: whenever we take the square roots on both sides of an equation,0100

when we are doing algebra, we put a plus/minus on one side.0105

So, for example, if we have x2 = 1, we take the square root of both sides; we get √(x2) = ±√1,0109

at which point we get x = ± 1, which we can unfold into x = +1 and x = -1, our two answers.0116

These things might get us thinking, though; it is easy to solve equations that are in this form, x2 = k.0125

We get x = ±√k; so that is pretty easy--if we could somehow get a quadratic to look like that, we would be doing pretty well.0132

So, what if we had a polynomial like x2 - 16 = 0?0140

Well, then it is really easy: we just toss that 16 over; we get x2 = 16.0143

The square root of both sides: √(x2) = ±√16; so we take the square root of both of those; x = ±4.0148

We managed to find the answer--nice--it worked really easily when we have this x2 - k = 0.0157

So, this method works great for x2 - k = 0, because we just move it over and get x2 = k,0166

at which point we take the square root and introduce that plus/minus.0174

But we couldn't find the roots of x2 - 2x - 3 with it, because we can't just move it over.0176

So, that is too bad...or could we?...maybe there is a way.0181

Let's say a little bird points out to us that x2 - 2x - 3 equals (x - 1)2 - 4.0186

Well, at this point, it is really easy to solve for the roots now.0195

We set x2 - 2x - 3 equal to 0, and then we use this piece of information right here.0197

We know that we can swap this and this; we have this right here, so we swap it out,0204

because we were told by that little bird (and we trust that little bird) that (x - 1)2 - 4 is the exact same thing.0210

So, at this point, we move our 4 over; we take the square root of both sides; √4 is 2, and we have ±,0216

because we have to introduce that plus/minus when we take the square root.0222

x - 1 = ±2; we move the 1 over, and we get x = 1 ± 2, which is equal to 3 and -1.0225

We have gotten both of the answers for this quadratic.0233

Great; what if there was some way that we could do this for any quadratic?0236

If we could get this form of something squared minus k, it would be easy to find roots for any quadratic.0241

So, let's try to do this on 4x2 + 24x + 9 = 0.0248

We will see if we can find a method for...we will call it completing the square,0252

because we are going from this form where there is a bunch of stuff to this nice thing that is squared.0257

So, we will call it completing the square, because once we have a square, minus just a constant,0262

it is really, really easy to be able to solve for what the roots have to be.0266

So, let's move the 9 over first; we will get 4x2 + 24x = -9.0271

Now, how can we get 4x2 + 24x to become (_x + _)2?0275

Well, there is that pesky 4, still in front of that x2; so let's just start by getting rid of that 4.0281

We divide out the 4, and we have x2 + 6x = -9/4.0287

4x2/4 becomes just x2; 24x/4 becomes 6x (6 times 4 is 24); that equals -9/4; great.0292

From this format, we want to get some (x + _)2, (x + r)2.0302

Now, notice: (x + r)2 is equal to x2 + 2rx + r2...0308

sorry, (x + r)2 becomes xr + rx; so we have two r's showing up on that x, 2r times x.0317

x2 + 2rx + r2: we need to figure out what goes in these blanks.0326

We have a blank here; x2 + 6x + effectively a blank...to be able to complete and get this here.0330

So, (x + _)2--how can we do this? Well, we will use this information that we just had here.0337

We realize that x2 + 6x + 9 = (x + 3)2.0343

We notice that if it is going to be a blank here, and it has to connect to here, well, that was 2r here; so it must be just 1r here.0347

So, if 2r is 6, then 1r is 3; and we check this out: x2 + 6x + 9 = (x + 3)2.0355

We check it: x times x is x2; x times 3 is 3x, plus 3 times x is another 3x, so 6x; great; plus 32...3 times 3 is 9; great.0361

It checks out; so we have x2 + 6x = -9/4; that is what our equation was.0372

How do we get a 9 on the left? Simple--we just add a 9 to both sides.0377

So, add 9 to both sides, because we figured out that we want it to look like this.0381

Since we want it to look like this, we make it look like this through basic algebra manipulation.0386

Add 9 to both sides; we get x2 + 6x + 9 = -9/4 + 9.0390

The left side is now (x + 3)2; we collapse it, and we have it equal to (-9 + 36)/4,0394

since 9 is equal to 9 times 4, over 4, which equals 36/4.0402

It connects to that other -9/4 by getting a common denominator and then adding to it.0408

