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INSTRUCTORS Carleen Eaton Grant Fraser Eric Smith
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Eric Smith

Eric Smith

Equations in Quadratic Form

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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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Equations in Quadratic Form

  • Some equations are similar to quadratic equations, but involve different powers other than two. For these we can try substituting a new variable in, and see if it becomes a quadratic.
  • When using a u-substitution, look for one term that has a power exactly twice as much as another power. This will give you a clue on what u-should replace in the original.
  • Don’t forget to return to the original variable, once you are done solving the quadratic. To do this, re-substitute in what u is equal to.

Equations in Quadratic Form

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 0:08
  • Equations in Quadratic Form 0:24
    • Using a Substitution
    • U-Substitution
  • Example 1 2:07
  • Example 2 5:36
  • Example 3 8:31
  • Example 4 11:14

Transcription: Equations in Quadratic Form

Welcome back to www.educator.com.0000

In this lesson we are going to go ahead and take a look at equations that are quadratic in form.0002

This will be some special types of problems and that they are not quite quadratic,0009

but we can manipulate them so that they at least look like quadratic.0013

The reason why that is important is because then we can use our quadratic solving techniques.0017

Let us go ahead and tackle them.0021

We call that an equation is a quadratic equation if it is on the form ax2 + bx + c =0.0027

As long as a, that first value is not 0.0036

If we can take an equation and write it in this form, then it is said to be in standard form.0041

We will see many different examples of quadratic equations and how to solve them.0047

There are a few different types of equations that are definitely not quadratic.0055

For example, this one is x4 is not quadratic.0060

This one has x2/3, the interesting thing about some of these equations is that you can treat them as quadratics during the solving process.0064

The way you do this is you use a substitution process to go ahead and turn it into a much more manageable form.0076

The way the substitution process works is that we choose part of the equation and we let that equal u.0088

We will go ahead and we rewrite the entire equation using this new u.0095

What you will find is that you should replace all of the original variables in the problem.0100

If you had x’s now you just have a whole bunch of u’s.0105

When you are done solving the equation though, we will make sure that we go back to the original variables.0110

It is kind of tricky what I'm describing here, I want you to see it in action.0116

You will see that it does make things much easier for solving these types of equations.0121

Let us use one like this example.0129

I want to solve x4 - 5x2 + 4 = 0.0132

This is not a quadratic because I’m dealing with x4 out front.0136

What I'm going to do is I’m going to turn it into a quadratic form.0141

I’m going to let u = x2.0146

The reason why I’m doing that is I'm hoping to replace this entire variable right there with u.0153

If that was the only thing I replaced I would still be in trouble.0160

I still have x's and u’s running around.0163

Let us take that little equation that we created u = x2.0166

Let us square both sides of that.0172

That would give us that u2 = x4 and that gives us a way that we can also swap out this other x over here.0176

x4 is the same as u2.0186

I can take our equation here and write it as u2 - 5u + 4 = 0.0192

The reason why that is just so important is because this new equation is quadratic in u.0204

I have open many different solving techniques that I can use on this thing.0212

I could try reverse foil to simply factor and use the 0 factor property.0216

I could try our quadratic formula and get the solution directly, but I have lots of things open to me.0221

This one is not too bad, so I'm just going to go ahead and factor it and then use that 0 factor property.0228

My first terms must be a u and u.0237

I need two things that multiply to get 4, but they add to give me -5.0241

I only have one option for that, just -4 and -1.0249

I know that u -4 could be 0 and u -1 = 0.0255

Solving each of the separately would give me u = 4 and u = 1.0261

Once we switch into a quadratic it is much more easier to manage.0268

The problem is that we solve it for u in and our original problem had x’s.0272

We are not exactly done with this problem yet.0277

At this stage we want to borrow what we called u and swap out for our original x's.0280

This says that x2 = 4.0289

This one is x2 = 1.0295

I'm left with two smaller quadratic equations and I will solve each of these directly.0301

I will simply take the square root of both sides, so + - square root of 4 and x = + - the square root of 1.0307

Good thing both of these can be simplified.0315

+ -2 and + -1, so I have four different solutions for this particular problem.0317

What got our foot in the door was being able to write all of these variables using a new variable and reducing those powers.0327

You can do this for a variety of types of equations to put them in a quadratic form.0338

What we want to recognize is what you should swap out for that u value.0344

You can usually use this technique if you recognize that one of the powers is twice as large as the other power.0350

2/3 is exactly twice as large as 1/3.0357

I’m going to use this to help me swap out my x's for u’s.0361

Let u =x1/3.0367

I remember that you can often figure out what the other one means to be by squaring both sides of this little equation.0373

u2 = (x1/3)2 which is the same as x2/3.0379

I can swap out both of these.0388

u2 – 2u -15 = 0.0392

Everything else is the same, but now I have those u’s in there and now it is quadratic in form.0402

