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INSTRUCTORS Carleen Eaton Grant Fraser Eric Smith
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Eric Smith

Eric Smith

Quadratic Formulas & Applications

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Table of Contents

I. Properties of Real Numbers
Basic Types of Numbers

30m 41s

Intro
0:00
Objectives
0:07
Basic Types of Numbers
0:36
Natural Numbers
1:02
Whole Numbers
1:29
Integers
2:04
Rational Numbers
2:38
Irrational Numbers
5:06
Imaginary Numbers
6:48
Basic Types of Numbers Cont.
8:09
The Big Picture
8:10
Real vs. Imaginary Numbers
8:30
Rational vs. Irrational Numbers
8:48
Basic Types of Numbers Cont.
10:55
Number Line
11:06
Absolute Value
11:44
Inequalities
12:39
Example 1
13:16
Example 2
17:30
Example 3
21:56
Example 4
24:27
Example 5
27:48
Operations on Numbers

19m 26s

Intro
0:00
Objectives
0:06
Operations on Numbers
0:25
Addition
0:53
Subtraction
1:33
Multiplication & Division
2:19
Exponents
3:24
Bases
4:04
Square Roots
4:59
Principle Square Roots
5:09
Perfect Squares
6:32
Simplifying and Combining Roots
6:52
Example 1
8:16
Example 2
12:30
Example 3
14:02
Example 4
16:27
Order of Operations

12m 6s

Intro
0:00
Objectives
0:06
The Order of Operations
0:25
Work Inside Parentheses
0:42
Simplify Exponents
0:52
Multiplication & Division from Left to Right
0:57
Addition & Subtraction from Left to Right
1:11
Remember PEMDAS
1:21
The Order of Operations Cont.
2:27
Example
2:43
Example 1
3:55
Example 2
5:36
Example 3
7:35
Example 4
8:56
Properties of Real Numbers

18m 52s

Intro
0:00
Objectives
0:07
The Properties of Real Numbers
0:23
Commutative Property of Addition and Multiplication
0:44
Associative Property of Addition and Multiplication
1:50
Distributive Property of Multiplication Over Addition
3:20
Division Property of Zero
4:46
Division Property of One
5:23
Multiplication Property of Zero
5:56
Multiplication Property of One
6:17
Addition Property of Zero
6:29
Why Are These Properties Important?
6:53
Example 1
9:16
Example 2
13:04
Example 3
14:30
Example 4
16:57
II. Linear Equations
The Vocabulary of Linear Equations

12m 22s

Intro
0:00
Objectives
0:09
The Vocabulary of Linear Equations
0:44
Variables
0:52
Terms
1:09
Coefficients
1:40
Like Terms
2:18
Examples of Like Terms
2:37
Expressions
4:01
Equations
4:26
Linear Equations
5:04
Solutions
5:55
Example 1
6:16
Example 2
7:16
Example 3
8:45
Example 4
10:20
Solving Linear Equations in One Variable

28m 52s

Intro
0:00
Objectives
0:08
Solving Linear Equations in One Variable
0:34
Conditional Cases
0:51
Identity Cases
1:09
Contradiction Cases
1:30
Solving Linear Equations in One Variable Cont.
2:00
Addition Property of Equality
2:10
Multiplication Property of Equality
2:43
Steps to Solve Linear Equations
3:14
Example 1
4:22
Example 2
8:21
Example 3
12:32
Example 4
14:19
Example 5
17:25
Example 6
22:17
Solving Formulas

12m 2s

Intro
0:00
Objectives
0:06
Solving Formulas
0:18
Formulas
0:26
Use the Same Properties as Solving Linear Equations
1:36
Addition Property of Equality
1:55
Multiplication Property of Equality
1:58
Steps to Solve Formulas
2:43
Example 1
3:56
Example 2
6:09
Example 3
8:39
Applications of Linear Equations

28m 41s

Intro
0:00
Objectives
0:10
Applications of Linear Equations
0:43
The Six-Step Method to Solving Word Problems
0:55
Common Terms
3:12
Example 1
5:03
Example 2
9:40
Example 3
13:48
Example 4
17:58
Example 5
23:28
Applications of Linear Equations, Motion & Mixtures

24m 26s

Intro
0:00
Objectives
0:21
Motion and Mixtures
0:46
Motion Problems: Distance, Rate, and Time
1:06
Mixture Problems: Amount, Percent, and Total
1:27
The Table Method
1:58
The Beaker Method
3:38
Example 1
5:05
Example 2
9:44
Example 3
14:20
Example 4
19:13
III. Graphing
Rectangular Coordinate System

22m 55s

Intro
0:00
Objectives
0:11
The Rectangular Coordinate System
0:39
The Cartesian Coordinate System
0:40
X-Axis
0:54
Y-Axis
1:04
Origin
1:11
Quadrants
1:26
Ordered Pairs
2:10
Example 1
2:55
The Rectangular Coordinate System Cont.
6:09
X-Intercept
6:45
Y-Intercept
6:55
Relation of X-Values and Y-Values
7:30
Example 2
11:03
Example 3
12:13
Example 4
14:10
Example 5
18:38
Slope & Graphing

