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INSTRUCTORS Carleen Eaton Grant Fraser
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Dr. Carleen Eaton

Dr. Carleen Eaton

Solving Quadratic Equations by Factoring

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Table of Contents

I. Equations and Inequalities
Expressions and Formulas

22m 23s

Intro
0:00
Order of Operations
0:19
Variable
0:27
Algebraic Expression
0:46
Term
0:57
Example: Algebraic Expression
1:25
Evaluate Inside Grouping Symbols
1:55
Evaluate Powers
2:30
Multiply/Divide Left to Right
2:55
Add/Subtract Left to Right
3:35
Monomials
4:40
Examples of Monomials
4:52
Constant
5:27
Coefficient
5:46
Degree
6:25
Power
7:15
Polynomials
8:02
Examples of Polynomials
8:24
Binomials, Trinomials, Monomials
8:53
Term
9:21
Like Terms
10:02
Formulas
11:00
Example: Pythagorean Theorem
11:15
Example 1: Evaluate the Algebraic Expression
11:50
Example 2: Evaluate the Algebraic Expression
14:38
Example 3: Area of a Triangle
19:11
Example 4: Fahrenheit to Celsius
20:41
Properties of Real Numbers

20m 15s

Intro
0:00
Real Numbers
0:07
Number Line
0:15
Rational Numbers
0:46
Irrational Numbers
2:24
Venn Diagram of Real Numbers
4:03
Irrational Numbers
5:00
Rational Numbers
5:19
Real Number System
5:27
Natural Numbers
5:32
Whole Numbers
5:53
Integers
6:19
Fractions
6:46
Properties of Real Numbers
7:15
Commutative Property
7:34
Associative Property
8:07
Identity Property
9:04
Inverse Property
9:53
Distributive Property
11:03
Example 1: What Set of Numbers?
12:21
Example 2: What Properties Are Used?
13:56
Example 3: Multiplicative Inverse
16:00
Example 4: Simplify Using Properties
17:18
Solving Equations

19m 10s

Intro
0:00
Translations
0:06
Verbal Expressions and Algebraic Expressions
0:13
Example: Sum of Two Numbers
0:19
Example: Square of a Number
1:33
Properties of Equality
3:20
Reflexive Property
3:30
Symmetric Property
3:42
Transitive Property
4:01
Addition Property
5:01
Subtraction Property
5:37
Multiplication Property
6:02
Division Property
6:30
Solving Equations
6:58
Example: Using Properties
7:18
Solving for a Variable
8:25
Example: Solve for Z
8:34
Example 1: Write Algebraic Expression
10:15
Example 2: Write Verbal Expression
11:31
Example 3: Solve the Equation
14:05
Example 4: Simplify Using Properties
17:26
Solving Absolute Value Equations

17m 31s

Intro
0:00
Absolute Value Expressions
0:09
Distance from Zero
0:18
Example: Absolute Value Expression
0:24
Absolute Value Equations
1:50
Example: Absolute Value Equation
2:00
Example: Isolate Expression
3:13
No Solution
3:46
Empty Set
3:58
Example: No Solution
4:12
Number of Solutions
4:46
Check Each Solution
4:57
Example: Two Solutions
5:05
Example: No Solution
6:18
Example: One Solution
6:28
Example 1: Evaluate for X
7:16
Example 2: Write Verbal Expression
9:08
Example 3: Solve the Equation
12:18
Example 4: Simplify Using Properties
13:36
Solving Inequalities

17m 14s

Intro
0:00
Properties of Inequalities
0:08
Addition Property
0:17
Example: Using Numbers
0:30
Subtraction Property
1:03
Example: Using Numbers
1:19
Multiplication Properties
1:44
C>0 (Positive Number)
1:50
Example: Using Numbers
2:05
C<0 (Negative Number)
2:40
Example: Using Numbers
3:10
Division Properties
4:11
C>0 (Positive Number)
4:15
Example: Using Numbers
4:27
C<0 (Negative Number)
5:21
Example: Using Numbers
5:32
Describing the Solution Set
6:10
Example: Set Builder Notation
6:26
Example: Graph (Closed Circle)
7:08
Example: Graph (Open Circle)
7:30
Example 1: Solve the Inequality
7:58
Example 2: Solve the Inequality
9:06
Example 3: Solve the Inequality
10:10
Example 4: Solve the Inequality
13:12
Solving Compound and Absolute Value Inequalities

25m

Intro
0:00
Compound Inequalities
0:08
And and Or
0:13
Example: And
0:22
Example: Or
1:12
And Inequality
1:41
Intersection
1:49
Example: Numbers
2:08
Example: Inequality
2:43
Or Inequality
4:35
Example: Union
4:45
Example: Inequality
5:53
Absolute Value Inequalities
7:19
Definition of Absolute Value
7:33
Examples: Compound Inequalities
8:30
Example: Complex Inequality
12:21
Example 1: Solve the Inequality
12:54
Example 2: Solve the Inequality
17:21
Example 3: Solve the Inequality
18:54
Example 4: Solve the Inequality
22:15
II. Linear Relations and Functions
Relations and Functions

32m 5s

Intro
0:00
Coordinate Plane
0:20
X-Coordinate and Y-Coordinate
0:30
Example: Coordinate Pairs
0:37
Quadrants
1:20
Relations
2:14
Domain and Range
2:19
Set of Ordered Pairs
2:29
As a Table
2:51
Functions
4:21
One Element in Range
4:32
Example: Mapping
4:43
Example: Table and Map
6:26
One-to-One Functions
8:01
Example: One-to-One
8:22
Example: Not One-to-One
9:18
Graphs of Relations
11:01
Discrete and Continuous
11:12
Example: Discrete
11:22
Example: Continous
12:30
Vertical Line Test
14:09
Example: S Curve
14:29
Example: Function
16:15
Equations, Relations, and Functions
17:03
Independent Variable and Dependent Variable
17:16
Function Notation
19:11
Example: Function Notation
19:23
Example 1: Domain and Range
20:51
Example 2: Discrete or Continous
23:03
Example 3: Discrete or Continous
25:53
Example 4: Function Notation
30:05
Linear Equations

14m 46s

Intro
0:00
Linear Equations and Functions
0:07
Linear Equation
0:19
Example: Linear Equation
0:29
Example: Linear Function
1:07
Standard Form
2:02
Integer Constants with No Common Factor
2:08
Example: Standard Form
2:27
Graphing with Intercepts
4:05
X-Intercept and Y-Intercept
4:12
Example: Intercepts
4:26
Example: Graphing
5:14
Example 1: Linear Function
7:53
Example 2: Linear Function
9:10
Example 3: Standard Form
10:04
Example 4: Graph with Intercepts
12:25
Slope

23m 7s

Intro
0:00
Definition of Slope
0:07
Change in Y / Change in X
0:26
Example: Slope of Graph
0:37
Interpretation of Slope
3:07
Horizontal Line (0 Slope)
3:13
Vertical Line (Undefined Slope)
4:52
Rises to Right (Positive Slope)
6:36
Falls to Right (Negative Slope)
6:53
Parallel Lines
7:18
Example: Not Vertical
7:30
Example: Vertical
7:58
Perpendicular Lines
8:31
Example: Perpendicular
8:42
Example 1: Slope of Line
10:32
Example 2: Graph Line
11:45
Example 3: Parallel to Graph
13:37
Example 4: Perpendicular to Graph
17:57
Writing Linear Functions

23m 5s

Intro
0:00
Slope Intercept Form
0:11
m and b
0:28
Example: Graph Using Slope Intercept
0:43
Point Slope Form
2:41
Relation to Slope Formula
3:03
Example: Point Slope Form
4:36
Parallel and Perpendicular Lines
6:28
Review of Parallel and Perpendicular Lines
6:31
Example: Parallel
7:50
Example: Perpendicular
9:58
Example 1: Slope Intercept Form
11:07
Example 2: Slope Intercept Form
13:07
Example 3: Parallel
15:49
Example 4: Perpendicular
18:42
Special Functions

31m 5s

Intro
0:00
Step Functions
0:07
Example: Apple Prices
0:30
Absolute Value Function
4:55
Example: Absolute Value
5:05
Piecewise Functions
9:08
Example: Piecewise
9:27
Example 1: Absolute Value Function
14:00
Example 2: Absolute Value Function
20:39
Example 3: Piecewise Function
22:26
Example 4: Step Function
25:25
Graphing Inequalities