I can use a lot of other techniques to go ahead and solve this one.0406

What shall we do with this one?0411

I think this is another one we can simply factor without too much problem.0413

My first terms better be u and now I need two things that will multiply and give me -15 but somehow add to get -2.0418

Let us use 5 and 3.0428

-5 and 3.0432

This will give me that u -5 = 0, and u + 3 = 0.0435

That comes from the 0 factor property.0443

Solving each of the separately I have u = 5 and u = - 3.0447

Each of these are looking pretty good.0454

We want to go back to our original variable.0457

I know exactly what u is, let us go ahead and put that in for both of these spots on here.0460

x1/3 x1/3 and this is equal to 5 and equal to -3.0467

To solve this directly from here, I think I can take both sides and cube it.0478

53 would be 125, -33 = -27.0491

Just like the foil, now we have our solutions to the original problem.0501

We work all the way back to x, which was the original variable.0507

Now some might actually be quadratic but you can still use this technique to put it in a much nicer form.0513

In this next one, we are dealing with a 3x - 12 + 2 × 3x - 1 = 8.0521

One way that you could handle this is simply to multiply everything out0529

and then get everything on one side sand set it equal to 0 and use the quadratic formula.0534

I'm not going to do that because I noticed that I actually have this common piece of 3x – 1.0539

I'm going to swap out that common piece and call it something else.0547

You will say let u = 3x - 1.0551

Now that we have that, this will become much easier to solve.0558

We will have a u2 + 2 × u = 8.0563

We can work to get everything on one side and now it is set equal to 0.0576

For this u, let us go ahead and reverse foil it.0585

u and u, need two numbers that would multiply to be -8 but add to be a positive 2, 4 and -2.0593

This gives us two solutions u = -4 or u = 2.0603

We must go back to that original variable.0616

We know what u is, u = 3x – 1.0620

We are going to put that in for both of our u’s.0624

3x – 1 = -4 and 3x - 1 = 2.0628

We can solve each of these separately.0641

Adding 1 to both sides would be -3 and divide both sides by 3 would give us -1.0646

There is one of our solutions right there.0654

For the other one, we will add 1 to both sides and get 3, then divide both sides by 3 and get 1.0660

We have our solutions for this problem.0669

Let us do one more and this would involve some negative exponents.0676

This one has 2 - y -61 - 1 = 6 × y – 6-2.0681

This one is a little jumbled up, it is difficult to figure out what we should swap out.0689

It is good to know that we do have a y -6 that seems to be a common.0693

That will be a part of what we end up exchanging with u.0698

Let us go ahead and get everything onto one side.0704

I’m going to move everything over to the right side 6y - 6-2.0707

I will add y -6-1 -2.0717

What I recognize here is that this -2 is exactly twice as large as that -1.0724

It is a pretty good indication that I will end up swapping on my u for this piece right here.0731

Let u = let us call this y – 6-1.0739

Let us go ahead and rewrite our equation using this.0750

We have 6u2 + u – 2.0755

What I can see here is that it is definitely quadratic and the numbers are much smaller, much nicer to deal with.0767

What do I do from here?0774

I still have to be able to figure out what the solutions of this quadratic are.0776

We got lots of tools available to us, let us go ahead and try the quadratic formula.0780

-b ±√4, (±√b2 – 4) × (a × c) ÷ 2a.0787

I think this will simplify nicely.0812

I have 4 × 6 = 24 × 2 = 48 ÷ 12 or -1 ± √49/12 or -1 ± 7/12.0816

u = -1 + 7/12 and I have u = -1- 7/12.0841

-1 + 7 = 6/12 would be -8/12.0856

Both of those reduce, this one to ½ .0863

The other one 4 goes into the top and 4 goes in the bottom, -3/2.0869

I know little bit more about what u is equal to.0876

Of course, we can not stop there, we must go back to our original variable.0880

We must work back all the way to those y’s.0886

Let us put those back in for u and see if we can solve this from here.0890

Let us see y - 6-1 = ½ and y - 6-1 = -2/3.0897

To solve this from here I think I will raise both sides to that -1 power.0912

That will give me y - 6 and then with that -1 exponent, that will change the location of the 1 and 2, 2/1.0920

Let us do the same thing with our other equation.0931

Raise both sides to -1, this is y - 6 = -3/2, since it changes the location of the -2 and the 3.0933

These are almost done.0945

With the one on the left here, let us add 6 to both sides.0948

This will give us y = 8 and with this other one let us add 6 to both sides, -3/2 + 6.0952

It looks like we do have to get a common denominator, but this one is not so bad.0965

It looks like it is just 9/2.0975

We have worked all the way back to our original variable, we know what the solutions are.0978

It can take quite a bit of work to end up swapping out the u and solving from there.0984

The important part is that if we do not swap out the u these we do not have a lot of other techniques to solve equations like this.0989

Make sure you properly identify what u you need to swap out and put it back in so you can get back to your original variable.0997

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