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INSTRUCTORS Carleen Eaton Grant Fraser
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Dr. Carleen Eaton

Dr. Carleen Eaton

Completing the Square

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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 (9)

1 answer

Last reply by: Dr Carleen Eaton
Wed Nov 6, 2013 12:48 AM

Post by Chateau Siqueira on September 27, 2013

Thank you for your lectures Dr. Eaton. My college algebra class is vary fast paced which sometimes I do not absorb all the material but then I come over here an it all makes sense! I appreciate your time.

0 answers

Post by julius mogyorossy on September 1, 2013

It seems to me that one of your solutions may not be a negative #, that is what I thought at first, that your solutions may not even be the same number, one positive, one negative, a simple example, (x-2)'2=4, it is easy hear to see what the solutions are, 4 and 0. If you worked out this problem it would be, x=2+-^4, the square root of 4 is 2 so that would be 2+2=4, 2-2=0, x=4, and x=0. But don't take my word on it, ask Dr. Carleen. (x-2)'2=4 is a question, asking you what two values for x in that equation =4.

0 answers

Post by julius mogyorossy on September 1, 2013

I meant to say (x-4)'2=5 reminds me of absolute value equations. I really understand what this is saying now, x=4+-^5 is just another way to say the same thing.

0 answers

Post by julius mogyorossy on September 1, 2013

The (x-4)'2=5 reminds me of inequalities.

1 answer

Last reply by: Dr Carleen Eaton
Tue Jun 4, 2013 8:10 PM

Post by Kavita Agrawal on June 3, 2013

At about 6 min., you said that -8^2 = 64. -8^2, however, equals -64 (because of the order of operations, and exponents come before multiplying.) I think that part would make more sense if it had parentheses around it.

0 answers

Post by julius mogyorossy on March 25, 2012

It seems that if you can divide, b, in to two equal parts, getting whole numbers, you then just multiply those two equal parts times each other to get the constant in the perfect square trinomial, the first operator can be positive or negative, the second one in the perfect square trinomial will always be positive, is this correct.

0 answers

Post by Ken Mullin on January 27, 2012

Nice review of completing thr square...
Espcially like the emphasis on ISOLATING (b^2)/4 and adding result to both sides.
Some textbooks use the reciprocal of 2 and so makes the operation seem more difficult than it otherwise is...

Completing the Square

  • You can solve quadratic equations by taking the square root of both sides of the equation. To do this, the quadratic expression must be a perfect square.
  • Complete the square to make the quadratic expression into a perfect square. Take one half of the coefficient of the linear term, square it, and add it to both sides of the equation.
  • If the coefficient of the quadratic term is not 1, you must first factor this coefficient out of the quadratic and linear term and then complete the square for the new quadratic expression which now has a coefficient of 1.