21m 42s

Intro
0:00
Graphing Linear Inequalities
0:07
Shaded Region
0:19
Using Test Points
0:32
Graph Corresponding Linear Function
0:46
Dashed or Solid Lines
0:59
Use Test Point
1:21
Example: Linear Inequality
1:58
Graphing Absolute Value Inequalities
4:50
Graph Corresponding Equations
4:59
Use Test Point
5:20
Example: Absolute Value Inequality
5:38
Example 1: Linear Inequality
9:17
Example 2: Linear Inequality
11:56
Example 3: Linear Inequality
14:29
Example 4: Absolute Value Inequality
17:06
III. Systems of Equations and Inequalities
Solving Systems of Equations by Graphing

17m 13s

Intro
0:00
Systems of Equations
0:09
Example: Two Equations
0:24
Solving by Graphing
0:53
Point of Intersection
1:09
Types of Systems
2:29
Independent (Single Solution)
2:34
Dependent (Infinite Solutions)
3:05
Inconsistent (No Solution)
4:23
Example 1: Solve by Graphing
5:20
Example 2: Solve by Graphing
9:10
Example 3: Solve by Graphing
12:27
Example 4: Solve by Graphing
14:54
Solving Systems of Equations Algebraically

23m 53s

Intro
0:00
Solving by Substitution
0:08
Example: System of Equations
0:36
Solving by Multiplication
7:22
Extra Step of Multiplying
7:38
Example: System of Equations
8:00
Inconsistent and Dependent Systems
11:14
Variables Drop Out
11:48
Inconsistent System (Never True)
12:01
Constant Equals Constant
12:53
Dependent System (Always True)
13:11
Example 1: Solve Algebraically
13:58
Example 2: Solve Algebraically
15:52
Example 3: Solve Algebraically
17:54
Example 4: Solve Algebraically
21:40
Solving Systems of Inequalities By Graphing

27m 12s

Intro
0:00
Solving by Graphing
0:08
Graph Each Inequality
0:25
Overlap
0:35
Corresponding Linear Equations
1:03
Test Point
1:23
Example: System of Inequalities
1:51
No Solution
7:06
Empty Set
7:26
Example: No Solution
7:34
Example 1: Solve by Graphing
10:27
Example 2: Solve by Graphing
13:30
Example 3: Solve by Graphing
17:19
Example 4: Solve by Graphing
23:23
Solving Systems of Equations in Three Variables

28m 53s

Intro
0:00
Solving Systems in Three Variables
0:17
Triple of Values
0:31
Example: Three Variables
0:56
Number of Solutions
5:55
One Solution
6:08
No Solution
6:24
Infinite Solutions
7:06
Example 1: Solve 3 Variables
7:59
Example 2: Solve 3 Variables
13:50
Example 3: Solve 3 Variables
19:54
Example 4: Solve 3 Variables
25:50
IV. Matrices
Basic Matrix Concepts

11m 34s

Intro
0:00
What is a Matrix
0:26
Brackets
0:46
Designation
1:21
Element
1:47
Matrix Equations
1:59
Dimensions
2:27
Rows (m) and Columns (n)
2:37
Examples: Dimensions
2:43
Special Matrices
4:22
Row Matrix
4:32
Column Matrix
5:00
Zero Matrix
6:00
Equal Matrices
6:30
Example: Corresponding Elements
6:36
Example 1: Matrix Dimension
8:12
Example 2: Matrix Dimension
9:03
Example 3: Zero Matrix
9:38
Example 4: Row and Column Matrix
10:26
Matrix Operations

21m 36s

Intro
0:00
Matrix Addition
0:18
Same Dimensions
0:25
Example: Adding Matrices
1:04
Matrix Subtraction
3:42
Same Dimensions
3:48
Example: Subtracting Matrices
4:04
Scalar Multiplication
6:08
Scalar Constant
6:24
Example: Multiplying Matrices
6:32
Properties of Matrix Operations
8:23
Commutative Property
8:41
Associative Property
9:08
Distributive Property
9:44
Example 1: Matrix Addition
10:24
Example 2: Matrix Subtraction
11:58
Example 3: Scalar Multiplication
14:23
Example 4: Matrix Properties
16:09
Matrix Multiplication

29m 36s

Intro
0:00
Dimension Requirement
0:17
n = p
0:24
Resulting Product Matrix (m x q)
1:21
Example: Multiplication
1:54
Matrix Multiplication
3:38
Example: Matrix Multiplication
4:07
Properties of Matrix Multiplication
10:46
Associative Property
11:00
Associative Property (Scalar)
11:28
Distributive Property
12:06
Distributive Property (Scalar)
12:30
Example 1: Possible Matrices
13:31
Example 2: Multiplying Matrices
17:08
Example 3: Multiplying Matrices
20:41
Example 4: Matrix Properties
24:41
Determinants

33m 13s

Intro
0:00
What is a Determinant
0:13
Square Matrices
0:23
Vertical Bars
0:41
Determinant of a 2x2 Matrix
1:21
Second Order Determinant
1:37
Formula
1:45
Example: 2x2 Determinant
1:58
Determinant of a 3x3 Matrix
2:50
Expansion by Minors
3:08
Third Order Determinant
3:19
Expanding Row One
4:06
Example: 3x3 Determinant
6:40
Diagonal Method for 3x3 Matrices
13:24
Example: Diagonal Method
13:36
Example 1: Determinant of 2x2
18:59
Example 2: Determinant of 3x3
20:03
Example 3: Determinant of 3x3
25:35
Example 4: Determinant of 3x3
29:22
Cramer's Rule

28m 25s

Intro
0:00
System of Two Equations in Two Variables
0:16
One Variable
0:50
Determinant of Denominator
1:14
Determinants of Numerators
2:23
Example: System of Equations
3:34
System of Three Equations in Three Variables
7:06
Determinant of Denominator
7:17
Determinants of Numerators
7:52
Example 1: Two Equations
8:57
Example 2: Two Equations
13:21
Example 3: Three Equations
17:11
Example 4: Three Equations
23:43
Identity and Inverse Matrices

22m 25s

Intro
0:00
Identity Matrix
0:13
Example: 2x2 Identity Matrix
0:30
Example: 4x4 Identity Matrix
0:50
Properties of Identity Matrices
1:24
Example: Multiplying Identity Matrix
2:52
Matrix Inverses
5:30
Writing Matrix Inverse
6:07
Inverse of a 2x2 Matrix
6:39
Example: 2x2 Matrix
7:31
Example 1: Inverse Matrix
10:18
Example 2: Find the Inverse Matrix
13:04
Example 3: Find the Inverse Matrix
17:53
Example 4: Find the Inverse Matrix
20:44
Solving Systems of Equations Using Matrices

22m 32s

Intro
0:00
Matrix Equations
0:11
Example: System of Equations
0:21
Solving Systems of Equations
4:01
Isolate x
4:16
Example: Using Numbers
5:10
Multiplicative Inverse
5:54
Example 1: Write as Matrix Equation
7:18
Example 2: Use Matrix Equations
9:12
Example 3: Use Matrix Equations
15:06
Example 4: Use Matrix Equations
19:35
V. Quadratic Functions and Inequalities
Graphing Quadratic Functions

31m 48s

Intro
0:00
Quadratic Functions
0:12
A is Zero
0:27
Example: Parabola
0:45
Properties of Parabolas
2:08
Axis of Symmetry
2:11
Vertex
2:32
Example: Parabola
2:48
Minimum and Maximum Values
9:02
Positive or Negative
9:28
Upward or Downward
9:58
Example: Minimum
10:31
Example: Maximum
11:16
Example 1: Axis of Symmetry, Vertex, Graph
12:41
Example 2: Axis of Symmetry, Vertex, Graph
17:25
Example 3: Minimum or Maximum
21:47
Example 4: Minimum or Maximum
27:09
Solving Quadratic Equations by Graphing

27m 3s

Intro
0:00
Quadratic Equations
0:16
Standard Form
0:18
Example: Quadratic Equation
0:47
Solving by Graphing
1:41
Roots (x-Intercepts)
1:48
Example: Number of Solutions
2:12
Estimating Solutions
9:23
Example: Integer Solutions
9:30
Example: Estimating
9:53
Example 1: Solve by Graphing
10:52
Example 2: Solve by Graphing
15:10
Example 1: Solve by Graphing
17:50
Example 1: Solve by Graphing
20:54
Solving Quadratic Equations by Factoring