27m 58s

Intro
0:00
Objectives
0:11
Slope and Graphing
0:48
Standard Form
1:14
Example 1
2:24
Slope and Graphing Cont.
4:58
Slope, m
5:07
Slope is Rise over Run
6:11
Don't Mix Up the Coordinates
8:20
Example 2
9:39
Slope and Graphing Cont.
14:26
Slope-Intercept Form
14:34
Example 3
16:55
Example 4
18:00
Slope and Graphing Cont.
19:00
Rewriting an Equation in Slope-Intercept Form
19:39
Rewriting an Equation in Standard Form
20:09
Slopes of Vertical & Horizontal Lines
20:56
Example 5
22:49
Example 6
24:09
Example 7
25:59
Example 8
26:57
Linear Equations in Two Variables

20m 36s

Intro
0:00
Objectives
0:13
Linear Equations in Two Variables
0:36
Point-Slope Form
1:07
Substitute in the Point and the Slope
2:21
Parallel Lines: Two Lines with the Same Slope
4:05
Perpendicular Lines: Slopes are Negative Reciprocals of Each Other
4:39
Perpendicular Lines: Product of Slopes is -1
5:24
Example 1
6:02
Example 2
7:50
Example 3
10:49
Example 4
13:26
Example 5
15:30
Example 6
17:43
IV. Functions
Introduction to Functions

21m 24s

Intro
0:00
Objectives
0:07
Introduction to Functions
0:58
Relations
1:03
Functions
1:37
Independent Variables
2:00
Dependent Variables
2:11
Function Notation
2:21
Function
3:43
Input and Output
3:53
Introduction to Functions Cont.
4:45
Domain
4:46
Range
4:55
Functions Represented by a Diagram
6:41
Natural Domain
9:11
Evaluating Functions
12:02
Example 1
13:13
Example 2
15:03
Example 3
16:18
Example 4
19:54
Graphing Functions

16m 12s

Intro
0:00
Objectives
0:09
Graphing Functions
0:54
Using Slope-Intercept Form
1:56
Vertical Line Test
2:58
Determining the Domain
4:20
Determining the Range
5:43
Example 1
6:06
Example 2
7:18
Example 3
8:31
Example 4
11:04
V. Systems of Linear Equations
Systems of Linear Equations

25m 54s

Intro
0:00
Objectives
0:13
Systems of Linear Equations
0:46
System of Equations
0:51
System of Linear Equations
1:15
Solutions
1:35
Points as Solutions
1:53
Finding Solutions Graphically
5:13
Example 1
6:37
Example 2
12:07
Systems of Linear Equations Cont.
17:01
One Solution, No Solution, or Infinite Solutions
17:10
Example 3
18:31
Example 4
22:37
Solving a System Using Substitution

20m 1s

Intro
0:00
Objectives
0:09
Solving a System Using Substitution
0:32
Substitution Method
1:24
Substitution Example
2:35
One Solution, No Solution, or Infinite Solutions
7:50
Example 1
9:45
Example 2
12:48
Example 3
15:01
Example 4
17:30
Solving a System Using Elimination

19m 40s

Intro
0:00
Objectives
0:09
Solving a System Using Elimination
0:27
Elimination Method
0:42
Elimination Example
2:01
One Solution, No Solution, or Infinite Solutions
7:05
Example 1
8:53
Example 2
11:46
Example 3
15:37
Example 4
17:45
Applications of Systems of Equations

24m 34s

Intro
0:00
Objectives
0:12
Applications of Systems of Equations
0:30
Word Problems
1:31
Example 1
2:17
Example 2
7:55
Example 3
13:07
Example 4
17:15
VI. Inequalities
Solving Linear Inequalities in One Variable

17m 13s

Intro
0:00
Objectives
0:08
Solving Linear Inequalities in One Variable
0:37
Inequality Expressions
0:46
Linear Inequality Solution Notations
3:40
Inequalities
3:51
Interval Notation
4:04
Number Lines
4:43
Set Builder Notation
5:24
Use Same Techniques as Solving Equations
6:59
'Flip' the Sign when Multiplying or Dividing by a Negative Number
7:12
'Flip' Example
7:50
Example 1
8:54
Example 2
11:40
Example 3
14:01
Compound Inequalities

16m 13s

Intro
0:00
Objectives
0:07
Compound Inequalities
0:37
'And' vs. 'Or'
0:44
'And'
3:24
'Or'
3:35
'And' Symbol, or Intersection
3:51
'Or' Symbol, or Union
4:13
Inequalities
4:41
Example 1
6:22
Example 2
9:30
Example 3
11:27
Example 4
13:49
Solving Equations with Absolute Values

