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INSTRUCTORS Carleen Eaton Grant Fraser
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Dr. Carleen Eaton

Dr. Carleen Eaton

Solving Radical Equations and Inequalities

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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 (4)

1 answer

Last reply by: Dr Carleen Eaton
Sun Jul 8, 2018 1:04 PM

Post by Okwudili Ezeh on June 27 at 02:32:28 PM

In example 4, why are you writing b as 2 rather than -2.

1 answer

Last reply by: Shiden Yemane
Sat Feb 10, 2018 11:04 AM

Post by Shiden Yemane on February 10 at 11:03:15 AM

On example 4, you wrote the radicand to be x+2. Shouldn't it be x+21?
Thanks

Solving Radical Equations and Inequalities

  • To solve a radical equation, raise both sides of the equation to the power equal to the index of the radical. This will eliminate at least one radical. If a radical remains in the new equation, repeat the process.
  • Always check for extraneous solutions – values for which one or more radicals in the original equation have a negative radicand. Exclude such values from the solution set.

Solving Radical Equations and Inequalities

Solve − 10 + √{v + 6} = − 5
  • Step 1 - Isolate the radical on one side of the equation
  • − 10 + √{v + 6} = − 5 = >√{v + 6} = 5
  • Step 2 - Raise both sides to the second power to eliminate the square root
  • ( √{v + 6} )2 = 52
  • v + 6 = 25
  • Step 3 - Solve
  • v = 19
  • Step 4 = Check for extraneous solutions.
  • √{v + 6} = 5
  • √{19 + 6} = 5
  • √{25} = 5
  • 5 = 5
Solution is not an extraneous solution, v = 19
Solve 0 = − 6 + √{35p + 1}
  • Step 1 - Isolate the radical on one side of the equation
  • 0 = − 6 + √{35p + 1} = > 6 = √{35p + 1}
  • Step 2 - Raise both sides to the second power to eliminate the square root
  • 62 = ( √{35p + 1} )2
  • 36 = 35p + 1
  • Step 3 - Solve
  • 35 = 35p
  • p = 1
  • Step 4 = Check for extraneous solutions.
  • 6 = √{35p + 1}
  • 6 = √{35(1) + 1}
  • 6 = √{36}
  • 6 = 6
Solution is not an extraneous solution, p = 1
Solve 0 = − 6 + √{[p/10]}
  • Step 1 - Isolate the radical on one side of the equation
  • 0 = − 6 + √{[p/10]} = > 6 = √{[p/10]}
  • Step 2 - Raise both sides to the second power to eliminate the square root
  • 62 = ( √{[p/10]} )2
  • 36 = [p/10]
  • Step 3 - Solve
  • 36(10) = [p/10]10 = > 36(10) = [p/]
  • p = 360
  • Step 4 = Check for extraneous solutions.
  • 6 = √{[p/10]}
  • 6 = √{[360/10]}
  • 6 = √{36}
  • 6 = 6
Solution is not an extraneous solution, p = 360
Solve − 1 = √{5x − 1} − √{8x + 1}
  • Step 1 - Isolate one radical to the left side to make solving easier.
  • − 1 = √{5x − 1} − √{8x + 1} → √{8x + 1} − 1 = √{5x − 1}
  • Step 2 - Raise both sides to the second power to eliminate the square roots. Multiply using FOIL or