19m 53s

Intro
0:00
Factoring Techniques
0:15
Greatest Common Factor (GCF)
0:37
Difference of Two Squares
1:48
Perfect Square Trinomials
2:30
General Trinomials
3:09
Zero Product Rule
5:22
Example: Zero Product
5:53
Example 1: Solve by Factoring
7:46
Example 1: Solve by Factoring
9:48
Example 1: Solve by Factoring
12:34
Example 1: Solve by Factoring
15:28
Imaginary and Complex Numbers

35m 45s

Intro
0:00
Properties of Square Roots
0:10
Product Property
0:26
Example: Product Property
0:56
Quotient Property
2:17
Example: Quotient Property
2:35
Imaginary Numbers
3:12
Imaginary i
3:51
Examples: Imaginary Number
4:22
Complex Numbers
7:23
Real Part and Imaginary Part
7:33
Examples: Complex Numbers
7:57
Equality
9:37
Example: Equal Complex Numbers
9:52
Addition and Subtraction
10:12
Examples: Adding Complex Numbers
10:25
Complex Plane
13:32
Horizontal Axis (Real)
13:49
Vertical Axis (Imaginary)
13:59
Example: Labeling
14:11
Multiplication
15:57
Example: FOIL Method
16:03
Division
18:37
Complex Conjugates
18:45
Conjugate Pairs
19:10
Example: Dividing Complex Numbers
20:00
Example 1: Simplify Complex Number
24:50
Example 2: Simplify Complex Number
27:56
Example 3: Multiply Complex Numbers
29:27
Example 3: Dividing Complex Numbers
31:48
Completing the Square

27m 11s

Intro
0:00
Square Root Property
0:12
Example: Perfect Square
0:38
Example: Perfect Square Trinomial
3:00
Completing the Square
4:39
Constant Term
4:50
Example: Complete the Square
5:04
Solve Equations
6:42
Add to Both Sides
6:59
Example: Complete the Square
7:07
Equations Where a Not Equal to 1
10:58
Divide by Coefficient
11:08
Example: Complete the Square
11:24
Complex Solutions
14:05
Real and Imaginary
14:14
Example: Complex Solution
14:35
Example 1: Square Root Property
18:31
Example 2: Complete the Square
19:15
Example 3: Complete the Square
20:40
Example 4: Complete the Square
23:56
Quadratic Formula and the Discriminant

22m 48s

Intro
0:00
Quadratic Formula
0:21
Standard Form
0:29
Example: Quadratic Formula
0:57
One Rational Root
3:00
Example: One Root
3:31
Complex Solutions
6:16
Complex Conjugate
6:28
Example: Complex Solution
7:15
Discriminant
9:42
Positive Discriminant
10:03
Perfect Square (Rational)
10:51
Not Perfect Square (2 Irrational)
11:27
Negative Discriminant
12:28
Zero Discriminant
12:57
Example 1: Quadratic Formula
13:50
Example 2: Quadratic Formula
16:03
Example 3: Quadratic Formula
19:00
Example 4: Discriminant
21:33
Analyzing the Graphs of Quadratic Functions

30m 7s

Intro
0:00
Vertex Form
0:12
H and K
0:32
Axis of Symmetry
0:36
Vertex
0:42
Example: Origin
1:00
Example: k = 2
2:12
Example: h = 1
4:27
Significance of Coefficient a
7:13
Example: |a| > 1
7:25
Example: |a| < 1
8:18
Example: |a| > 0
8:51
Example: |a| < 0
9:05
Writing Quadratic Equations in Vertex Form
10:22
Standard Form to Vertex Form
10:35
Example: Standard Form
11:02
Example: a Term Not 1
14:42
Example 1: Vertex Form
19:47
Example 2: Vertex Form
22:09
Example 3: Vertex Form
24:32
Example 4: Vertex Form
28:23
Graphing and Solving Quadratic Inequalities

27m 5s

Intro
0:00
Graphing Quadratic Inequalities
0:11
Test Point
0:18
Example: Quadratic Inequality
0:29
Solving Quadratic Inequalities
3:57
Example: Parameter
4:24
Example 1: Graph Inequality
11:16
Example 2: Solve Inequality
14:27
Example 3: Graph Inequality
19:14
Example 4: Solve Inequality
23:48
VI. Polynomial Functions
Properties of Exponents

19m 29s

Intro
0:00
Simplifying Exponential Expressions
0:09
Monomial Simplest Form
0:19
Negative Exponents
1:07
Examples: Simple
1:34
Properties of Exponents
3:06
Negative Exponents
3:13
Mutliplying Same Base
3:24
Dividing Same Base
3:45
Raising Power to a Power
4:33
Parentheses (Multiplying)
5:11
Parentheses (Dividing)
5:47
Raising to 0th Power
6:15
Example 1: Simplify Exponents
7:59
Example 2: Simplify Exponents
10:41
Example 3: Simplify Exponents
14:11
Example 4: Simplify Exponents
18:04
Operations on Polynomials

13m 27s

Intro
0:00
Adding and Subtracting Polynomials
0:13
Like Terms and Like Monomials
0:23
Examples: Adding Monomials
1:14
Multiplying Polynomials
3:40
Distributive Property
3:44
Example: Monomial by Polynomial
4:06
Example 1: Simplify Polynomials
5:47
Example 2: Simplify Polynomials
6:28
Example 3: Simplify Polynomials
8:38
Example 4: Simplify Polynomials
10:47
Dividing Polynomials

31m 11s

Intro
0:00
Dividing by a Monomial
0:13
Example: Numbers
0:26
Example: Polynomial by a Monomial
1:18
Long Division
2:28
Remainder Term
2:41
Example: Dividing with Numbers
3:04
Example: With Polynomials
5:01
Example: Missing Terms
7:58
Synthetic Division
11:44
Restriction
12:04
Example: Divisor in Form
12:20
Divisor in Synthetic Division
15:54
Example: Coefficient to 1
16:07
Example 1: Divide Polynomials
17:10
Example 2: Divide Polynomials
19:08
Example 3: Synthetic Division
21:42
Example 4: Synthetic Division
25:09
Polynomial Functions

22m 30s

Intro
0:00
Polynomial in One Variable
0:13
Leading Coefficient
0:27
Example: Polynomial
1:18
Degree
1:31
Polynomial Functions
2:57
Example: Function
3:13
Function Values
3:33
Example: Numerical Values
3:53
Example: Algebraic Expressions
5:11
Zeros of Polynomial Functions
5:50
Odd Degree
6:04
Even Degree
7:29
End Behavior
8:28
Even Degrees
9:09
Example: Leading Coefficient +/-
9:23
Odd Degrees
12:51
Example: Leading Coefficient +/-
13:00
Example 1: Degree and Leading Coefficient
15:03
Example 2: Polynomial Function
15:56
Example 3: Polynomial Function
17:34
Example 4: End Behavior
19:53
Analyzing Graphs of Polynomial Functions

33m 29s

Intro
0:00
Graphing Polynomial Functions
0:11
Example: Table and End Behavior
0:39
Location Principle
4:43
Zero Between Two Points
5:03
Example: Location Principle
5:21
Maximum and Minimum Points
8:40
Relative Maximum and Relative Minimum
9:16
Example: Number of Relative Max/Min
11:11
Example 1: Graph Polynomial Function
11:57
Example 2: Graph Polynomial Function
16:19
Example 3: Graph Polynomial Function
23:27
Example 4: Graph Polynomial Function
28:35
Solving Polynomial Functions

21m 10s

Intro
0:00
Factoring Polynomials
0:06
Greatest Common Factor (GCF)
0:25
Difference of Two Squares
1:14
Perfect Square Trinomials
2:07
General Trinomials
2:57
Grouping
4:32
Sum and Difference of Two Cubes
6:03
Examples: Two Cubes
6:14
Quadratic Form
8:22
Example: Quadratic Form
8:44
Example 1: Factor Polynomial
12:03
Example 2: Factor Polynomial
13:54
Example 3: Quadratic Form
15:33
Example 4: Solve Polynomial Function
17:24
Remainder and Factor Theorems