So, it is (-9 + 36)/4; we simplify that, and we get (x + 3)2 = 27/4; great.0414

If we wanted to, we could easily solve this.0420

So, we take the square root of both sides: we get ±√27/√4 = x + 3--easy.0422

We call this procedure, once again, completing the square, because we are going from a method0429

that doesn't really have this nice squared chunk to a thing that does have this nice squared chunk,0433

just minus some other factor or plus some other factor.0439

We can do this in general; we can do this to some general quadratic polynomial, ax2 + bx + c = 0.0443

We can do this in general, and basically follow the exact same method that we just did with numbers.0450

So, first we move the c over; just like we move the 9 over, we have -c now.0455

Since we eventually want something of the form (x + _)2, we don't want this pesky a getting in the way.0460

So, we divide both sides by a; b divided by a becomes over a; divided by a becomes just a 1 in front of that x2;0465

divide by a over here...-c/a; great, so we get x2 + b/a(x) = -c/a.0471

All right, next our goal is something of the form (x + _)2.0478

Now, we notice, once again: (x + r)2, whatever r is, equals x2 + 2rx + r2.0484

Now, we want to match up to this format; we already have b/a(x), and we want 2r(x).0491

The x2 here matches with the x2 here; the 2rx here matches with the b/a(x) here;0501

and the r2 that we haven't introduced yet0508

is what the blank is that we don't know what we are going to put in yet.0510

So, if 2r is the same thing as b/a, if b/a = 2r, then that means b/2a = r.0514

So now, with that in mind, we know that what we want to introduce is r2:0523

b/2a = r, so we want to add r2, or (b/2a)2, to both sides.0527

So, we add r2 = b2/4a2, and we complete the square.0535

x2 + b/a(x) + b2/4a2...that collapses into (x + b/2a)2.0539

Check that out really quickly: x times x becomes x2; great; x times b/2a becomes b/2a(x),0547

plus it will happen a second time; so b/2a + b/2a becomes 2b/2a, so just b/a, still times x;0556

b/2a times b/2a becomes b2, over...2 times 2 is 4...a times a is squared; so it is b2/4a2.0564

Great; that checks out; and we added this b2/4a2 to both sides; we can't just add it to one side.0571

And so, that will collapse into (b2 - 4ac)/4a2, because we have -c/a; so that becomes -4ac/4a2.0577

So, we can get them on a common denominator; so we have completed the square.0589

Great; that is a general form.0592

At this point, we have shown that any equation that starts as ax2 + bx + c = 0 is equivalent,0597

through completing the square, to this equation right here.0603

It is a little bit complex, but we just proved that we can just do that through basic algebraic manipulation.0606

At this point, it would be quite easy to solve a given quadratic for x by plugging in values for a, b, and c, then just doing a little algebra.0612

For example, if we have 4x2 + 8x + 2, then our a is equal to 4; our b equals 8; and our c equals 2.0618

Oh, let's color-code that; so a at 4 is red; b at 8 is blue; and green is c = 2; lovely.0627

This is x + b over 2a; so our a's are the red things; our blues are the b's; and our c is that green.0637

So, we follow this format; and we have blue 82 here; our a's...4 here and 4 here; 4 here;0650

and then finally, our green is here, and this coefficient here just stays here; this coefficient here just stays here;0666

this coefficient here just stays here; so if we wanted to, at this point, we could just do some arithmetic;0675

and we would be able to simplify that, and then we would be able to take the square root,0680

and basically just be able to solve for it, and we would be able to get the answer.0683

But we can go one step farther, and we can just set up a general formula to solve any quadratic, ax2 + bx + c = 0.0686

We are so close to this; and then we can just use that formula in the future, any time we want to find the roots of any quadratic.0693

So, at this point, we have shown that ax2 + bx + c = 0 is the same thing;0700

it is equivalent to the completed-square version of (x + b/2a)2 = (b2 - 4ac)/4a2.0705

So, we just take the square root of both sides to get to the x: x + b/2a = ±√(b2 - 4ac...0713

what is the square root of 4a2? 4 comes out as 2; a2 comes out as a.0719

So, we get the 2a on the bottom; so (x + b/2a) = ±√(b2 - 4ac), all over 2a.0724

Next, we isolate for x; we subtract the b/2a, plus or minus √(b2 - 4ac), all over 2a.0733