14m 12s

Intro
0:00
Objectives
0:08
Solve Equations with Absolute Values
0:18
Solve Equations with Absolute Values Cont.
1:11
Steps to Solving Equations with Absolute Values
2:21
Example 1
3:23
Example 2
6:34
Example 3
10:12
Inequalities with Absolute Values

17m 7s

Intro
0:00
Objectives
0:07
Inequalities with Absolute Values
0:23
Recall…
2:08
Example 1
3:39
Example 2
6:06
Example 3
8:14
Example 4
10:29
Example 5
13:29
Graphing Inequalities in Two Variables

15m 33s

Intro
0:00
Objectives
0:07
Graphing Inequalities in Two Variables
0:32
Split Graph into Two Regions
1:53
Graphing Inequalities
5:44
Test Points
6:20
Example 1
7:11
Example 2
10:17
Example 3
13:06
Systems of Inequalities

21m 13s

Intro
0:00
Objectives
0:08
Systems of Inequalities
0:24
Test Points
1:10
Steps to Solve Systems of Inequalities
1:25
Example 1
2:23
Example 2
7:28
Example 3
12:51
VII. Polynomials
Integer Exponents

44m 51s

Intro
0:00
Objectives
0:09
Integer Exponents
0:42
Exponents 'Package' Multiplication
1:25
Example 1
2:00
Example 2
3:13
Integer Exponents Cont.
4:50
Product Rule for Exponents
4:51
Example 3
7:16
Example 4
10:15
Integer Exponents Cont.
13:13
Power Rule for Exponents
13:14
Power Rule with Multiplication and Division
15:33
Example 5
16:18
Integer Exponents Cont.
20:04
Example 6
20:41
Integer Exponents Cont.
25:52
Zero Exponent Rule
25:53
Quotient Rule
28:24
Negative Exponents
30:14
Negative Exponent Rule
32:27
Example 7
34:05
Example 8
36:15
Example 9
39:33
Example 10
43:16
Adding & Subtracting Polynomials

18m 33s

Intro
0:00
Objectives
0:07
Adding and Subtracting Polynomials
0:25
Terms
0:33
Coefficients
0:51
Leading Coefficients
1:13
Like Terms
1:29
Polynomials
2:21
Monomials, Binomials, Trinomials, and Polynomials
5:41
Degrees
7:00
Evaluating Polynomials
8:12
Adding and Subtracting Polynomials Cont.
9:25
Example 1
11:48
Example 2
13:00
Example 3
14:41
Example 4
16:15
Multiplying Polynomials

25m 7s

Intro
0:00
Objectives
0:06
Multiplying Polynomials
0:41
Distributive Property
1:00
Example 1
2:49
Multiplying Polynomials Cont.
8:22
Organize Terms with a Table
8:23
Example 2
13:40
Multiplying Polynomials Cont.
16:33
Multiplying Binomials with FOIL
16:48
Example 3
18:49
Example 4
20:04
Example 5
21:42
Dividing Polynomials

44m 56s

Intro
0:00
Objectives
0:07
Dividing Polynomials
0:29
Dividing Polynomials by Monomials
2:10
Dividing Polynomials by Polynomials
2:59
Dividing Numbers
4:09
Dividing Polynomials Example
8:39
Example 1
12:35
Example 2
14:40
Example 3
16:45
Example 4
21:13
Example 5
24:33
Example 6
29:02
Dividing Polynomials with Synthetic Division Method
33:36
Example 7
38:43
Example 8
42:24
VIII. Factoring Polynomials
Greatest Common Factor & Factor by Grouping

28m 27s

Intro
0:00
Objectives
0:09
Greatest Common Factor
0:31
Factoring
0:40
Greatest Common Factor (GCF)
1:48
GCF for Polynomials
3:28
Factoring Polynomials
6:45
Prime
8:21
Example 1
9:14
Factor by Grouping
14:30
Steps to Factor by Grouping
17:03
Example 2
17:43
Example 3
19:20
Example 4
20:41
Example 5
22:29
Example 6
26:11
Factoring Trinomials

21m 44s

Intro
0:00
Objectives
0:06
Factoring Trinomials
0:25
Recall FOIL
0:26
Factor a Trinomial by Reversing FOIL
1:52
Tips when Using Reverse FOIL
5:31
Example 1
7:04
Example 2
9:09
Example 3
11:15
Example 4
13:41
Factoring Trinomials Cont.
15:50
Example 5
18:42
Factoring Trinomials Using the AC Method

30m 9s

Intro
0:00
Objectives
0:08
Factoring Trinomials Using the AC Method
0:27
Factoring when Leading Term has Coefficient Other Than 1
1:07
Reversing FOIL
1:18
Example 1
1:46
Example 2
4:28
Factoring Trinomials Using the AC Method Cont.
7:45
The AC Method
8:03
Steps to Using the AC Method
8:19
Tips on Using the AC Method
9:29
Example 3
10:45
Example 4
16:50
Example 5
21:08
Example 6
24:58
Special Factoring Techniques

