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INSTRUCTORS Carleen Eaton Grant Fraser Eric Smith
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Eric Smith

Eric Smith

Solving Radical Equations

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Table of Contents

I. Properties of Real Numbers
Basic Types of Numbers

30m 41s

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

19m 26s

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

12m 6s

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

18m 52s

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

12m 22s

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

28m 52s

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

12m 2s

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

28m 41s

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

24m 26s

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

22m 55s

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

27m 58s

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

20m 36s

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

21m 24s

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

16m 12s

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

25m 54s

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

20m 1s

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

19m 40s

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

24m 34s

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

17m 13s

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

16m 13s

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

14m 12s

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

17m 7s

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

15m 33s

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

21m 13s

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

44m 51s

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

18m 33s

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

25m 7s

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

44m 56s

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

28m 27s

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

21m 44s

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

30m 9s

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

30m 14s

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

23m 38s

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

29m 27s

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

16m 47s

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

29m 4s

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

26m 53s

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

21m 42s

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

26m 41s

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

20m 24s

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

18m 23s

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

16m 24s

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

18m 54s

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

20m 20s

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

27m 4s

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

14m 32s

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

15m 12s

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

17m 22s

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

19m 24s

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

15m 5s

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

29m 16s

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

1 answer

Last reply by: Professor Eric Smith
Wed Jul 30, 2014 2:45 PM

Post by Peter Spicer on July 22, 2014

Wouldn't you add 3 to both sides since we have an x-3?

Solving Radical Equations

  • A radical equation is an equation in which variables are present in the radical.
  • To solve a radical equation, isolate the radical and raise each side of the equation to an appropriate power.
  • You must check your solutions with these types of problems to ensure that they work in the original.
  • If there is more than one radical expression in the equation, isolate them one at a time. Also be very careful with raising each side to a power. Some instances require you to FOIL.

Solving Radical Equations

√{4x − 16} = 3√2
  • ( √{4x − 16} )2 = ( 3√2 )2
  • 4x − 16 = 9 ×2
  • 4x − 16 = 18
  • 4x = 34
x = 8[1/2]
√{10x + 12} = 8√3
  • 10x + 12 = 64 ×3
  • 10x + 12 = 192
  • 10x = 180
x = 18
4√5 = √{6y + 8}
  • 16 ×5 = 6y + 8
  • 80 = 6y + 8
  • 72 = 6y
12 = y
x = √{x + 30}
  • x2 = ( √{x + 30} )2
  • x2 = x + 30
  • x2 − x − 30 = 0
  • ( x + 5 )( x − 6 ) = 0
x = − 5,6
x = √{x + 56}
  • x2 = ( √{x + 56} )2
  • x2 = x + 56
  • x2 − x − 56 = 0
  • ( x + 7 )( x − 8 ) = 0
x = − 7,8
x = √{13x + 30}
  • x2 = ( √{13x + 30} )2
  • x2 = 13x + 30
  • x2 − 13x − 30 = 0
  • ( x + 2 )( x − 15 ) = 0
x = − 2,15
− 2 + √{x + 4} = x + 4
  • √{x + 4} = x + 4 + 2
  • √{x + 4} = x + 6
  • ( √{x + 4} )2 = ( x + 6 )2
  • x + 4 = x2 + 12x + 36
  • 4 = x2 + 11x + 36
  • 0 = x2 + 11x + 28
  • x2 + 11x + 28 = 0
  • ( x + 4 )( x + 7 ) = 0
x = − 4, − 7
1 + √{4x − 11} = x − 1
  • √{4x − 11} = x − 1 − 1
  • √{4x − 11} = x − 2
  • ( √{4x − 11} )2 = ( x − 2 )2
  • 4x − 11 = x2 − 4x + 4
  • − 11 = x2 − 8x + 4
  • 0 = x2 − 8x + 15
  • x2 − 8x + 15 = 0
  • ( x − 5 )( x − 3 )
x = 5,3
x − √{3 − 11x} = 3
  • − √{3 − 11x} = 3 − x
  • ( − √{3 − 11x} )2 = ( 3 − x )2
  • 3 − 11x = ( − x + 3 )2
  • 3 − 11x = x2 − 6x + 9
  • − 11x = x2 − 6x + 6
  • 0 = x2 + 5x + 6
  • 0 = ( x + 3 )( x + 2 )
x = − 3, − 2
x − √{5 − 7x} = 7
  • − √{5 − 7x} = 7 − x
  • ( − √{5 − 7x} )2 = ( − x + 7 )2
  • 5 − 7x = x2 − 14x + 49
  • − 7x = x2 − 14x + 44
  • 0 = x2 − 7x + 44
  • 0 = ( x − 11 )( x + 4 )
x = 11, − 4

*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

Lecture Slides are screen-captured images of important points in the lecture. Students can download and print out these lecture slide images to do practice problems as well as take notes while watching the lecture.