Completing the Square

Use the square root property to solve x2 − 10x + 25 = 11
  • Notice how the left side is a perfect square trinomial, therefore, no factoring is needed.
  • x2 − 10x + 25 = 11 = (x − 5)2 = 11
  • Take the square root of both sides
  • √{(x − 5)2} = ±√{11}
  • Simplify
  • (x − 5) = ±√{11}
x = 5 ±√{11}
Use the square root property to solve x2 + 4x + 4 = 3
  • Notice how the left side is a perfect square trinomial, therefore, no factoring is needed.
  • x2 + 4x + 4 = 3 = (x + 2)2 = 3
  • Take the square root of both sides
  • √{(x + 2)2} = ±√3
  • Simplify
  • (x + 2) = ±√3
x = − 2 ±√3
Complete the square x2 + 12x
  • You are looking for a number c such that x2 + 12x + c is a perfect square trinomial.
  • Recall that whenever you have a perfect square trinomial in standard form x2 + bx + c,c = [(b2)/4].
  • You can then use this information to factor completely x2 + bx + ( [b/2] )2 = x2 + bx + [(b2)/4] = ( x + [b/2] )2
  • Find c given b = 12
  • c = [(b2)/4] = [(122)/4] = [144/4] = 36
  • Factor Completely using the special rule
  • x2 + 12x + 36 = ( x + [b/2] )2 = ( x + [12/2] )2 = ( x + 6 )2
( x + 6 )2
Complete the square x2 + 8x
  • You are looking for a number c such that x2 + 12x + c is a perfect square trinomial.
  • Recall that whenever you have a perfect square trinomial in standard form x2 + bx + c,c = [(b2)/4].
  • You can then use this information to factor completely x2 + bx + ( [b/2] )2 = x2 + bx + [(b2)/4] = ( x + [b/2] )2
  • Find c given b = 8
  • c = [(b2)/4] = [(82)/4] = [64/4] = 16
  • Factor completely using the special rule
  • x2 + 8x + 16 = ( x + [b/2] )2 = ( x + [8/2] )2 = ( x + 4 )2
( x + 4 )2
Complete the square x2 + 3x
  • You are looking for a number c such that x2 + 12x + c is a perfect square trinomial.
  • Recall that whenever you have a perfect square trinomial in standard form x2 + bx + c,c = [(b2)/4].
  • You can then use this information to factor completely x2 + bx + ( [b/2] )2 = x2 + bx + [(b2)/4] = ( x + [b/2] )2
  • Find c given b = 3
  • c = [(b2)/4] = [(32)/4] = [9/4] = [9/4]
  • Factor completely using the special rule
  • x2 + 3x + [9/4] = ( x + [b/2] )2 = ( x + [3/2] )2
( x + [3/2] )2
Solve by completing the square x2 − 2x + 28 = 0
  • Isolate the variables on the left side of the equation. Subtract 28 from both sides.
  • x2 − 2x = − 28
  • Complete the square by adding [(b2)/4] to both sides of the equation.
  • x2 − 2x + [(b2)/4] = − 28 + [(b2)/4]
  • x2 − 2x + [(( − 2)2)/4] = − 28 + [(( − 2)2)/4]
  • x2 − 2x + [4/4] = − 28 + [4/4]
  • x2 − 2x + 1 = − 27
  • Factor the perfect square trinomial(left side) using the formula x2 + bx + [(b2)/4] = ( x + [b/2] )2
  • ( x + [b/2] )2 = ( x + [( − 2)/2] )2 = ( x − 1 )2
  • Solve using the Square Root Property
  • ( x − 1 )2 = − 27
  • √{( x − 1 )2} = ±√{ − 27}
  • (x − 1) = ±i√{27}
  • Simplify and reduce square roots
  • x = 1 ±i√{27} = 1 ±i√{9*3} = 1 ±i√9 √3 = 1 ±3i√3
x = 1 + 3i√3 and x = 1 − 3i√3
Solve by completing the square x2 − 18x + 82 = 0
  • Isolate the variables on the left side of the equation. Subtract 82 from both sides.
  • x2 − 18x = − 82
  • Complete the square by adding [(b2)/4] to both sides of the equation.
  • x2 − 18x + [(b2)/4] = − 82 + [(b2)/4]
  • x2 − 18x + [(( − 18)2)/4] = − 82 + [(( − 18)2)/4]
  • x2 − 18x + [324/4] = − 82 + [324/4]
  • x2 − 18x + 81 = − 82 + 81
  • x2 − 18x + 81 = − 1
  • Factor the perfect square trinomial(left side) using the formula x2 + bx + [(b2)/4] = ( x + [b/2] )2
  • ( x + [b/2] )2 = ( x + [( − 18)/2] )2 = ( x − 9 )2
  • Solve using the Square Root Property
  • ( x − 9 )2 = − 1
  • √{( x − 9 )2} = ±√{ − 1}
  • (x − 9) = ±i
  • Simplify
  • x = 9 ±i
x = 9 + i and x = 9 − i
Solve by completing the square x2 + 12x + 1 = − 10
  • Isolate the variables on the left side of the equation. Subtract 1 from both sides.
  • x2 + 12x = − 11
  • Complete the square by adding [(b2)/4] to both sides of the equation.
  • x2 + 12x + [(b2)/4] = − 11 + [(b2)/4]
  • x2 + 12x + [((12)2)/4] = − 11 + [((12)2)/4]
  • x2 + 12x + [144/4] = − 11 + [144/4]
  • x2 + 12x + 36 = − 11 + 36
  • x2 + 12x + 81 = 25
  • Factor the perfect square trinomial(left side) using the formula x2 + bx + [(b2)/4] = ( x + [b/2] )2
  • ( x + [b/2] )2 = ( x + [12/2] )2 = ( x + 6 )2
  • Solve using the Square Root Property
  • ( x + 6 )2 = 25
  • √{( x + 6 )2} = ±√{25}
  • (x + 6) = ±5
  • Simplify
  • x = − 6 ±5
x = − 1 and x = − 11
Solve by completing the square x2 + 14x + 38 = − 10
  • Isolate the variables on the left side of the equation. Subtract 38 from both sides.
  • x2 + 14x = − 48
  • Complete the square by adding [(b2)/4] to both sides of the equation.
  • x2 + 14x + [(b2)/4] = − 48 + [(b2)/4]
  • x2 + 14x + [((14)2)/4] = − 48 + [((14)2)/4]
  • x2 + 14x + [196/4] = − 48 + [196/4]
  • x2 + 14x + 49 = − 48 + 49
  • x2 + 14x + 49 = 1
  • Factor the perfect square trinomial(left side) using the formula x2 + bx + [(b2)/4] = ( x + [b/2] )2
  • ( x + [b/2] )2 = ( x + [14/2] )2 = ( x + 7 )2
  • Solve using the Square Root Property
  • ( x + 7 )2 = 1
  • √{( x + 7 )2} = ±√1
  • (x + 7) = ±1
  • Simplify
  • x = − 7 ±1
x = − 6 and x = − 8
Solve by completing the square 2x2 + 4x − 8 = − 2
  • Isolate the variables on the left side of the equation. Add 8 to both sides.
  • 2x2 + 4x = 6
  • In order to complete the square, the coefficient of x must equal 1.
  • Divide everything by 2
  • [(2x2)/2] + [4x/2] = [6/2] = x2 + 2x = 3
  • Complete the square by adding [(b2)/4] to both sides of the equation.
  • x2 + 2x + [(b2)/4] = 3 + [(b2)/4]
  • x2 + 2x + [((2)2)/4] = 3 + [((2)2)/4]
  • x2 + 2x + [4/4] = 3 + [4/4]
  • x2 + 2x + 1 = 3 + 1
  • x2 + 2x + 1 = 4
  • Factor the perfect square trinomial(left side) using the formula x2 + bx + [(b2)/4] = ( x + [b/2] )2
  • ( x + [b/2] )2 = ( x + [2/2] )2 = ( x + 1 )2
  • Solve using the Square Root Property
  • ( x + 1 )2 = 4
  • √{( x + 1 )2} = ±√4
  • (x + 1) = ±2
  • Simplify
  • x = − 1 ±2
x = 1 and x = − 3