30m 14s

Intro
0:00
Objectives
0:07
Special Factoring Techniques
0:26
Difference of Squares
1:46
Perfect Square Trinomials
2:38
No Sum of Squares
3:32
Special Factoring Techniques Cont.
4:03
Difference of Squares Example
4:04
Perfect Square Trinomials Example
5:29
Example 1
7:31
Example 2
9:59
Example 3
11:47
Example 4
15:09
Special Factoring Techniques Cont.
19:07
Sum of Cubes and Difference of Cubes
19:08
Example 5
23:13
Example 6
26:12
IX. Quadratic Equations
Solving Quadratic Equations by Factoring

23m 38s

Intro
0:00
Objectives
0:08
Solving Quadratic Equations by Factoring
0:19
Quadratic Equations
0:20
Zero Factor Property
1:39
Zero Factor Property Example
2:34
Example 1
4:00
Solving Quadratic Equations by Factoring Cont.
5:54
Example 2
7:28
Example 3
11:09
Example 4
14:22
Solving Quadratic Equations by Factoring Cont.
18:17
Higher Degree Polynomial Equations
18:18
Example 5
20:22
Solving Quadratic Equations

29m 27s

Intro
0:00
Objectives
0:12
Solving Quadratic Equations
0:29
Linear Factors
0:38
Not All Quadratics Factor Easily
1:22
Principle of Square Roots
3:36
Completing the Square
4:50
Steps for Using Completing the Square
5:15
Completing the Square Works on All Quadratic Equations
6:41
The Quadratic Formula
7:28
Discriminants
8:25
Solving Quadratic Equations - Summary
10:11
Example 1
11:54
Example 2
13:03
Example 3
16:30
Example 4
21:29
Example 5
25:07
Equations in Quadratic Form

16m 47s

Intro
0:00
Objectives
0:08
Equations in Quadratic Form
0:24
Using a Substitution
0:53
U-Substitution
1:26
Example 1
2:07
Example 2
5:36
Example 3
8:31
Example 4
11:14
Quadratic Formulas & Applications

29m 4s

Intro
0:00
Objectives
0:09
Quadratic Formulas and Applications
0:35
Squared Variable
0:40
Principle of Square Roots
0:51
Example 1
1:09
Example 2
2:04
Quadratic Formulas and Applications Cont.
3:34
Example 3
4:42
Example 4
13:33
Example 5
20:50
Graphs of Quadratics

26m 53s

Intro
0:00
Objectives
0:06
Graphs of Quadratics
0:39
Axis of Symmetry
1:46
Vertex
2:12
Transformations
2:57
Graphing in Quadratic Standard Form
3:23
Example 1
5:06
Example 2
6:02
Example 3
9:07
Graphs of Quadratics Cont.
11:26
Completing the Square
12:02
Vertex Shortcut
12:16
Example 4
13:49
Example 5
17:25
Example 6
20:07
Example 7
23:43
Polynomial Inequalities

21m 42s

Intro
0:00
Objectives
0:07
Polynomial Inequalities
0:30
Solving Polynomial Inequalities
1:20
Example 1
2:45
Polynomial Inequalities Cont.
5:12
Larger Polynomials
5:13
Positive or Negative Intervals
7:16
Example 2
9:01
Example 3
13:53
X. Rational Equations
Multiply & Divide Rational Expressions

26m 41s

Intro
0:00
Objectives
0:09
Multiply and Divide Rational Expressions
0:44
Rational Numbers
0:55
Dividing by Zero
1:45
Canceling Extra Factors
2:43
Negative Signs in Fractions
4:52
Multiplying Fractions
6:26
Dividing Fractions
7:17
Example 1
8:04
Example 2
14:01
Example 3
16:23
Example 4
18:56
Example 5
22:43
Adding & Subtracting Rational Expressions

20m 24s

Intro
0:00
Objectives
0:07
Adding and Subtracting Rational Expressions
0:41
Common Denominators
0:52
Common Denominator Examples
1:14
Steps to Adding and Subtracting Rational Expressions
2:39
Example 1
3:34
Example 2
5:27
Adding and Subtracting Rational Expressions Cont.
6:57
Least Common Denominators
6:58
Transitioning from Fractions to Rational Expressions
9:08
Identifying Least Common Denominators for Rational Expressions
9:56
Subtracting vs. Adding
10:41
Example 3
11:19
Example 4
12:36
Example 5
15:08
Example 6
16:46
Complex Fractions

18m 23s

Intro
0:00
Objectives
0:09
Complex Fractions
0:37
Dividing to Simplify Complex Fractions
1:10
Example 1
2:03
Example 2
3:58
Complex Fractions Cont.
9:15
Using the Least Common Denominator to Simplify Complex Fractions
9:16
Both Methods Lead to the Same Answer
10:07
Example 3
10:42
Example 4
14:28
Solving Rational Equations

16m 24s

Intro
0:00
Objectives
0:07
Solving Rational Equations
0:23
Isolate the Specified Variable
1:23
Example 1
1:58
Example 2
5:00
Example 3
8:23
Example 4
13:25
Rational Inequalities