  • any other method you prefer.
  • ( √{8x + 1} − 1 )2 = ( √{5x − 1} )2
  • ( √{8x + 1} − 1 )( √{8x + 1} − 1 ) = 5x − 1
  • Multiply on the left sde of the equation
  •  √{8x + 1}-1
    √{8x + 1}8x + 1 − √{8x + 1}
    -1 − √{8x + 1} + 1
  • 8x + 2 − 2√{8x + 1} = 5x − 1
  • Step 3 - Isolate radical to one side of the equation and raise both sides to the second power.
  • 3x + 3 = 2√{8x + 1}
  • ( 3x + 3 )2 = ( 2√{8x + 1} )2
  • 9x2 + 18x + 9 = 4(8x + 1)
  • 9x2 + 18x + 9 = 32x + 4
  • 9x2 − 14x + 5 = 0
  • Step 4 = Solve by factoring
  • 9x2 − 14x + 5 = (9x − 5)(x − 1) = 0
  • x = [5/9] and x = 1
  • Step 5 - Check Solutions
  • x=[5/9]
    − 1 = √{5x − 1} − √{8x + 1}
    − 1 = √{5[5/9] − 1} − √{8[5/9] + 1}
    − 1 = √{[25/9] − 1} − √{[40/9] + 1}
    − 1 = √{[16/9]} − √{[49/9]}
    − 1 = √{[16/9]} − √{[49/9]}
    − 1 = [4/3] − [7/3] = − [3/3] = − 1
  • x=1
    − 1 = √{5x − 1} − √{8x + 1}
    − 1 = √{5(1) − 1} − √{8(1) + 1}
    − 1 = √4 − √9
    − 1 = 2 − 3 = − 1
x = [5/9] and x = 1
Solve 1 = √{5 − 4x} − √{6 − 2x}
  • Step 1 - Isolate one radical to the left side to make solving easier.
  • 1 = √{5 − 4x} − √{6 − 2x} → 1 + √{6 − 2x} = √{5 − 4x}
  • Step 2 - Raise both sides to the second power to eliminate the square roots. Multiply using FOIL or
  • any other method you prefer.
  • ( 1 + √{6 − 2x} )2 = ( √{5 − 4x} )2
  • ( 1 + √{6 − 2x} )( 1 + √{6 − 2x} ) = 5 − 4x
  • Multiply on the left side of the equation
  •  √{6 − 2x}1
    1√{6 − 2x}1
    √{6 − 2x}6 − 2x√{6 − 2x}
  • 7 − 2x + 2√{6 − 2x} = 5 − 4x
  • Step 3 - Isolate radical to one side of the equation and raise both sides to the second power.
  • 2√{6 − 2x} = − 2x − 2
  • ( 2√{6 − 2x} )2 = ( − 2x − 2 )2
  • 4(6 − 2x) = 4x2 + 8x + 4
  • 24 − 8x = 4x2 + 8x + 4
  • 4x2 + 16x − 20 = 0
  • Step 4 = Solve by factoring, divide everything by 4.
  • [(4x2 + 16x − 20 = 0)/4]
  • x2 + 4x − 5 = (x + 5)(x − 1)
  • x = − 5 and x = 1
  • Step 5 - Check Solutions
  • x = −5
    1 = √{5 − 4x} − √{6 − 2x}
    1 = √{5 − 4( − 5)} − √{6 − 2( − 5)}
    1 = √{25} − √{16}
    1 = 5 − 4 = 1
    Checks!
  • x=1
    1 = √{5 − 4x} − √{6 − 2x}
    1 = √{5 − 4(1)} − √{6 − 2(1)}
    1 = √1 − √4
    1 = 1 − 2 = − 1
    Not a solution
x = − 5
Solve 3 = √{ − 2 − 9x} − √{ − 1 − x}
  • Step 1 - Isolate one radical to the left side to make solving easier.
  • 3 = √{ − 2 − 9x} − √{ − 1 − x} = > 3 + √{ − 1 − x} = √{ − 2 − 9x}
  • Step 2 - Raise both sides to the second power to eliminate the square roots. Multiply using FOIL or
  • any other method you prefer.
  • ( 3 + √{ − 1 − x} )2 = ( √{ − 2 − 9x} )2
  • ( 3 + √{ − 1 − x} )( 3 + √{ − 1 − x} ) = − 2 − 9x
  • Multiply on the left sde of the equation
  •  3√{ − 1 − x}
    393√{ − 1 − x}
    √{ − 1 − x}3√{ − 1 − x} − 1 − x
  • 8 − x + 6√{ − 1 − x} = − 2 − 9x