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

Answer

Completing the Square

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
  • Square Root Property 0:12
    • Example: Perfect Square
    • Example: Perfect Square Trinomial
  • Completing the Square 4:39
    • Constant Term
    • Example: Complete the Square
  • Solve Equations 6:42
    • Add to Both Sides
    • Example: Complete the Square
  • Equations Where a Not Equal to 1 10:58
    • Divide by Coefficient
    • Example: Complete the Square
  • Complex Solutions 14:05
    • Real and Imaginary
    • Example: Complex Solution
  • 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

Transcription: Completing the Square

Welcome to Educator.com.0000

In previous lessons, we talked about solving quadratic equations through factoring and through graphing.0002

Today, we are going to learn another method called completing the square.0008

And you may have learned this earlier on in Algebra I, so it may be review.0012

And if you need even more depth, you can look at the Algebra I series here at Educator.com.0017

Now, recall that some quadratic equations can be solved by taking the square root of both sides of the equation.0023

And this includes equations where the quadratic expression is a perfect square.0031

So, for example, if you have x2 - 8x + 16 = 5, you may recognize that this is a perfect square.0037

This trinomial is a perfect square trinomial.0050

It is actually equal to (x - 4)2, or (x - 4) (x - 4).0056

So, if I put this in this form and write it as (x - 4)2 = 5, I can then solve it by taking the square root of both sides.0063

So, take the square root of both sides of the equation to get x - 4 = ±√-5.0072

And recall that you need to take both the positive and negative square root of 5,0094

because, thinking back, if you had something such as x2 = 4,0101

and you were to take the square root of both sides, that would give you that x could equal 2,0108

or x could equal -2, because 22 equals 4, and (-2)2 equals 4.0114

So, it is the same concept here; it looks more complex, but it is the same concept.0122

On the left, I am taking the square root, and then on the right, I am taking the square root of a number;0126

and it could be both positive square root of 5, or the negative whatever-the-square-root-of-5-turns-out-to-be.0131

And I can't simplify that any further without a calculator; I can just leave it in this form, ±√5.0138

Since this is an irrational number, I can't get an exact value; I just need to leave it in this format.0146