18m 54s

Intro
0:00
Objectives
0:06
Rational Inequalities
0:18
Testing Intervals for Rational Inequalities
0:38
Steps to Solving Rational Inequalities
1:05
Tips to Solving Rational Inequalities
2:27
Example 1
3:33
Example 2
12:21
Applications of Rational Expressions

20m 20s

Intro
0:00
Objectives
0:07
Applications of Rational Expressions
0:27
Work Problems
1:05
Example 1
2:58
Example 2
6:45
Example 3
13:17
Example 4
16:37
Variation & Proportion

27m 4s

Intro
0:00
Objectives
0:10
Variation and Proportion
0:34
Variation
0:35
Inverse Variation
1:01
Direct Variation
1:10
Setting Up Proportions
1:31
Example 1
2:27
Example 2
5:36
Variation and Proportion Cont.
8:29
Inverse Variation
8:30
Example 3
9:20
Variation and Proportion Cont.
12:41
Constant of Proportionality
12:42
Example 4
13:59
Variation and Proportion Cont.
16:17
Varies Directly as the nth Power
16:30
Varies Inversely as the nth Power
16:53
Varies Jointly
17:09
Combining Variation Models
17:36
Example 5
19:09
Example 6
22:10
XI. Radical Equations
Rational Exponents

14m 32s

Intro
0:00
Objectives
0:07
Rational Exponents
0:32
Power on Top, Root on Bottom
1:05
Example 1
1:37
Rational Exponents Cont.
4:04
Using Rules from Exponents for Radicals as Exponents
4:05
Combining Terms Under a Single Root
4:50
Example 2
5:21
Example 3
7:39
Example 4
11:23
Example 5
13:14
Simplify Rational Exponents

15m 12s

Intro
0:00
Objectives
0:07
Simplify Rational Exponents
0:25
Product Rule for Radicals
0:26
Product Rule to Simplify Square Roots
1:11
Quotient Rule for Radicals
1:42
Applications of Product and Quotient Rules
2:17
Higher Roots
2:48
Example 1
3:39
Example 2
6:35
Example 3
8:41
Example 4
11:09
Adding & Subtracting Radicals

17m 22s

Intro
0:00
Objectives
0:07
Adding and Subtracting Radicals
0:33
Like Terms
1:29
Bases and Exponents May be Different
2:02
Bases and Powers Must be Same when Adding and Subtracting
2:42
Add Radicals' Coefficients
3:55
Example 1
4:47
Example 2
6:00
Adding and Subtracting Radicals Cont.
7:10
Simplify the Bases to Look the Same
7:25
Example 3
8:23
Example 4
11:45
Example 5
15:10
Multiply & Divide Radicals

19m 24s

Intro
0:00
Objectives
0:08
Multiply and Divide Radicals
0:25
Rules for Working With Radicals
0:26
Using FOIL for Radicals
1:11
Don’t Distribute Powers
2:54
Dividing Radical Expressions
4:25
Rationalizing Denominators
6:40
Example 1
7:22
Example 2
8:32
Multiply and Divide Radicals Cont.
9:23
Rationalizing Denominators with Higher Roots
9:25
Example 3
10:51
Example 4
11:53
Multiply and Divide Radicals Cont.
13:13
Rationalizing Denominators with Conjugates
13:14
Example 5
15:52
Example 6
17:25
Solving Radical Equations

15m 5s

Intro
0:00
Objectives
0:07
Solving Radical Equations
0:17
Radical Equations
0:18
Isolate the Roots and Raise to Power
0:34
Example 1
1:13
Example 2
3:09
Solving Radical Equations Cont.
7:04
Solving Radical Equations with More than One Radical
7:05
Example 3
7:54
Example 4
13:07
Complex Numbers

29m 16s

Intro
0:00
Objectives
0:06
Complex Numbers
1:05
Imaginary Numbers
1:08
Complex Numbers
2:27
Real Parts
2:48
Imaginary Parts
2:51
Commutative, Associative, and Distributive Properties
3:35
Adding and Subtracting Complex Numbers
4:04
Multiplying Complex Numbers
6:16
Dividing Complex Numbers
8:59
Complex Conjugate
9:07
Simplifying Powers of i
14:34
Shortcut for Simplifying Powers of i
18:33
Example 1
21:14
Example 2
22:15
Example 3
23:38
Example 4
26:33
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Lecture Comments (5)

1 answer

Last reply by: Professor Eric Smith
Tue Aug 14, 2018 10:34 AM

Post by Leo Jiang on August 13 at 11:42:55 PM

In Example 3, why do you have to FOIL for the term (x + 105)^2 in x^2 + (x + 105)^2 = 195^2? Couldn't you just distribute and get something like x^2 + x^2 + 11025 = 38025? Did you use FOIL becuase you could combine the like terms (x^2 + x^2 and 105x + 105x) after? Or to be able to use the quadratic formula?