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

Transcription: Solving Radical Equations

Welcome back to www.educator.com.0000

In this lesson we are going to go ahead and solve some radical equations.0003

We only have one thing to pickup in this lesson it is just all the nuts and bolts on how you solve on these radical equations.0009

A radical equation is an equation in which we have variables present in our radical.0018

That will be something like this, I have √x-5 - √x-3 = 1.0026

To solve these we essentially want to end up isolating these roots then raise each side of our equation to a power in order to get rid of them.0033

It has to be the appropriate power.0044

For dealing with square roots we will square both sides.0045

If we have say 4th root then we will raise both sides to the 4th power.0048

With these ones it is extremely important that you go ahead and you check your solutions when you are done.0054

In the solving process and you raise both sides to a power we may introduce solutions or possible solutions that do not work in the original.0060

Always check your solutions for these types of equations.0069

Let us get an idea of what I’m talking about with √3x + 1 - 4 = 0.0075

The very first thing you want to do when solving something like this is get that root all by itself.0082

Let us go to both sides of the equation so we can do that.0088

√3x+ 1 = 40092

We have isolated the root we are going to get rid of it by raising both sides to the power of two.0097

The reason why we are doing this is because we have a square root.0104

3x + 1 = 42 which is 160109

Now they got rid of your roots, we simply solve directly.0117

The type of equation that we might end up with could be quadratic, could be linear, but we use all of our other tools from here on out.0122

-1 from both sides we will get 15 and divide both sides by 3 will give us 5.0129

You will realize that this looks like a possible solution always go back to your original for these ones and check to see if it does work.0136

I’m going to write this out here -4 does that equals 0 or not.0146

Let us put in 5, 3 × 5 would be 15 and all of that is underneath the root and then 15 + 1 would be 16.0156

√16=4, 4 -4 does equal 0, I know that this checks out and x equals 5 is our solution.0172

This one, we will go ahead and try and solve it is 5 + √x+ 7 = x.0191

Isolate our radical by subtracting 5 from both sides of the equation √ x + 7 = x - 5.0200

To get rid of that radical let us go ahead and raise both sides to the power of 2.0213

On the left side we will just have x + 7.0222

Be very careful on what is going on over here, keep your eyes peeled because notice how we have a binomial and we are squaring it.0226

In fact, you should look at it like this x - 5 × x – 5.0238

That way you can remember that this is how it should be foiled instead of trying to do some weird distribution with the two.0244

It does not work like that.0250

Let us see what we have when we foil.0253

Our first terms are x2, outside terms - 5x, inside terms - 5x, and our last terms 25.0256

We can go ahead and combine to see what we will get x2 - 10x + 25.0270

You can see that this one is turning into a quadratic equation because we have x2 term.0285

Since it is quadratic we will get all of our x on to the same side and see if we can use some of our quadratic tools in order to solve it.0291

0 = x2 - 11x + 180301

How shall we solve this quadratic?0312

It is not too bad, it looks like we can use reverse foil to go ahead and break it up and see what our parts are.0315

x × x = x2 then 2 × 9 = 180322

I know that 2 and 9 are good candidate because If I add 2 and 9 I will get that 11.0330

Let us make these both negative.0335

Have it x = 2 or x = 9.0337

Two possible solutions and they are possible because they might not work.0342

Let us check in the original 5 + √ 2 + 7 does it equal 2?0347

Let us find out, 2 + √7 = √ 9, √9 is 3 so I get 8.0359

Unfortunately that looks like it is not the same as the other side.0372

This one does not work and we can mark it off our list.0377

Let us try the other one 5 + the square roots and we are testing out a 9 so we will put that in 9 and 9.0383

Underneath the square root, 9 + 7 would give us a 16 and √16 =4, I will get 9 which is the same thing as the other side.0396

That one looks good.0410

You will keep that one as your solution.0413

Be very careful and always check these types of ones back in the original, you will know which ones you should throw out.0416

If there happens to be more than one radical in your equation then we can solve these by isolating the radicals one at a time.0427