  • Step 3 - Isolate radical to one side of the equation and raise both sides to the second power.
  • 6√{ − 1 − x} = − 8x − 10
  • ( 6√{ − 1 − x} )2 = ( − 8x − 10 )2
  • 36( − 1 − x) = 64x2 + 160x + 100
  • − 36 − 36x = 64x2 + 160x + 100
  • 64x2 + 196x + 136 = 0
  • Step 4 = Solve by factoring, divide everything by 4.
  • [(64x2 + 196x + 136 = 0)/4]
  • 16x2 + 49x + 34 = (16x + 17)(x + 2)
  • x = − [17/16] and x = − 2
  • Step 5 - Check Solutions
  • x=−[17/16]
    3 = √{ − 2 − 9x} − √{ − 1 − x}
    3 = √{ − 2 − 9( − [17/16] )} − √{ − 1 − ( − [17/16] )}
    3 = √{ − 2 − 9( − [17/16] )} − √{ − 1 − ( − [17/16] )}
    3 = √{ − 2 + ( [153/16] )} − √{ − 1 + ( [17/16] )}
    3 = √{[121/16]} − √{[1/16]} = [11/4] − [1/4] = [10/4] = [5/2]
    Does Not Check!
  • x=−2
    3 = √{ − 2 − 9x} − √{ − 1 − x}
    3 = √{ − 2 − 9( − 2)} − √{ − 1 − ( − 2)}
    3 = √{ − 2 + 18} − √{ − 1 + 2}
    3 = √{16} − √1 = 4 − 1 = 3
    Checks! A solution
x = − 2
Solve 7 + √{x + 7} ≤ 14
  • Step 1: Isolate the square root on one side of the inequality
  • √{x + 7} ≤ 7
  • Step 2: Determine the domain of the inequality. You cannot have negative inside the radical.
  • x + 70
  • x − 7
  • Step 3: Raise both sides to the second power
  • ( √{x + 7} )2 ≤ ( 7 )2
  • x + 7 ≤ 49
  • x ≤ 41
  • Step 4: Combine the restrictions of the domain with previous answer
− 7 ≤ x ≤ 41
Solve √{ − 8 − 12x} − 5 ≤ 5
  • Step 1: Isolate the square root on one side of the inequality
  • √{ − 8 − 12x} ≤ 10
  • Step 2: Determine the domain of the inequality. You cannot have negative inside the radical.
  • − 8 − 12x0
  • x ≤ − [8/12]
  • x ≤ − [2/3]
  • Step 3: Raise both sides to the second power
  • ( √{ − 8 − 12x} )2 ≤ ( 10 )2
  • − 8 − 12x ≤ 100
  • − 12x ≤ 108
  • x − 9
  • Step 4: Combine the restrictions of the domain with previous answer
− 9 ≤ x ≤ − [2/3]
Solve √{3 − 39x} − 9 ≤ 0
  • Step 1: Isolate the square root on one side of the inequality
  • √{3 − 39x} ≤ 9
  • Step 2: Determine the domain of the inequality. You cannot have negative inside the radical.
  • 3 − 39x0
  • − 39x − 3
  • x ≤ − [3/( − 39)] = [1/13]
  • Step 3: Raise both sides to the second power
  • ( √{3 − 39x} )2 ≤ ( 9 )2
  • 3 − 39x ≤ 81
  • − 39x ≤ 78
  • x − 2
  • Step 4: Combine the restrictions of the domain with previous answer
− 2 ≤ x ≤ [1/13]
Solve 9√{12x − 3} − 81 ≤ 0
  • Step 1: Isolate the square root on one side of the inequality
  • 9√{12x − 3} ≤ 81
  • Step 2: Determine the domain of the inequality. You cannot have negative inside the radical.
  • 12x − 3 ≥ 0
  • 12x ≥ 3
  • x[3/12] = [1/4]
  • Step 3: Raise both sides to the second power, but divide both sides by 9 first.
  • [(9√{12x − 3} )/9] ≤ [81/9]
  • ( √{12x − 3} )2 ≤ ( 9 )2
  • 12x − 3 ≤ 81
  • 12x ≤ 84
  • x ≤ 7
  • Step 4: Combine the restrictions of the domain with previous answer
[1/4] ≤ x ≤ 7