31m 21s

Intro
0:00
Remainder Theorem
0:07
Checking Work
0:22
Dividend and Divisor in Theorem
1:12
Example: f(a)
2:05
Synthetic Substitution
5:43
Example: Polynomial Function
6:15
Factor Theorem
9:54
Example: Numbers
10:16
Example: Confirm Factor
11:27
Factoring Polynomials
14:48
Example: 3rd Degree Polynomial
15:07
Example 1: Remainder Theorem
19:17
Example 2: Other Factors
21:57
Example 3: Remainder Theorem
25:52
Example 4: Other Factors
28:21
Roots and Zeros

31m 27s

Intro
0:00
Number of Roots
0:08
Not Nature of Roots
0:18
Example: Real and Complex Roots
0:25
Descartes' Rule of Signs
2:05
Positive Real Roots
2:21
Example: Positve
2:39
Negative Real Roots
5:44
Example: Negative
6:06
Finding the Roots
9:59
Example: Combination of Real and Complex
10:07
Conjugate Roots
13:18
Example: Conjugate Roots
13:50
Example 1: Solve Polynomial
16:03
Example 2: Solve Polynomial
18:36
Example 3: Possible Combinations
23:13
Example 4: Possible Combinations
27:11
Rational Zero Theorem

31m 16s

Intro
0:00
Equation
0:08
List of Possibilities
0:16
Equation with Constant and Leading Coefficient
1:04
Example: Rational Zero
2:46
Leading Coefficient Equal to One
7:19
Equation with Leading Coefficient of One
7:34
Example: Coefficient Equal to 1
8:45
Finding Rational Zeros
12:58
Division with Remainder Zero
13:32
Example 1: Possible Rational Zeros
14:20
Example 2: Possible Rational Zeros
16:02
Example 3: Possible Rational Zeros
19:58
Example 4: Find All Zeros
22:06
VII. Radical Expressions and Inequalities
Operations on Functions

34m 30s

Intro
0:00
Arithmetic Operations
0:07
Domain
0:16
Intersection
0:24
Denominator is Zero
0:49
Example: Operations
1:02
Composition of Functions
7:18
Notation
7:48
Right to Left
8:18
Example: Composition
8:48
Composition is Not Commutative
17:23
Example: Not Commutative
17:51
Example 1: Function Operations
20:55
Example 2: Function Operations
24:34
Example 3: Compositions
27:51
Example 4: Function Operations
31:09
Inverse Functions and Relations

22m 42s

Intro
0:00
Inverse of a Relation
0:14
Example: Ordered Pairs
0:56
Inverse of a Function
3:24
Domain and Range Switched
3:52
Example: Inverse
4:28
Procedure to Construct an Inverse Function
6:42
f(x) to y
6:42
Interchange x and y
6:59
Solve for y
7:06
Write Inverse f(x) for y
7:14
Example: Inverse Function
7:25
Example: Inverse Function 2
8:48
Inverses and Compositions
10:44
Example: Inverse Composition
11:46
Example 1: Inverse Relation
14:49
Example 2: Inverse of Function
15:40
Example 3: Inverse of Function
17:06
Example 4: Inverse Functions
18:55
Square Root Functions and Inequalities

30m 4s

Intro
0:00
Square Root Functions
0:07
Examples: Square Root Function
0:16
Example: Not Square Root Function
0:46
Radicand
1:12
Example: Restriction
1:31
Graphing Square Root Functions
3:42
Example: Graphing
3:49
Square Root Inequalities
8:47
Same Technique
9:00
Example: Square Root Inequality
9:20
Example 1: Graph Square Root Function
15:19
Example 2: Graph Square Root Function
18:03
Example 3: Graph Square Root Function
22:41
Example 4: Square Root Inequalities
25:37
nth Roots

20m 46s

Intro
0:00
Definition of the nth Root
0:07
Example: 5th Root
0:20
Example: 6th Root
0:51
Principal nth Root
1:39
Example: Principal Roots
2:06
Using Absolute Values
5:58
Example: Square Root
6:18
Example: 6th Root
8:40
Example: Negative
10:15
Example 1: Simplify Radicals
12:23
Example 2: Simplify Radicals
13:29
Example 3: Simplify Radicals
16:07
Example 4: Simplify Radicals
18:18
Operations with Radical Expressions

41m 11s

Intro
0:00
Properties of Radicals
0:16
Quotient Property
0:29
Example: Quotient
1:00
Example: Product Property
1:47
Simplifying Radical Expressions
3:24
Radicand No nth Powers
3:47
Radicand No Fractions
6:33
No Radicals in Denominator
7:16
Rationalizing Denominators
8:27
Example: Radicand nth Power
9:05
Conjugate Radical Expressions
11:47
Conjugates
12:07
Example: Conjugate Radical Expression
13:11
Adding and Subtracting Radicals
16:12
Same Index, Same Radicand
16:20
Example: Like Radicals
16:28
Multiplying Radicals
19:04
Distributive Property
19:10
Example: Multiplying Radicals
19:20
Example 1: Simplify Radical
24:11
Example 2: Simplify Radicals
28:43
Example 3: Simplify Radicals
32:00
Example 4: Simplify Radical
36:34
Rational Exponents

30m 45s

Intro
0:00
Definition 1
0:20
Example: Using Numbers
0:39
Example: Non-Negative
2:46
Example: Odd
3:34
Definition 2
4:32
Restriction
4:52
Example: Relate to Definition 1
5:04
Example: m Not 1
5:31
Simplifying Expressions
7:53
Multiplication
8:31
Division
9:29
Multiply Exponents
10:08
Raised Power
11:05
Zero Power
11:29
Negative Power
11:49
Simplified Form
13:52
Complex Fraction
14:16
Negative Exponents
14:40
Example: More Complicated
15:14
Example 1: Write as Radical
19:03
Example 2: Write with Rational Exponents
20:40
Example 3: Complex Fraction
22:09
Example 4: Complex Fraction
26:22
Solving Radical Equations and Inequalities

31m 27s

Intro
0:00
Radical Equations
0:11
Variables in Radicands
0:22
Example: Radical Equation
1:06
Example: Complex Equation
2:42
Extraneous Roots
7:21
Squaring Technique
7:35
Double Check
7:44
Example: Extraneous
8:21
Eliminating nth Roots
10:04
Isolate and Raise Power
10:14
Example: nth Root
10:27
Radical Inequalities
11:27
Restriction: Index is Even
11:53
Example: Radical Inequality
12:29
Example 1: Solve Radical Equation
15:41
Example 2: Solve Radical Equation
17:44
Example 3: Solve Radical Inequality
20:24
Example 4: Solve Radical Equation
24:34
VIII. Rational Equations and Inequalities
Multiplying and Dividing Rational Expressions

40m 54s

Intro
0:00
Simplifying Rational Expressions
0:22
Algebraic Fraction
0:29
Examples: Rational Expressions
0:49
Example: GCF
1:33
Example: Simplify Rational Expression
2:26
Factoring -1
4:04
Example: Simplify with -1
4:19
Multiplying and Dividing Rational Expressions
6:59
Multiplying and Dividing
7:28
Example: Multiplying Rational Expressions
8:36
Example: Dividing Rational Expressions
11:20
Factoring
14:01
Factoring Polynomials
14:19
Example: Factoring
14:35
Complex Fractions
18:22
Example: Numbers
18:37
Example: Algebraic Complex Fractions
19:25
Example 1: Simplify Rational Expression
25:56
Example 2: Simplify Rational Expression
29:34
Example 3: Simplify Rational Expression
31:39
Example 4: Simplify Rational Expression
37:50
Adding and Subtracting Rational Expressions

55m 4s

Intro
0:00
Least Common Multiple (LCM)
0:27
Examples: LCM of Numbers
0:43
Example: LCM of Polynomials
4:02
Adding and Subtracting
7:55
Least Common Denominator (LCD)
8:07
Example: Numbers
8:17
Example: Rational Expressions
11:03
Equivalent Fractions
15:22
Simplifying Complex Fractions
21:19
Example: Previous Lessons
21:36
Example: More Complex
22:53
Example 1: Find LCM
28:30
Example 2: Add Rational Expressions
31:44
Example 3: Subtract Rational Expressions
39:18
Example 4: Simplify Rational Expression
38:26
Graphing Rational Functions