1 answer

Last reply by: Professor Eric Smith
Sat Dec 30, 2017 6:55 PM

Post by Juan Collado on December 27, 2017

SORRY
NEVER MIND
SQUARE ROOT OF R^2 IS R
I FEEL LOST IN THIS LESSON AND THE LAST ONE.

0 answers

Post by Juan Collado on December 27, 2017

In Example 2, you went from
_+ square root of 3V/pieH = Square root of R^2

to +- square root of 3V/pieH  = R
How do you just get rid of the square root on one side without doing the same to the other.

I always have problems with square roots

Quadratic Formulas & Applications

  • Some formulas are quadratic in nature if they contain a squared term.
  • Remember we can use the principle of square roots to help isolate the squared variable.
  • Some word problems lead to quadratic equations. These usually include problems with distance, right triangles, or area.
  • The Pythagorean Theorem says a2 + b2 = c2, where a and b are the legs of the right triangle and c is the longest side.

Quadratic Formulas & Applications

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
  • Objectives 0:09
  • Quadratic Formulas and Applications 0:35
    • Squared Variable
    • Principle of Square Roots
  • Example 1 1:09
  • Example 2 2:04
  • Quadratic Formulas and Applications Cont. 3:34
  • Example 3 4:42
  • Example 4 13:33
  • Example 5 20:50

Transcription: Quadratic Formulas & Applications

Welcome back to www.educator.com.0000

In this lesson we are going to take a look at some quadratic formulas and some applications of quadratics.0003

Specifically we will look at how you can recognize some quadratic equations or some formulas that have a quadratics in them.0012

Think of that squared term.0020

Some situations that tend to be very common with quadratic equations are the Pythagorean Theorem and the distance formula.0022

Look for some of these when we get to our application problems.0031

When looking at various different formulas sometimes they might involve a squared variable in them.0038

If you go through the process of trying to solve for that variable, you might have to try and get it all by itself.0044

Since it is going to be squared we will have to use the principle square roots to get it completely by itself.0052

Remember that when you are using the principle of square roots, not only will we have one solution that is positive,0058

we will also have another one that is negative.0064

Let us see this quick small formula.0072

In this formula I have E = m × c2.0075

Let us go ahead and try and solve for c.0080

As I work and try to get it all by itself, the first thing I will do is divide both sides by m.0084

E ÷ m = c.0092

To get it completely all alone, this is where we will go ahead and take the square root of both sides.0096

From the principle of square roots I will have ± √e÷m.0104

You can think of this as two different ones.0111

That c = √e÷m and c = -√e÷m.0114

In this other formula you will see that we have pretty much the same situation.0127

We are looking to solve for the r and r2.0132

We have to do a little bit more work to get it all by itself, but it is not too bad.0136

Let us start this one off by multiplying both sides by 3.0142

We are doing that just to get rid of that 1/3 that I see out front.0145

1/3 pi r2h.0154

This will give me 3v = and then the 3 and 1/3 will cancel each other out, pi r2 h.0161

To get a little bit closer to getting it all by itself, let us divide both sides by pi and h.0171

3v pi × h = r2.0180

We have not used that principles of square roots yet but now it is going to play a large part0187

because in order to get rid of that square, we will square root both sides.0192

Remember that we need a + and -.0196

Our formula is ±√3v÷pi h.0207

In some applications of quadratics we will have to do a little bit more work and set it up ourselves.0215

Some things that will help with the setup process is to watch for things such as right triangles or maybe even problems involving area.0222

The reason why this is important is these situations often lead to things that are being squared.0230

Here is a quick example.0236

When looking at a right triangle, triangle with a right angle in it, the sides are related.0238

If I take the two legs of the triangle, they are related to the hypotenuse in this way right here at a2 + b2 =c2.0247

This is known as the Pythagorean Theorem.0258

If you have a problem that deals with a right triangle, you can be sure that this will probably show up along the way.0266

If we are going to end up solving it with all the squared terms in there, we are going to need many of our quadratic tools to tackle it.0275

Let us see that triangle as we look at this first application problem.0286

This one says that we have two ships and they are leaving port at the same time.0290

One is headed due south and the other one is headed due east.0294

Several hours after departure the ships are 195 miles apart.0298

The ship traveling south travel 105 miles farther than the other one, how many miles did they each travel individually.0304

Let us see if we can get a diagram of what is going on here.0313

Maybe here I have my port, we will make a nice illusion here and I have two ships.0317

One is traveling due south and the other one is going due east.0325

What I know so far is that after a few hours that they are exactly 195 miles apart.0336

We would be talking about this distance right here.0344

You can see that what we have created here is one of those right triangles.0353

We cannot solve that yet, we need a little bit more information about the legs of the triangle.0358

The ship traveling south traveled 105 miles farther than the other one.0362

We have no idea how far the ship going east went, we better call that x.0368

x is the distance the east ship travelled.0375

The other one is exactly 105 miles more.0392

This gives me information about all sides of that triangle.0400

I can relate them using the Pythagorean Theorem.0404

That would be one of our legs square + the value of the other leg square = the value of the hypotenuse square.0409