*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 Radical Equations and Inequalities

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
  • Radical Equations 0:11
    • Variables in Radicands
    • Example: Radical Equation
    • Example: Complex Equation
  • Extraneous Roots 7:21
    • Squaring Technique
    • Double Check
    • Example: Extraneous
  • Eliminating nth Roots 10:04
    • Isolate and Raise Power
    • Example: nth Root
  • Radical Inequalities 11:27
    • Restriction: Index is Even
    • Example: Radical Inequality
  • 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

Transcription: Solving Radical Equations and Inequalities

Welcome to Educator.com.0000

In previous lessons, we have learned techniques for working with radicals.0003

Now, we are going to put those together and use them to solve radical equations and inequalities.0007

First, the definition of a radical equation: recall that radical equations contain radicals with variables in the radicand.0012

For example, it is something like the fourth root of x + 5 equals 3, or 6 + √(2x + 1) = 4.0020

These are both examples of radical equations.0033

If you had a radical, and there was just a number under it, like the square root of 5, plus 2x,0036

this is not a radical equation, because there is no variable under the radical sign.0047

OK, in order to solve these equations, we are going to use a technique learned earlier on in Algebra I and reviewed now, and also go into more complex problems.0056

For example, if we had the square root of x + 2, minus 6, equals -1; what we are going to do is0066

raise each side of the equation to a power that is the index of the radical.0077

So, the index of the radical here is 2; so I am going to raise both sides to the second power; in other words, I am going to square both sides.0083

But before you do that, first isolate the radical; then raise both sides to a power equal to the index of the radical.0090

First, I am isolating this by adding 6 to both sides; so, -1 + 6 is going to give me 5.0110

Now, I am going to square both sides; I am going to raise both sides to that index power, which is 2.0120

Well, since the index is 2, and it is being raised to the second power, this equals the radicand, and this equals 25.0126

Subtracting 2 from both sides gives me x = 23.0136

So, to solve radical equations, you isolate the radical, then raise both sides to a power equal to the value of the index; then, solve the equation.0140

Now, you see here that it says sometimes this must be done twice in order to eliminate all radicals.0152

That is a more complex situation that we are going to look at right now.0158

Let's say I am given √(x + 3) - 2 = √(x - 5).0163

I can't just isolate the radical, because there is more than one radical.0171

So in this case, I just begin by raising both sides to a power that is equal to the index.0175

Since the index is 2, I am just going to first square both sides; I can't isolate.0183

Squaring both sides is going to give me this; and if I look at this left side, I am squaring a difference; that is analogous to this situation:0193

(a - b)2, which turns out to be a2 - 2ab + b2.0209

I just substitute in here; a is the square root of x + 3, and b is 2.0215

So, that makes my work on the left much quicker, instead of having to use FOIL, use the distributive property,0222

multiply everything out, add...I just say, "OK, the square root of x + 3, squared, minus 2 times a times b..."0227

a is the square root of x + 3, and then b is 2.0239

And then, finally, I get b2.0248

On the right, I raise this to the index power, so I get the radicand, x - 5.0256

OK, I squared the square root of x + 3; the square root of (x + 3)2 is simply the radicand, x + 3.0262

Here, I have 2 times 2 is 4, times the square root of x + 3.0272

Here I have -2 times -2 is 4.0279

Now, you can see that the radical on the right was eliminated.0285

However, I still have a radical on the left; so that is why I am going to need to repeat this process.0288