57m 13s

Intro
0:00
Rational Functions
0:18
Restriction
0:34
Example: Rational Function
0:51
Breaks in Continuity
2:52
Example: Continuous Function
3:10
Discontinuities
3:30
Example: Excluded Values
4:37
Graphs and Discontinuities
5:02
Common Binomial Factor (Hole)
5:08
Example: Common Factor
5:31
Asymptote
10:06
Example: Vertical Asymptote
11:08
Horizontal Asymptotes
20:00
Example: Horizontal Asymptote
20:25
Example 1: Holes and Vertical Asymptotes
26:12
Example 2: Graph Rational Faction
28:35
Example 3: Graph Rational Faction
39:23
Example 4: Graph Rational Faction
47:28
Direct, Joint, and Inverse Variation

20m 21s

Intro
0:00
Direct Variation
0:07
Constant of Variation
0:25
Graph of Constant Variation
1:26
Slope is Constant k
1:35
Example: Straight Lines
1:41
Joint Variation
2:48
Three Variables
2:52
Inverse Variation
3:38
Rewritten Form
3:52
Examples in Biology
4:22
Graph of Inverse Variation
4:51
Asymptotes are Axes
5:12
Example: Inverse Variation
5:40
Proportions
10:11
Direct Variation
10:25
Inverse Variation
11:32
Example 1: Type of Variation
12:42
Example 2: Direct Variation
14:13
Example 3: Joint Variation
16:24
Example 4: Graph Rational Faction
18:50
Solving Rational Equations and Inequalities

55m 14s

Intro
0:00
Rational Equations
0:15
Example: Algebraic Fraction
0:26
Least Common Denominator
0:49
Example: Simple Rational Equation
1:22
Example: Solve Rational Equation
5:40
Extraneous Solutions
9:31
Doublecheck
10:00
No Solution
10:38
Example: Extraneous
10:44
Rational Inequalities
14:01
Excluded Values
14:31
Solve Related Equation
14:49
Find Intervals
14:58
Use Test Values
15:25
Example: Rational Inequality
15:51
Example: Rational Inequality 2
17:07
Example 1: Rational Equation
28:50
Example 2: Rational Equation
33:51
Example 3: Rational Equation
38:19
Example 4: Rational Inequality
46:49
IX. Exponential and Logarithmic Relations
Exponential Functions

35m 58s

Intro
0:00
What is an Exponential Function?
0:12
Restriction on b
0:31
Base
0:46
Example: Exponents as Bases
0:56
Variables as Exponents
1:12
Example: Exponential Function
1:50
Graphing Exponential Functions
2:33
Example: Using Table
2:49
Properties
11:52
Continuous and One to One
12:00
Domain is All Real Numbers
13:14
X-Axis Asymptote
13:55
Y-Intercept
14:02
Reflection Across Y-Axis
14:31
Growth and Decay
15:06
Exponential Growth
15:10
Real Life Examples
15:41
Example: Growth
15:52
Example: Decay
16:12
Real Life Examples
16:30
Equations
17:32
Bases are Same
18:05
Examples: Variables as Exponents
18:20
Inequalities
21:29
Property
21:51
Example: Inequality
22:37
Example 1: Graph Exponential Function
24:05
Example 2: Growth or Decay
27:50
Example 3: Exponential Equation
29:31
Example 4: Exponential Inequality
32:54
Logarithms and Logarithmic Functions

45m 54s

Intro
0:00
What are Logarithms?
0:08
Restrictions
0:15
Written Form
0:26
Logarithms are Exponents
0:52
Example: Logarithms
1:49
Logarithmic Functions
5:14
Same Restrictions
5:30
Inverses
5:53
Example: Logarithmic Function
6:24
Graph of the Logarithmic Function
9:20
Example: Using Table
9:35
Properties
15:09
Continuous and One to One
15:14
Domain
15:36
Range
15:56
Y-Axis is Asymptote
16:02
X Intercept
16:12
Inverse Property
16:57
Compositions of Functions
17:10
Equations
18:30
Example: Logarithmic Equation
19:13
Inequalities
20:36
Properties
20:47
Example: Logarithmic Inequality
21:40
Equations with Logarithms on Both Sides
24:43
Property
24:51
Example: Both Sides
25:23
Inequalities with Logarithms on Both Sides
26:52
Property
27:02
Example: Both Sides
28:05
Example 1: Solve Log Equation
31:52
Example 2: Solve Log Equation
33:53
Example 3: Solve Log Equation
36:15
Example 4: Solve Log Inequality
39:19
Properties of Logarithms

28m 43s

Intro
0:00
Product Property
0:08
Example: Product
0:46
Quotient Property
2:40
Example: Quotient
2:59
Power Property
3:51
Moved Exponent
4:07
Example: Power
4:37
Equations
5:15
Example: Use Properties
5:58
Example 1: Simplify Log
11:17
Example 2: Single Log
15:54
Example 3: Solve Log Equation
18:48
Example 4: Solve Log Equation
22:13
Common Logarithms

25m 23s

Intro
0:00
What are Common Logarithms?
0:10
Real World Applications
0:16
Base Not Written
0:27
Example: Base 10
0:39
Equations
1:47
Example: Same Base
1:56
Example: Different Base
2:37
Inequalities
6:07
Multiplying/Dividing Inequality
6:21
Example: Log Inequality
6:54
Change of Base
12:45
Base 10
13:24
Example: Change of Base
14:05
Example 1: Log Equation
15:21
Example 2: Common Logs
17:13
Example 3: Log Equation
18:22
Example 4: Log Inequality
21:52
Base e and Natural Logarithms

21m 14s

Intro
0:00
Number e
0:09
Natural Base
0:21
Growth/Decay
0:33
Example: Exponential Function
0:53
Natural Logarithms
1:11
ln x
1:19
Inverse and Identity Function
1:39
Example: Inverse Composition
1:55
Equations and Inequalities
4:39
Extraneous Solutions
5:30
Examples: Natural Log Equations
5:48
Example 1: Natural Log Equation
9:08
Example 2: Natural Log Equation
10:37
Example 3: Natural Log Inequality
16:54
Example 4: Natural Log Inequality
18:16
Exponential Growth and Decay

24m 30s

Intro
0:00
Decay
0:17
Decreases by Fixed Percentage
0:23
Rate of Decay
0:56
Example: Finance
1:34
Scientific Model of Decay
3:37
Exponential Decay
3:45
Radioactive Decay
4:13
Example: Half Life
5:33
Growth
9:06
Increases by Fixed Percentage
9:18
Example: Finance
10:09
Scientific Model of Growth
11:35
Population Growth
12:04
Example: Growth
12:20
Example 1: Computer Price
14:00
Example 2: Stock Price
15:46
Example 3: Medicine Disintegration
19:10
Example 4: Population Growth
22:33
X. Conic Sections
Midpoint and Distance Formulas

32m 42s

Intro
0:00
Midpoint Formula
0:15
Example: Midpoint
0:30
Distance Formula
2:30
Example: Distance
2:52
Example 1: Midpoint and Distance
4:58
Example 2: Midpoint and Distance
8:07
Example 3: Median Length
18:51
Example 4: Perimeter and Area
23:36
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
Solving Quadratic Systems

47m 4s

Intro
0:00
Linear Quadratic Systems
0:22
Example: Linear Quadratic System
0:45
Solutions
2:49
Graphs of Possible Solutions
3:10
Quadratic Quadratic System
4:10
Example: Elimination
4:21
Solutions
11:39
Example: 0, 1, 2, 3, 4 Solutions
11:50
Systems of Quadratic Inequalities
12:48
Example: Quadratic Inequality
13:09
Example 1: Solve Quadratic System
21:42
Example 2: Solve Quadratic System
29:13
Example 3: Solve Quadratic System
35:02
Example 4: Solve Quadratic Inequality
40:29
XI. Sequences and Series
Arithmetic Sequences

21m 16s

Intro
0:00
Sequences
0:10
General Form of Sequence
0:16
Example: Finite/Infinite Sequences
0:33
Arithmetic Sequences
0:28
Common Difference
2:41
Example: Arithmetic Sequence
2:50
Formula for the nth Term
3:51
Example: nth Term
4:32
Equation for the nth Term
6:37
Example: Using Formula
6:56
Arithmetic Means
9:47
Example: Arithmetic Means
10:16
Example 1: nth Term
12:38
Example 2: Arithmetic Means
13:49
Example 3: Arithmetic Means
16:12
Example 4: nth Term
18:26
Arithmetic Series