Look, they are already in common denominators; so we get x = [-b ± √(b2 - 4ac)]/2a.0740

We have the quadratic formula, an easy way to solve for the roots of any quadratic polynomial.0750

So, as long as we have some quadratic polynomial like this, we just plug into this thing and do some arithmetic.0756

It might get ugly; it might require a calculator; it might not be really easy arithmetic.0761

But there is not much thinking that we have to do; there is no difficult cleverness of figuring out just the right way to factor it.0766

We just plug in and go, and an answer will pop out.0772

Now, I am not a big fan of memorizing a lot of things; I think, for the most part, that you want to understand how to get to these things.0775

But the quadratic formula is going to come up so often that you are going to end up needing to see0780

this [-b ± √(b2 - 4ac)]/2a...I am going to have to recommend that you probably want to memorize this thing.0785

Memorize the quadratic formula, because it will show up a lot.0793

And even if your teacher doesn't absolutely require you to have it memorized in another class,0797

you are going to end up seeing this so often, and you are going to have to solve for so many quadratics,0802

that you want to just have this ready, so that you can pull it out any time.0807

You will just remember and think, "Oh, yes, I am trying to look for the roots of a quadratic; I can solve this through the quadratic formula."0810

It comes up a lot, so it is good to just have it memorized.0816

All right, follow the format if you are going to use the formula.0819

It is really important to note that, if you want to use the quadratic formula,0822

the polynomial must be set up in this format, ax2 + bx + c = 0.0825

It absolutely has to be set up in this format.0830

For example, if we have 2x2 - 47x + 23, then our a equals 2; our b equals -470831

(because notice that here it is a +, but here it is a negative, so it must be a part of the number); and then finally, our c equals 23.0840

So, a = 2; b = -47; c = 23; great.0849

But it would be totally wrong, absolutely wrong, to say x2 + 3x - 4 = -2x + 8 gives us a = 1, b = 3, c = -4.0857

We have this equals...stuff over here; it has to equal 0; otherwise it doesn't work.0867

It has to be in this format of ax2 + bx + c = 0.0873

That is how we derive completing the square; that is how we derive the quadratic formula.0877

If it is not in this format, it breaks down entirely; we can't use the quadratic formula.0882

So, we have to put it into this format before we can use the quadratic formula.0886

It absolutely has to be in the form; otherwise, it just doesn't work.0890

How many roots does a quadratic have?0894

Now, of course, not all quadratics have the same number of roots.0896

The graphs below show the three possibilities: we have one where it intersects it twice (one here, one here--two roots);0899

we have it where it intersects it just once (it barely grazes and touches, barely just hitting it once);0908

or we have absolutely none, where it never manages to cross the x-axis.0914

And of course, these could all flip the other way; we could have it going down this way.0919

We could have it barely touching on this side; and we could also have it crossing over like this.0924

There is no guarantee that it has to be pointed or look like these.0931

But the idea is just that these are the numbers of times it could cut: it could cut on both sides, cut just once at the tip,0933

or cut not at all, because it never manages to cross it.0940

So, the quadratic formula actually manages to show us which one of these situations we are in.0943

The way we do this is through the discriminant.0948

Remember the formula: -b ± √(b2 - 4ac), all over 2a.0950

You are going to hear that a lot; it is good to memorize.0955

The expression b2 - 4ac is the discriminant; it tells us how many roots there are.0957

b2 - 4ac being greater than 0 means that there are two roots.0963

If it is equal to 0, then we have one root; and if it is less than 0, we have 0 roots,0971

each corresponding to those colors on the last picture, as well.0977

So, the discriminant tells us how many roots--what kind of situation we are in0981

for how often our parabola is going to actually manage to cross over that x-intercept and touch that x-axis.0984

Why or how does it work--what is going on here?0990

Let's look at the quadratic formula once again.0992

Remember: our discriminant is the b2 - 4ac part, the part underneath the square root.0995

It is under the square root; each of the three cases we just saw correlates to how many answers come out of the square root.1000

We have this plus or minus here; so if b2 - 4ac is positive, two are going to come out, because of the plus or minus.1007

If we have the square root of 4 in there, then we have plus 2 and minus 2 (the square root of 4 is 2, so we have ± 2).1014