I squared both sides, and then I need to repeat if radicals remain.0293

I squared both sides, and I am going to need to repeat; let's simplify first.0305

This is 3 + 4, so this gives me x + 7, minus 4√(x + 3), equals x - 5.0309

I am going to move this 7 over to the right by subtracting 7 from both sides; and that is going to be x - 5 - 7, so x - 12.0320

Now, the next thing is: to isolate this, I want to subtract x from each side; so x - x...that drops out.0332

x -x...that drops out, too: conveniently, the x's dropped out, making this even simpler to work with.0339

I still want to isolate this completely; so, to do that, I need to divide both sides by -4.0351

Now, I am left with something easier to work with, because I end up with just the radical isolated on the left, and then I just have a number on the right.0365

So, I am going to repeat my process, and I am going to square this; and let's start right up here with that second process, √(x + 3)2 = 32.0376

I raised this radical to the index power; that is going to leave me with the radicand, x + 3; and on the right, I have 9.0395

This gives me x = 6; this was pretty complex.0404

And I solved it by first squaring both sides, which got rid of the radical on the right.0407

Then, I simplified the left, found that the x's dropped out, combined my constants, and ended up with this.0412

To completely isolate the radical, I divided by 4; and when I ended up with this, then I have something that is more familiar,0424

where I have an isolated radical expression on the left, and squared both sides, solved, and determined that x equals 6.0431

OK, recall that sometimes, when we use this method of squaring both sides, we end up with a solution;0442

and when we check the solution, it actually does not satisfy the original equation.0449

This is a result of the squaring step of using this technique; and these solutions are called extraneous solutions.0456

Therefore, you always must check the solutions you get from this technique back in the original equation or inequality.0464

If the solution exists, and you use this technique, the solution will be among your results.0474

And I say "among your results" because you might have the valid solution, plus some extraneous solutions.0478

There may be no solution; you may just end up with one or two extraneous solutions; you may end up with one or more valid solutions.0485

There is no way of knowing; but if the solution exists, it will be somewhere among the solutions you end up with.0492

The only way to find it, though, is to check all the solutions to see what you have.0497

For example, we did, earlier on, that example (which was a little bit complicated): √(x + 3) - 2 = √(x - 5).0503

And when we worked that out, I came out with the solution x = 6.0521

I can't just assume it is a valid solution; it actually may not be valid.0525

The only way to know is to go back and substitute in.0529

Everywhere there is an x, I am going to make it a 6; and I am going to see if my equation holds up.0534

This is going to give me √(x + 3), which is √9, minus 2, equals √(6 - 5); so that becomes √1.0542

The square root of 9 is 3; minus 2 equals the square root of 1, which is 1.0557

1 does equal 1; this is true, so the solution, which is x = 6, is valid.0564

It is possible sometimes that you will get a solution, and you check it, and you get something strange like 4 = 5, which is not true.0574

That means that this answer, this solution, does not satisfy the equation; and therefore, it was an extraneous solution0583

that popped up as a result of squaring both sides.0591

I could have ended up with two solutions or three solutions; and some were extraneous/some were valid/none were valid.0596

Again, the only way to tell is to check.0601

Eliminating nth roots: we have been doing this, but just to bring it out and explain exactly what we were doing:0605

to eliminate an nth root, you isolate the root (I talked about isolating the radicals),0611

and then raise both sides of the equation to the power n.0618

That is what we did when we took something like √(x + 2) - 4 = 8, and we first isolated, which would give me 12.0622

And then, I said, "OK, n is 2, so I am going to square it; I am going to raise it to the second power."0638

So, this is exactly what we have been doing: isolate the root, and then raise both sides of the equation to that index power.0649

And the same technique could apply to if you are using a higher index; but it does become more complicated.0657

OK, and again, an important last step is to check solutions in the original equation to make sure that they are valid.0666