So, this is going to give me x = 4 ± √5.0155

And I could put this in this form; I could leave it; or I could say x = 4 + √5, and x = 4 - √5.0159

These are both solutions to this quadratic equation.0172

Now, how do you recognize perfect squares?0176

If you have a perfect square trinomial, and it is in the form ax2 + bx + c,0181

then what you can do is realize that the constant equals b divided by 2, squared; or sometimes we just say it is b2 over 4.0196

So, if you are looking, and you are not sure if it is a perfect square trinomial, look at the constant term.0211

And here, I have a = 1 for a coefficient; b = -8; and c = 16.0216

So, what I want to do is see if b2/4 equals 16.0224

And if I look, I see this (-8)2/4, and that equals 64 divided by 4, which is 16.0231

So, this checks out; so if you are not sure if you have a perfect square trinomial, you can just look at the constant.0247

If the rest of it checks out as perfect squares, and then you are not sure,0252

then you can check and say, "OK, does the constant equal b2/4? Yes, it does."0258

So, I knew I had a perfect square trinomial; and that makes it easy to find the square root of both sides and find the two solutions for this quadratic equation.0266

OK, sometimes you don't have a perfect square trinomial; in that case, what you can do is actually complete the square.0277

You can make it into a perfect square.0286

Now, what I just said is that the constant term in a perfect square trinomial equals b squared over 2, or I like to just use it as b2/4.0288

But it is the same thing either way.0299

So, for example, let's say I am given x2 - 8x, similar to the last one (but that one was already a perfect square).0303

So, I am just given this portion; and I want to complete the square.0313

So, in order to complete the square, what I need is a constant term.0320

And I know that, in a perfect square trinomial, the constant is going to equal b2/4.0327

So, really what I am looking for is this.0333

Here, a equals 1; b equals -8; therefore, I have x2 - 8x + -82/4.0339

x2 - 8x + 8 times 8...that is 64...over 4 gives me x2 - 8x + 16.0356

And now, I have a perfect square trinomial.0368

So, last time I talked about if you had something and you thought it was a perfect square trinomial;0373

you could check it out by seeing if the constant is equal to this.0379

Conversely, if you don't have a perfect square, and you need to complete the square--0385

you just have this part, and you want to get the constant term to complete the square--0388

you can use this knowledge to find what that constant should be; and then, from there, we can solve equations.0393

Now, to solve any quadratic equation, you can complete the square by adding this, which is equal to b2/4, to both sides of the equation.0402

In the last slide, I talked about completing the square; but if you have an equation, you need to keep the equation balanced.0416

So, you can't just complete the square, and then do nothing to the other side.0421

So, let's take an example here, slightly different than the other one, but related.0425

This time, I have x2 + 6x + 8 = 0; so now I have an equation.0431

Now, the first thing to do is: if you are dealing with an equation, get the constants on one side.0440

I am going to get the constants on the right, because I want to clear out0450

and just have these x variable terms here on the left, so I can complete the square on that.0454

I am going to subtract 8 from both sides to get x2 + 6x = -8.0463

Now, I want to complete the square, and I know that, to complete the square, I am going to need x2 + 6x;0476

but I need a constant term, and the constant term is going to be b2/4.0483

Now, I have to add to both sides; add b2/4 to both sides.0491

This is the step that sometimes gets forgotten, and then you get an incorrect answer.0501

OK, this is going to give me x2 + 6x +...well, b is 6; now, I have to add that over here, as well; OK.0506

This is going to give me 36 divided by 4, which comes out to 9.0533

So, coming up here, x2 + 6x + 9 = -8 + 9; or x2 + 6x + 9 = 1.0541

Now, I have a perfect square trinomial, and I can go and use techniques used previously in order to solve this--0560

such as taking the square root of both sides.0569

(x + 3)2 (which is the same as this trinomial) equals 1.0575

Now, we take the square root of both sides to solve this to get x + 3 = ±√1.0581

So, the square root of 1 is 1, so I get x + 3 = ±1.0590

From there, you can just solve; I get x + 3 = 1, so x = -3 + 1; x = -2;0598

and I also have x + 3 = -1; therefore, I am going to get x = -4.0609

OK, so I am really focusing just on these first steps; but to finish it out, what you ended up with is x = -2 and x = -4 as the solutions.0619