So, we have a plus version and a minus version; that is two different worlds, so we get two different answers.1021

But if we have b2 - 4ac equals 0, then ± 0 is just 0.1028

So either way, it is just the same world; √0...plus or minus 0...it doesn't really matter if we go with the plus or the minus; we get the same thing.1033

So, we just have one root--only one possibility.1042

Finally, if b2 - 4ac is negative (that is, less than 0), the square root fails entirely, so there are no answers.1045

It is impossible to take the square root of a negative number, remember, because any positive squared becomes positive;1052

any negative squared becomes negative; so there is no number out there that, when you square it, will become a negative.1058

So, there are no answers if b2 - 4ac is negative--at least, there are no real answers.1064

We will talk about how things get a little shady once we get into complex numbers.1068

But for right now, the discriminant tells us that there are no answers if b2 - 4ac is negative.1071

Really, we can just look at how this interacts with square root--how many things can come out of ± square root.1075

If two things can come out, because it is a positive number under the square root, then we have two possibilities, two answers.1083

If 0 is under the square root, then there is one possibility, because it is just one possibility underneath √0.1089

So, ±0 is just one thing; if it is ± the square root of a negative number,1097

then that is impossible to do in the first place, so we have no possibilities under it; we have no answers.1100

All right, we are ready for some examples.1105

Complete the square on the polynomial 3x2 - 30x + 87 to give an equivalent expression.1107

Now, we didn't formally talk about completing the square when it wasn't equal to 0.1111

But we can follow the exact same method: 3x2 - 30x + 87--the first thing we do is...we want to look at that 87 as being off on the side.1115

It is still part of the expression, but it is not what we really want to work with, fundamentally.1125

Now, we have a 3 in here, so we want to pull this 3; we will pull it out; we get 3(x2 -...30...pulling out a 3 gets 10x) + 87 (is still over there).1128

3(x2 - 10x); we check--yes, it checks out; we still have the same thing there.1140

The next step: we want to figure out + _...what blanks can go in there?1145

Well, remember: if it was (x + r)2, then that would become x2 + 2rx + r2.1150

So, what makes up our 2r right now? Our 2r here is our -10.1158

So, if 2r = -10, then our r by itself would be equal to -5; so we want to get a 25 inside.1162

So, we have 3(x2 - 10x +...[we want r2 equal to]25); -5 times -5 is 25, so it is +25 on the inside.1171

So, we want that to show up; of course, if we just change our expression around--we just put a number in there--1184

it is not the same expression; we just broke our mathematics--it doesn't work like that.1189

So, we have to make sure that however much we put in on one place, we take out of somewhere else.1192

I can have any number...5...and I could add 3 and subtract 3, and it would have no effect.1197

I would still be left with 5, because the same thing is adding 0.1202

So, if we add 25 into the inside of our quantity, how much do we need to take away on the outside?1205

Well, putting in 25 on the inside...remember, it is 3 times (....+ 25).1211

Well, that is going to be ..... + 75; 3 times 25 goes to connect like that; 2 times 25 means we need to take away 751218

on the outside to keep our scale balanced; we put 25 into the inside of the parentheses;1233

3 times 25 is 75; so we need to take 75 out.1240

We put a total of 75 in the expression; so if we take a total of 75 out, our scale remains balanced.1243

- 75...and then we still have to bring on what was in the expression before, + 87.1249

So, 3(x2 - 10x + 25)...(x + r)2...our r equals -5, so we have (x - 5)2.1255

Let's check and make sure that is still correct: x2, x times x, x2.1264

x times -5 is -5x; -5 times x is -5x again, so double that: -10x...it checks out still; -5 times -5 is 25; great, it checks out.1268

Minus 75 + 87; that becomes + 12; and now we have something that is equivalent.1277

3(x - 5)2 + 12...let's check to make sure that is correct.1287

3(x - 5)2 would become x2 - 10x + 25, plus 12; 3x2 - 30x + 75 + 12; 3x2 - 30x + 87.1293

Great--it checks out; that is the same thing as what we started with.1311

All right, the next one--the second example: Solve -x2 + 10x - 20 = 4x - 16.1314

So, we have that nice, fancy quadratic formula; let's try it out.1320

The first thing, though: it is not currently equal to 0--it is not set to 0,1324

so we need to get the whole thing so it looks like that ax2 + bx + c = 0.1329

So, let's move things around: we subtract 4x; we add 16; we get -x2 + 6x - 4 = 0.1333