Looking over, we can see that because of that squared terms we definitely have a quadratic equation.0430

We are going to use our quadratic tools to tackle it.0435

Let us go ahead and copy that formula down so we can look at it.0444

If we are going to solve this we need to foil things out and get everything over on one side.0460

That way we could do something like the quadratic formula or even factoring.0466

I'm going to foil this out and see what we get.0472

Our first terms x × x =x2.0480

Outside would be 105x and so would be the inside terms.0485

Our last terms would take 105 × 105.0491

On the other side we have 1952.0503

The numbers are quite large.0514

There are lots of things in here that we can go ahead and combine.0516

I have some x2 that will go ahead and put together.0519

I have some x’s and then we will move this 38,025 to the other side so we can combine it with our other constant.0522

Let us see what this gives us.0534

2x2 + 210x = -27,000.0538

Now it definitely looks more like a quadratic, we are getting closer.0557

Always check to see if you can pull out a common factor.0560

One thing I see here is everything is divisible by 2.0563

Let us divide by 2.0567

x2 + 105x - 13,500.0573

We have several options open to us.0588

We could try factoring but I’m a little hesitant because the numbers are quite large.0590

I’m going to go ahead and jump to something like a quadratic formula and go more direct root.0596

I will identify a, b, and c.0602

Negative the value of b ±√(b2 )-4 ×a ×c ÷ 2a.0607

Do not be intimidated by those large numbers just work carefully through the problem and try not to make a mistake.0628

Let us take care of that 105 being squared underneath the square root.0637

I get -105 ±√11025 and we have 4 × 13, 500.0647

All of that is being divided by 2.0669

We will go ahead and add these together and end up taking their square root.0678

I’m getting -105 ± and I borrowed down the square root in here so 255 ÷ 2.0692

I have two things to consider either -105 + 255 ÷ 2 or -105 – 255 ÷ 2.0705

That will be 150 ÷ 2 that will simply be 75.0724

The other one being -360 ÷ 2 =-180.0731

Remember when we first identified what x was, it was the distance that our east ship was traveling.0742

We have a bit of a problem with one of our solutions.0748

Notice how this one turned out to be -180.0752

We can not have our ship traveling a negative distance.0755

We are going to get rid of that as one of our possible solutions.0758

It just does not make sense in the context of the problem.0761

We will keep that the east ship went 75 miles.0763

That is only half of the problem.0768

The other ship went exactly 105 more so we can add that to this and now get the distance of both ships traveled.0771

We get 75 + 105 = 180 that would be our south ship 180 miles and the other one we will say as our east ship 75 miles of travel.0786

We have the distance for both of them.0809

Let us try and set up another one of these and see how it works.0816

Harold and his wife have recently installed a built-in rectangular swimming pool.0821

It measures 18 ft × 22 ft.0826

They want to take this swimming pool and add a nice decorative tile boarding, a nice uniform width all around the pool.0829

The question is how wide can they make the title border if they only purchased enough to cover 176 ft2?0836

I think in this one, we need to draw a picture as well.0845

What we have here is a nice rectangular pool and we know about its dimensions.0849

This pool is 22 × 18 ft.0858

What we are going to do with this pool is actually build a nice border that goes around it.0864

The thing is that we do not know how wide this border should be.0874

That is what we are trying to figure out, how wide is this border?0877

Let us set that as our unknown.0881

x is the width of the border.0885

I want to be able to figure out how the area of just the border is related to the 176.0897

Let us see what we can do in here.0905

Well, if I'm looking at the area of just this tiled part, I could look at that as the area of the entire rectangle0907

and then subtract out that area of the pool in between.0917

Let us try and do that.0921

What is the total length of the large rectangle and the total length and width of the rectangle?0924

Since we are adding these little border pieces on the outside, this would be 22 +.it has 1, 2 widths of x.0932

That would be 18 + 2x.0944

The large rectangles area would be (22 + 2x) × (18 + 2x), that is the area of the large rectangle.0949

We will subtract out the area of the other one.0964

The difference of those two rectangles should give us just the area of the border, 176.0972

Now we have our quadratic equation that we can go ahead and solve from here.0981

Let us copy that down.0990

At first it does not look like a quadratic but notice I will end up multiplying these x’s here1010

and will get that x2 term, so this is going to be quadratic after all.1017

We have some multiplication to do, let us go ahead and start foiling things out so we can eventually get everything over on one side.1022

Our first term is 22 × 18 = 396.1034

Our outside terms 44x, inside terms 36x and our last terms 4x2.1040

I have 18 × 22= 396, equals 176.1057

There are no squared terms that need and there is a few other things that we can combine.1066

Here we have 396 and -396, those will go away.1072

We can go ahead and put these two together.1077

The largest term in here is 4x2 that takes care of that one.1083

Then I will combine together my x’s 44 + 36 = 80x.1090

It takes care of both of those since that 396 is cancel out and1100

we will get the 176 on the same side as everything else by subtracting it from both sides of the equation.1106