21m 36s

Intro
0:00
What are Arithmetic Series?
0:11
Common Difference
0:28
Example: Arithmetic Sequence
0:43
Example: Arithmetic Series
1:09
Finite/Infinite Series
1:36
Sum of Arithmetic Series
2:27
Example: Sum
3:21
Sigma Notation
5:53
Index
6:14
Example: Sigma Notation
7:14
Example 1: First Term
9:00
Example 2: Three Terms
10:52
Example 3: Sum of Series
14:14
Example 4: Sum of Series
18:13
Geometric Sequences

23m 3s

Intro
0:00
Geometric Sequences
0:11
Common Difference
0:38
Common Ratio
1:08
Example: Geometric Sequence
2:38
nth Term of a Geometric Sequence
4:41
Example: nth Term
4:56
Geometric Means
6:51
Example: Geometric Mean
7:09
Example 1: 9th Term
12:04
Example 2: Geometric Means
15:18
Example 3: nth Term
18:32
Example 4: Three Terms
20:59
Geometric Series

22m 43s

Intro
0:00
What are Geometric Series?
0:11
List of Numbers
0:24
Example: Geometric Series
1:12
Sum of Geometric Series
2:16
Example: Sum of Geometric Series
2:41
Sigma Notation
4:21
Lower Index, Upper Index
4:38
Example: Sigma Notation
4:57
Another Sum Formula
6:08
Example: n Unknown
6:28
Specific Terms
7:41
Sum Formula
7:56
Example: Specific Term
8:11
Example 1: Sum of Geometric Series
10:02
Example 2: Sum of 8 Terms
14:15
Example 3: Sum of Geometric Series
18:23
Example 4: First Term
20:16
Infinite Geometric Series

18m 32s

Intro
0:00
What are Infinite Geometric Series
0:10
Example: Finite
0:29
Example: Infinite
0:51
Partial Sums
1:09
Formula
1:37
Sum of an Infinite Geometric Series
2:39
Convergent Series
2:58
Example: Sum of Convergent Series
3:28
Sigma Notation
7:31
Example: Sigma
8:17
Repeating Decimals
8:42
Example: Repeating Decimal
8:53
Example 1: Sum of Infinite Geometric Series
12:15
Example 2: Repeating Decimal
13:24
Example 3: Sum of Infinite Geometric Series
15:14
Example 4: Repeating Decimal
16:48
Recursion and Special Sequences

14m 34s

Intro
0:00
Fibonacci Sequence
0:05
Background of Fibonacci
0:23
Recursive Formula
0:37
Fibonacci Sequence
0:52
Example: Recursive Formula
2:18
Iteration
3:49
Example: Iteration
4:30
Example 1: Five Terms
7:08
Example 2: Three Terms
9:00
Example 3: Five Terms
10:38
Example 4: Three Iterates
12:41
Binomial Theorem

48m 30s

Intro
0:00
Pascal's Triangle
0:06
Expand Binomial
0:13
Pascal's Triangle
4:26
Properties
6:52
Example: Properties of Binomials
6:58
Factorials
9:11
Product
9:28
Example: Factorial
9:45
Binomial Theorem
11:08
Example: Binomial Theorem
13:48
Finding a Specific Term
18:36
Example: Specific Term
19:26
Example 1: Expand
24:39
Example 2: Fourth Term
30:26
Example 3: Five Terms
36:13
Example 4: Three Iterates
45:07
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Lecture Comments (10)

2 answers

Last reply by: DJ Sai
Mon Sep 3, 2018 12:12 PM

Post by Jerry Xu on July 31 at 02:59:15 PM

All examples say example 1...
also 3rd example (3x^2 = 27) would be easier i you just divide both sides by 3 to get x^2 = 9

0 answers

Post by Mia Moore on November 2, 2016

omg I'm ur biggest fan... like totes fangirling right now #beliebe

0 answers

Post by Mia Moore on November 2, 2016

hello everyone

1 answer

Last reply by: Dr Carleen Eaton
Thu Jan 7, 2016 6:28 PM

Post by Manuel Ramirez on November 20, 2015

5xsquare-12=11x what happen if u got the x on the other side

2 answers

Last reply by: Mia Moore
Wed Nov 2, 2016 6:21 PM

Post by Manuel Ramirez on November 20, 2015

3xsquare-9x=0... having trouble

Solving Quadratic Equations by Factoring

  • Use factoring techniques to factor a quadratic expression. For trinomials, use the guess and check technique. Factor a greatest common factor first. Look for a perfect square trinomial and the difference of two squares.
  • Use the zero product property to set each factor equal to 0 and find the solutions.