Great; so we are now set up--we have a = -1, b = 6, c = -4; we are in that format.1348

Our normal format is ax2 + bx + c = 0; so we have that parallel.1361

What is our formula? The roots are going to be x = [-b ± √(b2 - 4ac)]/2a.1367

All right, so we start plugging into that: we have x =...plug in our blue -6, our b; plus or minus the square root...1382

b2 is 62; minus 4 times a (-1) times c (-4)...keep that going;1397

the colors are getting a little bit crazy here, but the basic idea going on is still the same;1410

-6 ± √...2 times...and the red one here, a; and notice these coefficients--they just stick around the whole time; -4, -4.1417

They just stick around no matter what; the ± moves down; the negative moves down; so those things are always there.1429

At this point, all we have to do is solve it out; so x = [-6 ± √(36...(-4)(-1)(-4)...1436

two of those cancel out, but we are still left with a negative, so...- 16, over 2 times...1447

we are going to replace that with what it should become, 2 times -1; I got confused by all the colors.1455

2 times -1 becomes -2; so we have x = -6/-2 ± √(20)/-2.1461

x =...this becomes positive 3, plus or minus the square root of 20 (is equal to √(4)(5)); we get 2√5.1475

So, we replace that down here; so we have 3 ± 2√5, all over -2.1491

That is 3; now, the negative here hits that plus/minus, but all that is going to do is cause the plus to become a negative,1499

and the negative to become a plus, so it didn't really do anything: plus/minus is the same thing as minus/plus.1505

We are basically where we were before: √5.1510

So, our answers are going to be x = 3 - √5 and 3 + √5; those are the solutions to that,1513

because we found the roots to when we turn it into the format that we could use it on; great.1522

A man standing on the top of a 127-meter-tall cliff throws a ball directly down at 10 meters per second.1530

The height of the ball above the ground is given by: height at time t equals -4.9t2 - 10t + 127, where t is in seconds.1536

How long does it take for the ball to hit the ground?1546

We have this man; he is standing on top of a cliff; and for some reason, he throws a ball down.1548

All right, the ball is going down; it is moving down towards the ground down here.1555

Let's see what this means: what is ground?1561

Ground is h =...what does that mean?1567

Well, we notice that if we plug in 0, then that is going to be just as he threw it, which would be...1570

the 0's would cancel out; the t would cancel with the t here;1574

and we would be left with just 127, which is where he starts at, 127 meters high.1576

So, that makes sense--that the ground is going to be 0 meters high; it makes intuitive sense.1583

So, we know that what we are looking for is when the ball hits the ground.1587

When does the ball hit the ground? The ball hits the ground at h = 0.1591

That is what the height we are looking for is: ground is normally put at h = 0.1599

You can sometimes move it around; but for the most part, you will end up seeing,1603

in any word problem where they are talking about the height, that the ground level--1606

and normally, whatever our base level is considered--is a height of 0.1608

How long does it take for the ball to hit the ground?1613

Well, that means, if we are looking for when h equals 0, that we are going to use that in here in our functions.1615

So, 0 = -4.9t2 - 10t + 127; we plug this into our quadratic formula.1620

We are not looking at x anymore; we are looking at when t is going to give our roots.1631

So, t = -b, -(-10), plus or minus the square root of (-10)2, minus 4ac, 4 times -4.9 times 127, all over 2a, 2 times -4.9.1634

t = +10 ± √(100 -...that will end up...we work that out with a calculator...2489.2), over -9.8.1659

So, at this point, that is not super easy to work out; so we are going to start plugging into a calculator.1674

We won't actually do that here; but we are going to basically plug two different expressions into our calculator.1679

The plus version is...one mistake there...that was -4 times -4.9, so it becomes +.1683

Otherwise it would be impossible, because we would have a negative underneath our square root.1697

That is what made me catch that.1700

2589.2 divided by -9.8...and 10 plus 10 minus √2589.2/-9.8.1702

We plug that into our calculator, and we get two different answers: t = -6.213 and 4.172.1717

Now, at first, that should set off some alarm bells in our head.1728

We throw a ball down a cliff...we imagine this in our head; that is the very first step--we imagine it in our head.1731