To solve a quadratic equation when you don't already have a perfect square trinomial involved,0629

get the constants on the right and the variables on the left.0634

Then, complete the square by adding b2/4 to both sides.0637

Once you have done that, you have a perfect square here on the left.0642

Turn this into this form, and then take the square root of both sides to get your solutions.0646

Sometimes, the coefficient of x2--the leading coefficient--is not 1.0658

So far, we have been working with situations where we just had x2.0664

If the coefficient is not 1, you need to take an extra step,0668

and you need to divide both sides of the equation by the coefficient0671

in front of the x2 term in order to make that coefficient 1.0675

Then, you just go on and complete the square as usual.0680

For example, if I was given 2x2 - 12x + 4 = 0, I need to make this coefficient 1.0684

So, I can do that by dividing both sides by 2.0694

And that is going to cancel out right here to give me x2 - 6x + 2 = 0.0704

OK, now I am going to go ahead and complete the square.0714

Remember that my first step is to get the x variables on the left, and I am going to subtract 2 from both sides to get the constants on the right.0719

Then, I need to complete the square: x2 - 6x...I need to now add b2/4 to both sides to complete the square.0730

This is x2 - 6x + b2/4 equals -2 + b2/4.0743

OK, x2 - 6x...well, b is -6, so that is + (-6)2/4...= -2 + (-6)2/4.0755

This gives me x2 - 6x; this is 36 divided by 4; and I am just going to simplify that to 9.0770

And if I had not added to both sides, the equation would not be balanced; I would not get the correct solutions.0781

OK, I am coming up here to get x2 - 6x + 9 = 7.0786

Well, now I have a perfect square trinomial, which is actually (x - 3) (this is a negative right here) squared equals 7.0794

Remember to take the square root of both sides to get x - 3 = ±√7.0802

Isolate the x to get x = 3 ± √7.0810

So, the two solutions are x = 3 + √7 and x = 3 - √7.0815

OK, so it is the same as what we just did with completing the square, and then taking the square root of both sides;0823

but with an extra step, because in order to complete the square, you need for the coefficient x to be 1.0828

If it is not 1, if you have a leading coefficient that is not 1, begin by dividing both sides by that coefficient.0833

OK, sometimes the solutions to a quadratic equation may be complex numbers.0844

Just to review: complex numbers consist of a real part and an imaginary part, something such as 3 + 2i.0851

So, this is in the form a + bi, where a is the real part, and 2i is the imaginary part.0857

It turns out that, for quadratic equations, solutions may end up in this form.0867

Let's look at a situation where that could occur.0872

Let's say I am given x2 + 2x + 10 = 0.0875

And I am going to solve this by completing the square.0882

So first, I am going to isolate these x variables on the left and the constants on the right by subtracting 10 from both sides.0885

Now, I need to complete the square.0892

I want to add b2/4 to both sides of this equation.0899

Well, b is 2, so that is going to give me 22/4 = -10 + 22/4.0910

So, this is x2 + 2x...well, 2 times 2 is 4; divided by 4--that is just 1.0925

So, this is -10 + 1; this gives me x2 + 2x + 1 = -9.0932

So, I now have a perfect square trinomial; and because I have that, I will just go ahead my usual way and take the square root of both sides.0942

So, let's come up here to finish this and take the square root of both sides, and we rewrite it.0951

This is a perfect square: it is (x + 1)2 = -9.0964

The square root of both sides gives me x + 1 = √-9.0970

Now, before learning about complex numbers, we would have had to stop here and say,0975

"OK, this is undefined; there is no real number solution; we don't know what to do with this."0978

But now that we have discussed imaginary numbers, we do know what to do with this.0984

So, let's look at how we can handle this.0987

We know that this equals √-1 times 9; and remember the positive and negative results for this: you need to take both: ±√-9.0994

OK, using the product property, this is going to equal this.1009

Recall that, with complex numbers, the square root of -1 is equal to i.1018

OK, so this is going to give me x + 1 = ± i times √9, which is 3.1036

Therefore, now this is something I can work with: x = (I am going to subtract -1) ± 3i.1051

All right, so here we have a situation where the solution to the quadratic equation is a complex number.1061

And the reason is: you are taking the square roots.1067

When you are taking the square roots, if you end up with a negative number and you take the square root of that, you are going to get an imaginary number.1070

So here, I ended up getting all the way down by my usual method, completing the square,1078

then taking the square root of both sides; and I ended up with this.1083

But I recognized that the square root of -9 is equal to 3i; so then, I could simply say that my solutions are x = -1 + 3i, and x = -1 - 3i.1086