Solving Quadratic Equations by Factoring

Factor x2 − 9
  • Notice that this problem can be done using Difference of Squares
  • x2 − y2 = (x + y)(x − y)
x2 − 9 = x2 − 32 = (x + 3)(x − 3)
Factor k2 + 8k + 7
  • Remember that when our quadratic expression is in standard form such that a = 1, we are
  • always looking for two numbers n and m such that:
  • n*m = 7
  • n + m = 8
  • The only numbers that work are 1 and 7 because
  • 1*7 = 7 and 1 + 7 = 8
Final solution is then (k + 1)(k + 7)
Factor 4x3 − 144x
  • Since this problem is not a trinomial, the best way to solve it is to find the GCF.
  • The GCF is 4x. Which leaves you with:
  • 4x(x2 − 36)
  • Notice how the factored term can be factored further by differences of squares.
4x(x2 − 36) = 4x(x2 − 62) = 4x(x + 6)(x − 6)
Solve 3n4 + 12n3 − 36n2 = 0 by factoring
  • Find the GCF
  • The GCF is 3n2
  • That leaves 3n2(n2 + 4n − 12).
  • Now, given that trinomial is in standard form and a = 1, find the two numbers n and m such that
  • n*m = − 12
  • n + m = 4
  • the only two numbers that work in this case are 6 and − 2 because
  • 6* − 2 = − 12
  • 6 - 2 = 4
  • That leaves you with 3n2(n + 6)(n − 2) = 0
  • Using the Zero Product Property, that leaves you with three equations.
  • 3n2 = 0, n + 6 = 0, n − 2 = 0
  • Solve
n = 0, n = − 6, n = 2
Solve 2r3 − 4r2 = 0 by factoring
  • Find the GCF
  • The GCF is 2r2
  • That leaves you with 2r2(r − 2) = 0
  • Use the zero product property to solve for r.
  • 2r2 = 0, r − 2 = 0
  • Solve
r = 0, r = 2
Solve 9x3 + 3x2 − 30x = 0 by factoring.
  • Find the GCF
  • The GCFis 3x
  • 3x(3x2 + x − 10) = 0
  • Notice that the resulting trinomial, although in standard form, a is no longer equal to 1.
  • That requires an additional step we'll call Dividing by a, a in this case equals to 3.
  • Look for a number n and m such that
  • n*m = a*c = 3* − 10 = − 30
  • n + m = b = 1
  • The only numbers that work are − 5 and 6 because
  • − 5*6 = − 30
  • − 5 + 6 = 1 which is exactly what we need.
  • Now that we've found n and m, we have
  • 3x(x − 5)(x + 6) = 0 but because a is not 1, we must divide the numbers we found by a
  • 3x(x − 5)(x + 6) = 3x(x − [5/3])(x + [6/3])
  • Simplify
  • 3x(x − [5/3])(x + [6/3]) = 3x(3x − 5)(x + 2) = 0. Notice how the 3 in the denominator went infront of the x since [5/3] cannot be reduced.
  • Solve using Zero Product Property.
  • 3x = 0, 3x − 5 = 0 x + 2 = 0
  • Solve
x = 0, x = [5/3], x = − 2
Solve 5x3 − 18x2 + 9x = 0 by factoring.
  • Find the GCF
  • The GCFis x
  • x(5x2 − 18x + 9) = 0
  • Notice that the resulting trinomial, although in standard form, a is no longer equal to 1.
  • That requires an additional step we'll call Dividing by a, a in this case equals to 5.
  • Look for a number n and m such that
  • n*m = a*c = 5*9 = 45
  • n + m = b = − 18
  • The only numbers that work are − 3 and − 15 because
  • − 3* − 15 = 45
  • − 3 + ( − 15) = − 18 which is exactly what we need.
  • Now that we've found n and m, we have
  • x(x − 3)(x − 15) = 0 but because a is not 1, we must divide the numbers we found by a
  • x(x − 3)(x − 15) = x(x − [3/5])(x − [15/5]) = 0
  • Simplify
  • x(x − [3/5])(x − [15/5]) = x(5x − 3)(x − 3) = 0 Notice how the 5 in the denominator went infront of the x since [3/5] cannot be reduced.
  • Solve using Zero Product Property.
  • x = 0, 5x − 3 = 0, x − 3 = 0
  • Solve
x = 0, x = [3/5], x = 3
Solve 3x3 − 11x2 − 20x = 0 by factoring.
  • Find the GCF
  • The GCFis x
  • x(3x2 − 11x − 20) = 0
  • Notice that the resulting trinomial, although in standard form, a is no longer equal to 1.
  • That requires an additional step we'll call Dividing by a, a in this case equals to 3.
  • Look for a number n and m such that
  • n*m = a*c = 3* − 20 = − 60
  • n + m = b = − 11
  • The only numbers that work are 4 and − 15 because
  • 43* − 15 = − 60
  • 4 + ( − 15) = − 11 which is exactly what we need.
  • Now that we've found n and m, we have
  • x(x + 4)(x − 15) = 0 but because a is not 1, we must divide the numbers we found by a
  • x(x + 4)(x − 15) = x(x + [4/3])(x − [15/3]) = 0
  • Simplify
  • x(x + [4/3])(x − [15/3]) = x(3x + 4)(x − 5) = 0 Notice how the 3 in the denominator went infront of the x since [4/3] cannot be reduced.
  • Solve using Zero Product Property.
  • x = 0, 3x + 4 = 0, and x − 5 = 0
  • Solve
x = 0, x = − [4/3], and x = 5
Solve 4x2 + 10x + 6 = 0 by factoring.
  • Find the GCF
  • The GCFis 2
  • 2(2x2 + 5x + 3) = 0
  • Notice that the resulting trinomial, although in standard form, a is no longer equal to 1.
  • That requires an additional step we'll call Dividing by a, a in this case equals to 2.
  • Look for a number n and m such that
  • n*m = a*c = 2*3 = 6
  • n + m = b = 5
  • The only numbers that work are 3 and 2 because
  • 3*2 = 6
  • 3 + 2 = 5 which is exactly what we need.
  • Now that we've found n and m, we have
  • 2(x + 3)(x + 2) = 0 but because a is not 1, we must divide the numbers we found by a
  • 2(x + 3)(x + 2) = 2(x + [3/2])(x + [2/2]) = 0
  • Simplify
  • 2(x + [3/2])(x + [2/2]) = 2(2x + 3)(x + 1) = 0 Notice how the 2 in the denominator went infront of the x since [3/2] cannot be reduced.
  • Solve using Zero Product Property.
  • 2 = 0, 2x + 3 = 0, and x + 1 = 0
  • Solve
x = − [3/2] and x = − 1
Solve 9x2 − 24x + 12 = 0 by factoring.
  • Find the GCF
  • The GCFis 3
  • 3(3x2 − 8x + 4) = 0
  • Notice that the resulting trinomial, although in standard form, a is no longer equal to 1.
  • That requires an additional step we'll call Dividing by a, a in this case equals to 3.
  • Look for a number n and m such that
  • n*m = a*c = 3*4 = 12
  • n + m = b = − 8
  • The only numbers that work are − 2 and − 6 because
  • − 2* − 6 = 12
  • − 2 + ( − 6) = − 8 which is exactly what we need.
  • Now that we've found n and m, we have
  • 3(x − 2)(x − 6) = 0 but because a is not 1, we must divide the numbers we found by a
  • 3(x − 2)(x − 6) = 3(x − [2/3])(x − [6/3]) = 0
  • Simplify
  • 3(x − [2/3])(x − [6/3]) = 3(3x − 2)(x − 2) = 0 Notice how the 3 in the denominator went infront of the x since [2/3] cannot be reduced.
  • Solve using Zero Product Property.
  • 3 = 0, 3x − 2 = 0, and x − 2 = 0
  • Solve
x = [2/3] and x = 2

*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

Solving Quadratic Equations by Factoring

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
  • Factoring Techniques 0:15
    • Greatest Common Factor (GCF)
    • Difference of Two Squares
    • Perfect Square Trinomials
    • General Trinomials
  • Zero Product Rule 5:22
    • Example: Zero Product
  • Example 1: Solve by Factoring 7:46
  • Example 1: Solve by Factoring 9:48
  • Example 1: Solve by Factoring 12:34
  • Example 1: Solve by Factoring 15:28

Transcription: Solving Quadratic Equations by Factoring

Welcome to Educator.com.0000

In a previous lesson, we talked about solving quadratic equations by graphing.0002

And we are going to go on to discuss another technique, an algebraic technique, which is solving quadratic equations by factoring.0007

So, first I am starting out with just a brief review of the different factoring techniques.0015

And each of these is covered in-depth in the Algebra I series.0022

So, if you are unsure on any of these, make sure you review that before you go on,0025

because we are just using these techniques as a tool, now, to actually solve quadratic equations.0029

OK, first, remember the greatest common factor: the greatest common factor of two or more numbers is the product of their common prime factors.0036

So, the GCF of two or more numbers is the product of their common prime factors.0048

And, in factoring, often, when you are factoring an equation, if there is a GCF, you want to factor that out first,0067

because then you are working with something less complicated.0074

For example, if you had 4x2 + 16x + 36, you will recognize that there is a GCF of 4.0077

So, I am going to factor that out and get 4 times x squared, plus 4x, plus 9.0087

Factoring out the 4 gives me x2 + 4x + 9; and this is much easier to work with.0097

So, remember to factor out the GCF first, if there is one.0103

Recall the difference of two squares: you will recognize this, because it is going to be in the form a2 - b2.0109

So, an example would be x2 - 4; and that is equal to (x + 2) (x - 2).0121

So, this is the difference of two squares; here is the plus and the minus.0134

And so, when you are factoring, if you recognize this, you can quickly factor it as the difference of two squares,0139

knowing that it fits into this formula, a2 - b2.0144

Perfect square trinomials: perfect square trinomials would be the result of squaring a binomial.0150

For example, if you have x2 + 6x + 9, this is actually equal to (x + 3)2.0164

So, if you take a binomial and square it, you will get a perfect square trinomial.0175

Take a binomial; multiply it times itself; that is what a perfect square trinomial is.0180

And so, recognizing that makes for easy factoring.0185

General trinomials are a little more complex to factor; again, you can review all of this in Algebra I--techniques for factoring.0189

And an example of a general trinomial would be x2 + x - 12;0198

it doesn't fit into any of these special cases, like the difference of two squares, or perfect square trinomials.0203

So, I have to do a little more work in factoring that out.0210

Recall from earlier lessons that you need to look at the signs.0213

I have a positive sign here; I have a negative here.0216

And if I have a negative, that must be the result of multiplying a positive and a negative.0218

I then look at what the factors of 12 are; and the factors of 12 are 1 and 12, 2 and 6, and 3 and 4.0226

And one will be positive, and one will be negative; and I need to have factors that sum to this middle term, which is actually 1.0239

And I can try different combinations; and I know that -1 and 12 is not going to work; -12 and 1 certainly will not work.0249

And I go on down, and from that, you can see that, if I want them to sum to 1, I am going to need factors that are close to each other.0259

And the closest two I have here are 3 and 4.0268

So, if I make 4 positive and 3 negative, then when I add the outer term and the inner terms, I will get my middle term x.0271

And you can always check this using the FOIL method: First terms (that is x2),0286

Outer terms (-3x), Inner terms (4x), and then the Last is -12.0292

Simplifying this, I get my original back.0303

General trinomials take a little more work, and you can always check those by multiplying it back out to make sure you did it correctly.0306

So, we are making sure you have all these and know how to use them well.0313

And then, we are going to be using them today to actually solve some quadratic equations.0317

Now, once you have factored, you need to use the zero product rule to actually find the solutions.0322