You throw a ball down some tall height, and eventually it hits the ground.1736

All right, that is the end; it doesn't hit the ground at two different times.1739

And so, what does this mean?--we have two different answers here, so which one is the correct answer, -6.213 or 4.172?1743

Which one is right? We think; we know that this is t given in seconds, so we know what happens.1751

He throws the ball down, and so forward in time, it is falling; we are going by this equation.1758

But what about a negative time--did he throw the ball before 0 seconds?1764

No, he throws the ball at 0 seconds; it starts at his height at 0 seconds.1768

So, it must be that h(t) is only true--its domain is only 0 to positive infinity.1773

And actually, it is not even going to be true after 4.172, because all of a sudden the ground gets in the way and stops this equation from being true.1781

So, h(t) is only true from t = 0 until the ball hits the ground--until whatever t the ball hits the ground at.1788

So, that is sort of an implicit thing that we hadn't explicitly stated; but we had to understand what is going on,1799

because otherwise we will get two answers, and one of them is going to be wrong.1804

We have to realize what is going on; we don't want to just blindly do what the formula told us.1807

We want to think about what this represents; word problems require thinking.1812

So, -6.213 and 4.172...we realize it is only true from t = 0 to higher numbers, until the ball hits the ground.1816

The ball hits the ground at 4.172; and -6.213 is an extraneous solution--it is impossible to look at those times,1823

because that is back before this function ever ended up even being used.1831

The function comes into existence only once the ball is thrown at time t = 0.1837

So, negative times are completely extraneous--we can't use those answers.1842

And we get 4.172 seconds as the correct answer; great.1846

The final example: Two cars are approaching a right-angle intersection on straight roads.1853

The first one is coming from the north at a constant speed of 30 meters per second,1857

while the second one is from the east at a constant speed of 25 meters per second.1861

If both cars are currently 200 meters from the intersection, how much time is there until they have a distance of 90 meters between them?1865

This is the classic nightmare word problem with so many things here--what are we going to do?1871

Well, we just start figuring out what it is telling us; then we will work on the math.1876

So, the first thing we do is try to make a picture of a right-angle intersection.1880

We know what an intersection looks like on the street.1884

Two streets intersect one another; we know that they came together at a right angle, because it says "right-angle intersection."1888

Great; they are on straight roads, so we are guaranteed the fact that straight lines come out of it; so this makes sense.1894

The first one is coming from the north; well, let's put north as being this way.1899

This car is up here; it is coming from the north at a constant speed of 30 meters per second.1904

It is going down 30 meters per second; where is it right now?1909

We know it is 200 meters away at the start; what about the other one?1913

The other one is coming from the east (east will be over here), and it is going at 25 meters per second.1919

How far away is it? We figured out the 200 meters to get the first; both cars are currently 200 meters, so it is 200 here, as well.1928

How much time is there until they have a distance of 90 meters between them?1937

We also want to be able to introduce...the other thing that we have to figure out is what it means by "distance between them."1940

If we have a point here and a point here, then the distance is just the distance between those two points.1947

But we don't have any information about the distance between them; we are not told how far away they are.1955

But we are told how far they are from this intersection in the middle.1959

If we have...here is x and here is y...look, we can use the Pythagorean theorem: x2 + y2 = d2.1964

Great; so we have a way of relating these two things.1973

But if we are just frozen at 200, 2002 + 2002 = distance squared,1976

then we are going to get something that is never going to be 90, because we are just looking at a single snapshot in time.1980

We also have to have a way of their movement, their motion, affecting this.1985

So, they start at 200 away; but then they start to get closer and closer to that intersection.1988

It is going to be 200 - 30t, because it is going to be that, as they get closer to the intersection,1993

as more time goes on, more of their distance from the intersection will disappear.2000

So, the 200 - 30t...and then, the other one is 200 - 25t, because the red one, the north car,2004

is coming at 30 meters per second; so for every second that goes by, it will have moved 30 meters2014

towards the intersection, so it will be 200 - 30t; and the blue car, the east car, will be 200 - 25t,2019

because for every second, it moves 25 meters towards the intersection: 200 less 25 meters.2026

OK, so at this point, we have a real understanding of what is going on.2034

We can now put this d in here; and we can connect all of these ideas.2036

So, we have 200 - 30t is what describes the red car, our north car.2043

And 200 - 25t is what describes the east car, our blue car.2049

And we are looking for when the distance is equal to 90.2055

So remember: we had x2 + y2 = d2 from the fact that this is just a nice, normal Pythagorean triangle.2060