They may be valid; they may be extraneous.0683

All right, we talked about radical equations; and radical inequalities are very similar.0687

So, these are inequalities with variables in the radicand.0693

And just as we talked about with equations (radical equations have variables in the radicand), radical inequalities0697

are the same thing, only we are working with greater than or less than, or greater than or equal to, and all that, instead of equals.0703

We have this restriction that we have seen before with radicals: if the index is even,0713

we have to determine the values of the variable for which the radicand is greater than or equal to 0.0717

In other words, we have to restrict the solution set to values that will not make the radicand negative,0723

because if the radicand is negative, and we have an even index, we will end up with something that is not a real number.0732

So, after we determined that restriction, we solved algebraically, using the same techniques that we have talked about before with radicals.0739

Let's use an example: the square root of 3x + 4, minus 2, is greater than 5.0747

Before I even begin trying to solve this, I am going to figure out what my excluded values are.0754

So, I know that 3x + 4 has to be greater than or equal to 0.0760

If this ended up being negative, and I have an even index here, 2, then I can end up with a complex number as a solution.0765

And I don't want that; we are just working with real numbers.0773

So, I am going to go ahead and say 3x, and I am going to subtract 4 from both sides to get 3x ≥ -4.0777

And then, I am going to divide both sides by 3; so x ≥ -4/3.0785

This is just saying that I am restricting the domain to these values.0792

Any value of x that is less than -4/3 is an excluded value.0797

So, x less than -4/3 are excluded.0803

I haven't even begun to solve this yet; I was just figuring out what the restrictions are on my solution set.0810

OK, so now, I am going to go ahead and actually solve this algebraically, using techniques that we used with radical equations.0818

The first one here is to isolate the radical on one side of this inequality.0828

And I am going to do that by adding 2 to both sides.0834

So, this is going to give me √(3x + 4) > 7.0843

Next, I am going to raise both sides to the power equal to the index, which, in this case, would be the power 2.0852

So, squaring both sides is going to give me...the square root squared is going to give me the radicand,, 3x + 4 > 7 squared, or 49.0860

Now, I just solve: 3x...subtracting 4 from both sides, the 4 drops out from here...49 - 4 is 45.0875

3x > 45; and I want to isolate the x, so I am going to divide by 3; 3 is going to drop out--I get x > 45/3, or x>15.0885

And I check and say, "OK, is this allowed?"0901

Yes, it is, because if x is greater than 15, it is going to be greater than -4/3.0904

So, I am not going into values that are not allowed.0912

And then, you could check this by picking a value greater than 15,0916

and putting it in here, back into the original, and making sure that the inequality is true--that it holds up.0921

I could pick a value such as 20 or 16, put it in here, solve this, and make sure that I ended up with something greater than 5,0927

because again, when we use this technique of squaring both sides, we need to check the answers.0936

OK, first example: I am going to solve this radical equation by the technique of first isolating the radical.0942

So, all I have to do in order to achieve that is subtract 8 from both sides to get √(3x - 2) = 7.0957

Then, I am going to square both sides, because the index is 2.0965

So, if I raise both sides to the second power, then I am going to get the radicand from here,0969

because the square root of 3x - 2, times itself, is 3x - 2.0976

On the right, I get 72, is 49.0983

Now, I just solve using my usual algebraic techniques, adding 2 to both sides.0986

3x = 51; now, dividing both sides by 3, x = 51/3; and if you do the arithmetic on that, you will find that x equals 17.0993

Another important step is checking to make sure this is a valid solution.1007

So, I am going to go all the way back to this original; and I am going to let x equal 171010

and substitute that in to get 8 + √3 times 17 minus 2 equals 15.1017

3 times 17 is 51; 51 minus 2 is 49; so, this is going to give me 8 + √49 (which is 7) = 15.1028

Since 15 = 15, then x = 17 is valid, because x = 17 actually satisfied this equation--the two sides were equal.1048