And what the zero product rule says is that, for any number a and b, if ab is zero, then either a is zero or b is zero--0331

because if a equals 0, then you would get 0 times b; that would work as a solution;0339

if b is 0, then 0 times a would give you 0, and that is also a solution.0346

For example, if I was given x2 - 16 = 0 and asked to solve that, I would first recognize that it is0353

in the form a2 - b2, and that it is therefore the difference of two squares.0361

That allows me to factor it pretty quickly into (x + 4) (x - 4).0372

So, I am factoring this, and it still equals 0.0380

Now, use the zero product rule: the zero product rule tells me that, if this is 0 or this is 0, then this entire thing will equal 0, which is what I want.0383

So, if x + 4 equals 0, this will be solved; if x - 4 equals 0, this will be solved.0397

So, I am going to set this factor equal to 0 and solve for x; I am going to set this factor equal to 0 and solve for x.0407

And that is going to give me...let's see...x = 4.0418

And if you wanted to check that, you could go back up here and say, "OK, let x equal -4."0428

So, that is -42 - 16 = 0; that is 16 - 16 = 0, and that checks out; that is a valid solution.0434

I could do the same thing for 4; x = -4, or I could say x = 4; and that is going to give me 42 - 16 = 0.0445

16 - 16 does equal 0, so there are two solutions here.0454

And I was able to find those using factoring and the zero product rule.0458

So, trying this out: first I am just asked to factor; and recall that the first thing you want to do is factor out any greatest common factor,0467

because that is going to make whatever is left much easier to work with.0478

And I see that I have a greatest common factor of 4.0482

This factors into 4 times x2 - x - 6.0490

Now, all I have here is a general trinomial, so I want to think about what I have.0496

And I want to make sure that I bring my 4 along with me, because that is part of the solution.0502

I have a negative sign here; and the only way you are going to end up with that is if one of these is positive and one is negative.0510

Now, I am going to think about what my factors of 6 are; I have 1 and 6, and 2 and 3.0521

I need factors that sum to a middle term of -1; and that is not a very large number, so I am going to look for factors that are close together.0529

I am going to try these first: now one is negative, and the other is positive.0536

Since this is negative, I am going to look for making the larger number negative; so let me try if I have -3 plus 2.0542

That equals -1, and that gives me the coefficient of that middle term; so this is what I have.0549

And I can always check that by using FOIL to go back and multiply these out;0566

and then I would have to multiply the 4 back into it.0570

But this is factored; so I first factored out the GCF, and then I saw that I can take it farther, because this is not factored out all the way.0574

It is a general trinomial that factors into (x + 2) (x - 3), times that greatest common factor, 4.0581

OK, now we are asked to actually solve; and this is 3x2 = 27.0589

Before you can solve a quadratic equation, you need to put it in standard form.0597

Recall that standard form is ax2 + bx + c = 0.0605

So, I have 3x2 = 27; to put this in standard form, I am going to subtract 27 from both sides.0613

And I have an ax2 term; b must be 0, because there is no x term; and then I have a c of -27.0625

So, I am going to solve by factoring; I have this in standard form--now I am going to factor out the GCF.0634

And the GCF is 3; I am going to pull that out.0643

Now, I am looking at this, and I see that what I have is something in the form a2 - b2, which is the difference of two squares.0653

Always make sure you bring this GCF down--don't leave it behind.0665

(x - 3) (x + 3) = 0; so now, I am going to use the zero product property (or zero product rule) to find my solutions.0669

And the zero product rule says that if a times b equals 0, then a equals 0 or b equals 0, or they both have to be 0.0687

So, first I am going to have 3 times (x - 3) equals 0; and if I divide 0/3, I am just going to get x - 3 = 0, or x = 3.0699

So, that is one solution; the other solution is going to be x + 3 = 0.0717

Using the zero product rule, that tells me that x equals -3; so my two solutions are that x equals 3 and x equals -3.0722

These are my two solutions for this quadratic equation.0733

And I found that just by factoring: x = 3 and x = -3; and I made my factoring a lot easier by first pulling out the GCF.0743

Again, solve by factoring: and we have another situation where it is not in standard form.0756

So, I am going to put it in standard form, which is ax2 + bx + c = 0.0760

All right, that is going to give me 4x2 - 24x + 36 = 0, just subtracting 24x from both sides.0772

This is another situation where I have a greatest common factor; so I have a GCF equal to 4--factor that out.0784

Pull that out: that is 4, times x2 - 6x + 9, equals 0.0792

Again, I have something much easier to work with since I have pulled that out.0800

So, figuring out how to work with this, I am going to go ahead and factor this out, because it is not all the way factored.0805

And here, I have (x - 3) times (x -3); and it is actually a perfect square trinomial; this is really just (x - 3)2.0817

OK, so this is a perfect square trinomial, because you have x2, and then (just check this using FOIL)0838

I would have x2 - 3x - 3x + 9; so I know I did that correctly.0850

Now, I am going to use the zero product rule, which tells me that, if one of these is equal to 0, then the entire thing is equal to 0.0862

So, I can use that to find the solution.0879

Now, you can just go ahead and divide both sides by 4, in which case this will move over to here, and that just stays 0.0881

Really, I just need to work with this and this: x - 3 = 0, and this is the same thing: x - 3 = 0.0889

So, actually, when I figure this out, I just get the same thing for both; and it is x = 3.0899

So, I only have one solution, or one real root, in this case; so, x equals 3.0906

And again, put it in standard form; factor out the greatest common factor; complete your factoring;0912

and then use the zero product rule to determine that there is one real solution, and that is that x equals 3.0919

OK, again, solve by factoring; this is already in standard form; however, I have to get rid of this greatest common factor, which is 2.0929

So, I am pulling that out to get 2x2 + 5x - 12.0938

When the leading coefficient is not 1, the factoring is a bit more difficult.0945

So, let's move over here and work on this.0950

When the leading coefficient is 2, I am going to have something in this form.0953

And I have a negative here, so I also know that one of these is going to be a positive, and one is going to be a negative.0959

But I don't actually know which one yet; but I know I am going to have +/-, or I am going to have this.0965

OK, let's look at some factors of 12: factors of 12 would be 1 and 12, 2 and 6, and 3 and 4.0974

Now, 5 is not that large of a number, especially when we think about the fact that we are going to have to be also multiplying by 2.0985

So, if I go and use something like 6, and it ends up getting multiplied by 2, it is going to be very large; the difference is going to be great.0992

I want factors that are smaller, since the difference between those, even with this 2x thrown in, is only going to be 5x.0999

So, I am going to start with these, because they don't have a large difference between them.1008

So, I am going to start out just trying (2x + 3) (x - 4) and seeing what I get.1012

I am not worried about the first term; I am worried about the outer terms added to the inner terms.1022

And see if I get the correct middle term.1029

I am looking for the middle term equal to 5x.1031

This is going to give me 2x times -4; that is -8x, plus 3x; that is 5x.1036

I have the right idea, but I actually have the wrong signs here.1049

So, I am going to try reversing the signs, because (this is -5x) I want this to be 5x, because here I have -8x + 3x is -5x.1053

So, the same idea: let's try different signs, though.1063

This time, I am making this negative and this positive; this is going to give me 2x(4); that is going to give me +8x; -3x is 5x.1070

So, this is correct; I got the correct middle term, so this is the correct factoring.1083

And this can be a lot of work to factor these; so it is important to go logically--1088

for example, seeing that I don't have a very large term here, especially when I am dealing with the 2x also1093

(it is going to amplify things), to look for factors that are not very far apart.1099

OK, so now, I am back here, and I am solving by factoring.1103

This is going to give me (2x - 3) (x + 4) = 0.1108

Dividing both sides by 2, this 2 is just going to drop out.1117

So, when I use the zero product property, I am going to get 2x - 3 = 0, and I am also going to get x + 4 = 0.1123

So, I just need to go ahead and solve those to get 2x = 3, or x = 3/2.1133

Here, I just have x + 4 = 0, and that is simple: it is x = -4.1151

I have two solutions: x = 3/2 and x = -4.1159

I solved this by pulling out the greatest common factor, factoring that out, then factoring this trinomial into this,1170

and using the zero product rule to give me 3/2 for a solution from here, and x = -4 for a solution from here.1177

Thanks for visiting Educator.com, and I will see you next lesson!1189

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