We can use the Pythagorean theorem here: the square of the two sides is equal to the square of the hypotenuse.2068

So, we have 902 = (200 - 30t)2 (our red car--our north car) + (200 - 25t)2 (our east car--our blue car); great.2075

And now we are trying to solve this and figure out when t is going to make this true.2086

At what t will that equation there be true?2089

So, we just start working it out; now, this is going to get pretty big pretty fast, because we have big numbers.2092

But luckily, we have access to calculators in this world.2097

So, 902...plug that in; that comes out as 8100; 200 times 200 becomes 40000; minus 30t...2099

200 times -30t, plus -30t times 200; 200 times -30t is -6000, but we have to double that,2110

so it is -12000t; minus 30t times -30t...+ 900 t2.2116

The next one is + 40000 again for the other portion; 200 times -25 becomes -5000.2126

But then, we also have to have -25 times 200 again the other way, so it is -500 doubled, so -10000t.2135

Plus -25 times -25, so + 625t2...2144

All right, we simplify this; this looks like something that could eventually turn into a quadratic.2152

So, we say, "Oh, it is time to use the quadratic formula!"2158

So, we need to get into that form: we will subtract 8100 over, so -8100, -8100 over here...2160

We look at 625t2 + 900t2; we get 1525t2.2170

Next, -10000t; -12000t; -22000t; 40000 and 40000...minus 81000, plus 71900; wow.2179

But we are in a position where we can now use the quadratic formula.2193

Once again, it is a good thing that we have calculators; otherwise this would be really difficult.2195

But we can use the quadratic formula now.2199

So, t = -b, -(-22000), so 22000, plus or minus the square root of b2, 220002;2201

we will drop the negative sign, just because it is going to get squared anyway; minus 4, times 1525, times 71900.2213

At this step, we might toss those parts in, just the part underneath the square root, use the discriminant,2225

and make sure that there is an answer; it will turn out that there is an answer, so we will just keep going.2230

And that is going to be divided by 2 times 1525.2234

I won't work this all out here; I am going to trust the fact that you can do the two different versions.2239

Remember: there is a plus version, and then there is a minus version.2243

So, we do both of the versions, and we will end up getting t = 5.004 seconds and 9.423 seconds.2247

In the last one, the problem with the falling rock, where he threw the rock down the cliff,2260

there was only one answer that was true out of the two things that came out of it.2264

But what about this one--is one of them wrong and the other one right?2268

Is it only possible for one of them to happen first?2271

Well, if that is the case--if we are only looking for what is the immediate time, the soonest time,2273

when they are 90 meters away from each other, then it is 5.004 seconds.2278

However, if we think about what is going on, let's try to visualize this.2284

We have a car in the north and a car in the east.2288

They start very far away from one another, but as they get closer and closer to one another,2293

at some point, their distance...this one is going faster; this one is going slower; they are going to pass,2298

so that their distance is close enough for it to be...at 5.04 seconds, they are now 90 meters away.2304

Now, they pass, and they end up being very close briefly.2311

But then, they keep going; and at some point on the reverse side, their distance begins to grow now, after they pass the intersection.2314

So, they actually start to get farther away.2320

After they pass the intersection, eventually it is going to be that they are now 90 meters away from each other, once again, at 9.423 seconds.2322

And then, if they keep going on and on and on, they will never end up being 90 meters away from each other.2330

But this is the first time they are 90 meters away, and then this is the second time.2334

Now, it is quite likely that a question phrased in this way would only be asking for the first one.2340

But we are actually able to find out both of the times that they are 90 meters away, assuming they maintain constant speeds and straight roads.2345

That is pretty cool; all right, I hope you have a good sense of how to complete the square2352

and how important and useful the quadratic formula can be.2356

Make sure you memorize it; I know--I hate memorizing things, too; but it ends up coming up so often.2358

You really have to have it memorized: [-b ± √(b2 - 4ac)]/2a.2362

Make sure you get that one burned in your memory, because it will show up a lot.2368

And we will see you later; next time, we will talk more about the general nature of quadratics and parabolas2373

and see how what we just did in this lesson will end up connecting to that one, as well.2378

See you at Educator.com later--goodbye!2382