And that is a set of complex conjugates.1107

So, let's use the square root property to solve this.1112

I am recognizing that this is a perfect square trinomial; so I don't have to complete the square--it is already done for me.1115

Since this is negative here, this is (x - 3)2 = 7.1123

And if I take the square root of both sides, I am going to get x - 3 = ±√7, so x = 3 ± √7.1127

Or, you could write this out as x = 3 + √7, and x = 3 - √7, as the solutions.1141

When it is already a perfect square trinomial, it is much easier.1151

In this example, I am going to have to complete the square.1156

And this is just asking me to complete the square of x2 - 10x.1161

So, in order to do that, I need a constant.1166

And for a perfect square trinomial, the constant is going to equal b2/4, so that is x2 - 10x + b2/4.1169

And since this is in standard form, that is ax2 + bx + c.1183

That means that b equals -10; so x2 - 10x + (-10)2, all divided by 4.1189

OK, so it is x2 - 10x +...-10 squared is 100, divided by 4; x2 - 10x + 25.1203

This is a perfect square; this is the same as (x - 5)2.1217

So, to complete the square, you need to find a constant term; and the constant term is going to be equal to b2/4.1223

So, b equals -10; substituting in here gave me a perfect square trinomial.1232

Solve by completing the square: the first step is to isolate the x variables on the left side of the equation.1241

So, subtract 12 from both sides.1251

Then, complete the square by adding b2/4 to both sides; and it is very important that you do this to both sides, to keep the equation balanced.1257

OK, b is 6, so this gives me x2 + 6x + 62/4 = -12 + 62/4, which is 36/4; 36/4 is just 9.1271

We will find a bit more on the right; now I have my perfect square trinomial on the left.1306

So, I am going to come up here and just rewrite that, because I was asked to do more than just complete the square; I actually have to solve the equation.1321

Now, I am going to write this as (x + 3)2 = -3.1331

In order to solve this, I am going to take the square root of both sides.1337

x + 3 = ±√-3; from previous lessons, I know that I can change this to -1 times the square root of 3.1342

And recall that i is equal to the square root of -1, so this is i√3.1359

OK, this gives me x = -3 ± i√3; or I can write it out as x = -3 + i√3, and x =....1369

These are both solutions; they are not real-number solutions, but they are solutions.1385

Solving by completing the square means isolating the x variable terms on the left and the constants on the right.1393

Adding b2/4 to both sides gave me a perfect square trinomial on the left, which was (x + 3)2, and -3 on the right.1404

Solving by taking the square root of both sides gave me this.1414

I then simplified this, since the square root of -1 is i, into i√3; and then, I isolated x to get x = -3 + i√3, x = -3 - i√3.1419

Again, we are going to solve by completing the square.1437

And my first step is always to get the variable terms on the left and the constants on the right, then complete the square.1441

Subtract 18 from both sides; now, here, before I go any farther, I have to look and see that my x2 term is not 1.1451

So, if the coefficient of x2 is not 1, divide both sides of the equation...1471

And the reason is that I need to do that in order to complete the square and go on as usual.1488

So, I am going to divide both sides here (and I could have done it up here, but it is easier to do it at this point, when I am simplified a bit) by 3.1495

These cancel out, and now my x2 coefficient is 1.1507

12 divided by 3 gives me -4x; and 6 divided by 3 is 2.1511

Now, I can go about completing the square, because my leading coefficient is 1.1517

I am doing that by adding b2/4 to both sides: b is -4, so this gives me x2 - 4x, and this is...1532

4 times 4 is 16, divided by 4; so that is just 4; and this is also going to be 4.1555

OK, coming up here to finish, this is going to give me x2 - 4x + 4 = 6.1563

I now have my perfect square trinomial here, and it is (x - 2)2; it is a negative 2, because the middle term is negative.1576

And I am going to find the square root of both sides; so x equals 2 plus or minus the square root of 6.1585

This problem took an extra step: after isolating my variables on the left and my constants on the right,1597

I saw that my leading coefficient was not 1, so I had to divide both sides of the equation by that coefficient 3.1604

Once I was there, then I went about completing the square by adding this term to both sides,1611

getting a perfect square trinomial (which is this), and taking the square root of both sides to get my solutions.1618

Thanks for visiting Educator.com; and I will see you for the next lesson.1626

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