OK, the second example is a similar idea; but instead of working with square roots, now the index is 4.1065

But I am going to use the same technique: I am going to first isolate this radical.1072

And I am going to do that by subtracting 4 from both sides.1077

6 - 4 is 2, so now I have that the fourth root of 2x - 8 equals 2.1086

In this case, since the index power is 4, I am going to raise both sides to the fourth power.1094

OK, I have the fourth root of 2x - 8 raised to the fourth power.1104

If the index and the power you are raising it to are the same, this, then, equals the radicand.1110

2 to the fourth power is...2 times 2 is 4, times 2 is 8, times 2 is 16.1119

Next, I just isolate the x, using algebraic techniques: I am going to add 8 to both sides--that gives me 2x = 24.1129

Dividing both sides by 2 gives me x = 12.1140

And we can't forget the important step of checking our solution back in the original equation.1144

I am going to check, and this is the fourth root of 2...I am going to check x = 12.1150

The fourth root of 2 times 12, minus 8, plus 4, equals 6; let's see if that holds up.1162

2 times 12 is 24, minus 8, plus 4, equals 6.1173

This is going to give me the fourth root of 24 - 8, which is 16.1180

The fourth root of 16 is 2, because 24 = 16; so the fourth root of 16 is 2.1187

That leaves me with 2 + 4 = 6; and 6 does equal 6, so x = 12 is valid.1202

It is a valid solution, because it satisfies this equation.1212

When x is 12, the equation holds true.1216

In our third example, we have an inequality: we have to take that extra step of excluding values--finding what the excluded values are.1224

3x - 6 must be greater than or equal to 0; it needs to be a non-negative number,1235

because since this is an even power, if this radical is the square root of a negative number,1242

then I could end up with an imaginary number, which I don't want.1251

So, since this is an even index, I am going to take the extra step and define the values of x that I am allowed to work with.1255

OK, 3x - 6 must be greater than or equal to 0; adding 6 to both sides gives me 3x ≥ 6.1263

Now, dividing both sides by 3, x ≥ 2.1275

So, I have not solved it yet; all I have said is that the only values that are even allowable to use are these.1280

Now, I am going to go about solving it; so let's rewrite this here.1287

2 plus the square root of 3x minus 6 is less than or equal to 8.1291

Just like with radical equations, I start out by isolating the radical.1296

So, I am going to subtract 2 from both sides; my index value is 2--it is just the square root.1301

So, I am going to square both sides.1309

The square root of something squared is just that value; so this becomes the radicand, because the square root of 3x - 6, times itself, is 3x - 6.1317

It is less than or equal to 36.1331

Now, I am going to add 6 to both sides: 36 + 6 is 42.1334

Divide both sides by 3: 42 divided by 3 is actually 14, so x is less than or equal to 14.1343

I am not quite done: I need to put all of this together, because if I just said,1355

"OK, x is less than or equal to 14," I might say, "Oh, OK, x can be 0," but it actually can't,1359

because I have this restriction right here on the domain that it has to be greater than or equal to 2.1365

Therefore, I am going to write out the whole thing together, saying that x needs to be greater than or equal to 21370

in order to even be a value that will give me a real number.1380

But to actually solve this inequality, to find the solution set...the solution set only encompasses those values of x that are less than or equal to 14.1386

So, in this case, I actually had to put this all together.1394

Now, in order to check this, one way to check it is to choose a value that is within the solution set (for example, 5--1400

5 is greater than or equal to 2, and less than or equal to 14).1408

So, I am going to check with x = 5, and I am going to see if that satisfies this inequality.1412

2 + √(3x - 6) ≤ 8: if I am going to let x equal 5, and just make sure that this thing holds up,1420

this is going to give me 15 - 6 under the radical, which is...15 - 6 is 9, so I have the square root of 9 there,1434

which is 3; so I end up with 5 ≤ 8, and this is true.1448

So, this helped to verify that I actually have a valid solution.1457

Again, when you are using this technique of squaring both sides, extraneous solutions can occur.1466