Okay, now it is looking much better.1113

Definitely you will it is quadratic.1116

There is my 4x2 at the very beginning.1117

I need to just work on solving it.1120

The numbers are getting large, but I think everything in here is divisible by 4.1123

Let us go ahead and divide everything by 4.1129

It should make things much smaller.1135

Definitely looking much better now .1147

This one, we can solve using our quadratic techniques, quadratic formula, factoring.1149

In this one I think I see a way that this could factor.1155

Let us sit down a couple of parentheses and try reverse foil.1160

Two things that will multiply to give me x2 would have to be x and x.1165

Some possibilities for the second term would be 1 and 44, 2 and 22.1171

I'm pretty sure it is going to be those so that I can get the 20 in the middle, 22 and -2.1178

Now that we have both of our factors we will take each of the factors and set them equal to 0.1193

Solving each of these separately I have x = -22 and x = 2.1205

If you think back when we are first deciding what x was, x represents the width of the border.1213

We can not have a negative width for our border.1220

It simply does not make sense.1224

We can have a positive border that will work out just fine.1227

Let us say that the border is 2 ft wide.1231

It does take a little bit of work to set these equations up but definitely watch for quadratic to show up in there somewhere.1243

One last problem.1252

In this one we have a rectangular piece of metal and it is 15 inches longer than it is wide.1254

There are going to be little squares that are 3 inches long cut from the four corners and we are going to fill up the flaps to form an open box.1260

The volume of this box is supposed to be a 1092 in3.1269

What were the original dimensions of the piece of metal?1275

We have a very interesting situation going on here.1279

We are starting off with a nice rectangular piece of metal, but then what is going to happen1282

when little squares are cut out from each of the corners.1289

Our metal is starting to look more like this.1295

Then it is folded up into a type of open box.1299

We want to look at some of the variables present and see what we can set up now.1317

Also what do we know about the original piece of metal?1323

It was 15 inches longer than it is wide.1327

I have no idea how wide it is but the length is going to be 15 inches wider.1330

W is the width of the metal.1336

We are going to go along and cut out these little 3 inch corners.1348

Consider when we do that, these new sides here represent the new lengths on the box.1357

How long are those pieces going to be?1367

This first one started off as W but we cut a couple of 3 inches off of it.1372

This length is W – 6, 2 of those 3.1378

We did the same thing for the other length so W + 15 – 6 = W + 9.1384

Look at our correspondence to the folded up box, it means that this length down here is W – 6.1393

This one over here is our W + 9.1400

That is all the information we need.1406

What about the height of this box?1408

That comes from our corners and we know that they are 3 inches.1412

We know the height is 3.1417

Once we have all of our dimensions we can finally relate it to the volume.1421

The volume comes from multiplying the length × the width × the height of this box.1426

Volume = length × width × height or 1092 = a length of W + 9, a width of W - 6 and the height of 3.1433

Let us go ahead and borrow that equation and see what we can do.1453

1092 = W + 9W - 6 and 3.1464

Let us go ahead and start off by maybe multiplying things out using foil.1484

W × W = W2 I have my outside terms and my inside terms, and -54.1495

Things are looking a little bit better but I still have that 3 out there.1510

Unfortunately 1092 is divisible by 3, let us go ahead and divide both sides by 3 so we can get rid of it.1515

It is looking much better.1534

Combine our middle terms here W2 + 3W1535

I’m going to go ahead and subtract this 364 from both sides and we will have that term on there as well.1548

-418 looking good.1568

I just have to simply try and solve this quadratic equation.1574

What do we got?1581

I could try and use something like the AC method.1584

I could try and factor this.1587

We have lots of different options to actually try and break this down.1589

My first leading term here is just got a 1W2, let us see what possibilities we have to multiply and get that 418.1598

As soon as we find them we can say that this is factored.1607

It could be 1418, could be 2209, we got a lot of different things.1613

Packing up a lot of things go in for 18, it looks like 19 does.1626

19 and 22 do, which is good because I'm trying to that middle term 3.1642

Let us go ahead and borrow 22 – 19.1648

Remember you can also use the quadratic formula on this now we will also find what we need.1655

We did two things here, either that W + 22 = 0 or that W -19 = 0.1662

I know that -22 could be W or that 19 could be W.1671

What is W exactly?1678

Remember it represents the width of our original piece of metal.1680

We do not want to deal with a negative width because that does not make sense in the context of the problem, let us get rid of that.1685

But if it is perfectly okay to have the width at 19.1692

At least we know about one of the dimensions of our piece of metal.1699

If you want the other dimension, remember that it is just a little bit larger.1707

In fact, it is exactly 15 larger.1713

You take 19 add 15 and see what we get.1716

It looks like the other length will be 34 inches.1723

When dealing with applications of quadratics, it does take a little bit of work to set them up1730

but when you do get to that quadratic part use all tools available to you for solving those quadratics.1736

Thank you for watching www.educator.com.1742

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