Do not try and take care of them both at once.0434

Just focus on one, get rid of that radical and focus on the other one and then get rid of that radical.0436

It does not matter which radical you choose first.0442

If you have a bunch of them just go ahead choose one and go after it.0446

Be very careful as you raise both sides to a power since you have more than one radical in there,0450

when you raise both sides to power it will often end having to get foiled.0456

Let us see exactly what I'm talking about with this step, but will have to be extremely careful to make sure we multiply correctly.0460

Always make sure you check your solutions to see that they work in the original.0468

But this one has two radicals in it.0474

I have √x - 3 + √x + 5 = 40477

It does not matter which radical you choose at the very beginning and I'm going to choose this one.0483

I’m going to work to isolate it and get all by itself x + 5 = 4 – and I have subtracted the other radical to the other side.0490

Since √x+ 5 is the one I’m trying to get rid of, I’m going to square both sides to get rid of it.0505

That will leave me with just x + 5 on that left side.0514

Be very careful what happens over here on the other side, it is tempting to try and distribute the 2 but that is not how those work.0520

In fact if we are going to take look at this as two binomial so we can go ahead and distribute.0528

Let us go ahead and take care of this very carefully.0542

Our first terms 4 × 4 = 16, our outside terms we are taking 4 × -√x – 3, -4 × √x - 3.0545

Inside terms would be the same thing -4 √x - 3 and now we have our last terms.0560

negative × negative = positive and then we have √x -3 × √x -3.0570

That will give us √x - 32 since it will be multiplied by themselves.0581

When you do that step the first time, it usually looks like you have made things way more complicated.0590

I mean, we are trying to get rid of our root but now it looks like I have 3 of them running around the page.0596

It is okay we will be able to simplify and in the end we will end up with just one root,0600

which is good because originally we started with two and if I get it down to one we are moving forward with this problem.0605

Let us see what we can do.0612

On that left side I have 16 these radicals are exactly the same, I will just put their coefficients together -8 √x -30614

The last one square of square root would be x -3.0626

Now you can start to combine a few other things.0634

I will drop these parentheses here, if I subtract x from both sides that will take care of both of those x.0639

Let us see I can go ahead and subtract my 3 from the 16, 5 =13 - 8√x -3.0650

Let us go ahead and subtract a 13 from both sides, -13 – 13, -8 = -8√x -3.0665

It looks like we can divide both sides by -8, 1 =√x -3.0682

That is quite a bit of work, but all of that work was just to get rid of one of those radicals and we have accomplished that.0689

We got rid of that one.0695

But we still have the other radical here to get rid of.0696

We go through the same process of getting it all alone on one side, isolating it, squaring both sides get rid of it, and solving the resulting equation.0700

This one is already isolated we are good there.0709

I will simply move forward by squaring both sides.0711

1 = x -3 let us get some space.0716

If I add 3 both sides this will be 4 = x.0724

Even after going through all of that work and just when you think you have found a solution, we got to check these things.0729

We have to make sure that it will work in the original.0735

4 - 3 + √4 + 5 we are checking does that equal 4.0741

Let us see what we have.0756

4 - 3 would give us √1 , 4 + 5 would be 9, so√4 =1 and √9 = 3 and fortunately 4 does equal 4.0758

We know that this solution checks out, 4 =x.0774

If you have more than one of those radicals just try and get rid of them one at a time.0781

You might be faced with some higher roots and that is okay.0788

You will end up simply using a higher power on both sides of your equation in order to get rid of them.0791

In this one I have the 3rd root of 7x - 8 = 3rd root of 8x+ 2.0796

If I'm going to get rid of these cube roots I will raise both sides of it to the power of 3.0802

Leaving me with 7x - 8 = 8x + 20810

I can work on just giving my x together.0818

-7x from both sides, -8 = x + 2.0822

We will subtract 2 from both sides and this will give us -10 = x.0833

Let us quickly jump back up here to the top and see if that checks out.0841

3rd root of 7 × putting that -10 - 8 and 8 × -10 + 20845

Let us see if they are equal.0866

This time I’m dealing with 3rd root of (-70-8) to be the 3rd root of -78.0868

The other side I have -80 + 2 which is the 3rd root of -78.0880

It looks like the two agree.0887

I know that the -10 is my solution.0890

With those radicals try and isolate them, and then get rid of them by raising them to a power.0894

Always check your solutions with these ones to make sure they work in the original.0899

Thanks for watching www.educator.com.0904

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