OK, this time, I actually have two radicals; so it is not going to be possible to just isolate the radical.1474

Therefore, my first step is just going to be to get rid of at least one of the radicals by squaring both sides.1481

So, I am going to go ahead and square both sides, and recall that, on the left, if I have something1489

such (a - b)2, it is going to give me a2 - 2ab + b2.1501

And by remembering that, I will save myself the work of having to use the distributive property and figure this whole thing out.1507

I am just going to go ahead and say, "OK, a equals the square root of x plus 21, and b equals 2."1514

So, on the left, I am going to get a2...the square root of x plus 21, squared...minus 2 times a1523

(the square root of x plus 21), times b (which is 2); and then, for the last term, I will get b2, which is 22.1537

On the right, I have a -1 in front of this; when I square that, -1 times -1 becomes 1.1551

So, I am going to end up with a positive expression on the right, and the square root of x + 5 squared...1558

since the index is the same as the power, this becomes the radicand; so on the right, I end up with x + 5.1568

OK, simplifying: over here, I have a square root raised to the second power; the square root squared gives me the radicand.1576

So, I can see I have gotten rid of this radical on the right; and this is not a radical.1586

However, in the middle, this middle term is -2, times 2 is -4, times the square root of x plus 21.1592

I am still left with this radical; here, 2 times 2 is 4; simplify before you proceed.1600

This gives me 2 and 4, so that is x + 6 minus this equals x + 5.1612

So, I am going to go ahead and subtract this 6 from both sides...this is actually 21.1622

So, go ahead and simplify this; this is 21 plus 4, is actually going to be 25.1642

So, x + 21, the square root of that squared, gives me x + 21; so that is 25 right here; 21 + 4 is 25.1649

OK, simplify by subtracting 25 from both sides: 5 - 25 is -20.1662

In the next step, I would subtract x from each side; so when I do that, I get x - x; the x drops out of the left.1675

When I take x - x, the x also drops out of the right side.1687

Now, I have gotten rid of this radical on the right; on the left, I have a radical,1693

but I also have a -4 in front of it, so I need to isolate the radical by dividing both sides by -4.1698

So, this gives me -20 divided by -4 on the right; so the square root of x + 21 equals 5.1706

Since I still have a radical left, I have to go through this process again.1717

So, let's take this up here and repeat the process of squaring both sides.1721

The square root of x + 21, squared, gives me the radicand: x + 21 = 52...that is 25.1734

Subtracting 21 from both sides gives me 4; so I came up with this solution that x = 4, and I need to check that in the original.1743

So, check by inserting 4 for each x in the original.1755

That is going to give me the square root of 4 + 21, minus 2, equals negative...and then that is the square root of 4 + 5.1763

This gives me the square root of 25, minus 2, equals minus the square root of 9.1776

The square root of 25 is 5, minus 2 equals minus the square root of 9, which is 3; so that gives me -3.1783

This is 5 - 2 = -3; this is not true; therefore, the solution is not valid.1795

This solution, x = 4, is not valid.1807

Now, I said that, when you use this method, if there is a valid solution, you will come up with it.1809

You might have some extra solutions, but you will have a valid solution, if it exists.1815

What this tells me, since this is not valid, is that there is no valid solution here.1820

So, this was a pretty complex problem: there were two radicals, so we had to square to get rid of the radical on the right;1832

do a bunch of simplifying; isolate the remaining radical, which is on the left;1840

and repeat the process with that by squaring both sides to come up with x = 4.1847

And then, after all that work, we went and checked it, and found that x = 4 does not satisfy this equation--1853

that, when we use that, we end up with 3 = -3.1860

Had that negative not been there, we could have come up with a valid solution.1865

But with that negative in the original, the solution was not valid.1871

Therefore, there is no solution to this radical equation.1875

That concludes this lesson on radical equations and inequalities.1880

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