Solving Radical Equations

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11-9
Solving Radical Equations
Warm Up
Lesson Presentation
Lesson Quiz
Holt Algebra 1
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• Warm Up
• Solve each equation.
• 1. 3x 5 17
• 2. 4x 1 2x 3
• 3.
• 4. (x 7)(x 4) 0
• 5. x2 11x 30 0
• 6. x2 2x 15

4
2
35
7, 4
6, 5
5, 3
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Objective
Solve radical equations.
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Vocabulary
radical equation extraneous solution
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A radical equation is an equation that contains a
variable within a radical. In this course, you
will only study radical equations that contain
square roots.
Recall that you use inverse operations to solve
equations. For nonnegative numbers, squaring and
taking the square root are inverse operations.
When an equation contains a variable within a
square root, square both sides of the equation to
solve.
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(No Transcript)
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Example 1A Solving Simple Radical Equations
Solve the equation. Check your answer.
Square both sides.
x 25
Substitute 25 for x in the original equation.
?
Simplify.
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Example 1B Solving Simple Radical Equations
Solve the equation. Check your answer.
Square both sides.
100 2x
50 x
Divide both sides by 2.
Substitute 50 for x in the original equation.
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Simplify.
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Check It Out! Example 1a
Solve the equation. Check your answer.
Square both sides.
Simplify.
Substitute 36 for x in the original equation.
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6 6
Simplify.
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Check It Out! Example 1b
Solve the equation. Check your answer.
Square both sides.
81 27x
Divide both sides by 27.
3 x
Substitute 3 for x in the original equation.
Simplify.
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Check It Out! Example 1c
Solve the equation. Check your answer.
Square both sides.
3x 1
Divide both sides by 3.
Simplify.
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Some square-root equations do not have the square
root isolated. To solve these equations, you may
have to isolate the square root before squaring
both sides. You can do this by using one or more
inverse operations.
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Example 2A Solving Simple Radical Equations
Solve the equation. Check your answer.
Add 4 to both sides.
Square both sides.
x 81
9 4 5
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5 5
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Example 2B Solving Simple Radical Equations
Solve the equation. Check your answer.
Square both sides.
x 46
Subtract 3 from both sides.
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Example 2C Solving Simple Radical Equations
Solve the equation. Check your answer.
Subtract 6 from both sides.
Square both sides.
5x 1 16
5x 15
Subtract 1 from both sides.
x 3
Divide both sides by 5.
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Example 2C Continued
Solve the equation. Check your answer.
4 6 10
?
10 10
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Check It Out! Example 2a
Solve the equation. Check your answer.
Add 2 to both sides.
Square both sides.
x 9
?
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Check It Out! Example 2b
Solve the equation. Check your answer.
Square both sides.
Subtract 7 from both sides.
x 18
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Check It Out! Example 2c
Solve the equation. Check your answer.
Add 1 to both sides.
Square both sides.
3x 9
Subtract 7 from both sides.
x 3
Divide both sides by 3.
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Check It Out! Example 2c
Solve the equation. Check your answer.
Check
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3 3
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Example 3A Solving Radical Equations by
Multiplying or Dividing
Solve the equation. Check your answer.
Method 1
Divide both sides by 4.
Square both sides.
x 64
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Example 3A Continued
Solve the equation. Check your answer.
Method 2
Square both sides.
x 64
Divide both sides by 16.
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Example 3A Continued
Solve the equation. Check your answer.
Substitute 64 for x in the original equation.
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32 32
Simplify.
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Example 3B Solving Radical Equations by
Multiplying or Dividing
Solve the equation. Check your answer.
Method 1
Multiply both sides by 2.
Square both sides.
144 x
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Example 3B Continued
Solve the equation. Check your answer.
Method 2
Square both sides.
Multiply both sides by 4.
144 x
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Example 3B Continued
Solve the equation. Check your answer.
Check
Substitute 144 for x in the original equation.
Simplify.
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12 12
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Check It Out! Example 3a
Solve the equation. Check your answer.
Method 1
Divide both sides by 2.
Square both sides.
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Check It Out! Example 3a Continued
Solve the equation. Check your answer.
Method 2
Square both sides.
Divide both sides by 4.
x 121
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Check It Out! Example 3a Continued
Solve the equation. Check your answer.
Substitute 121 for x in the original equation.
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Simplify.
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Check It Out! Example 3b
Solve the equation. Check your answer.
Method 1
Multiply both sides by 4.
Square both sides.
64 x
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Check It Out! Example 3b Continued
Solve the equation. Check your answer.
Method 2
Square both sides.
Multiply both sides by 16.
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Check It Out! Example 3b Continued
Solve the equation. Check your answer.
Substitute 64 for x in the original equation.
Simplify.
?
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Check It Out! Example 3c
Solve the equation. Check your answer.
Method 1
Multiply both sides by 5.
Square both sides.
Divide both sides by 4.
x 100
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Check It Out! Example 3c Continued
Solve the equation. Check your answer.
Method 2
Square both sides.
Multiply both sides by 25.
4x 400
Divide both sides by 4.
x 100
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Check It Out! Example 3c Continued
Solve the equation. Check your answer.
Substitute 100 for x in the original equation.
Simplify.
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4 4
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Example 4A Solving Radical Equations with Square
Roots on Both Sides
Solve the equation. Check your answer.
Square both sides.
2x 1 x 7
Add 1 to both sides and subtract x from both
sides.
x 8
?
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Example 4B Solving Radical Equations with Square
Roots on Both Sides
Solve the equation. Check your answer.
Square both sides.
5x 4 6
Add 4 to both sides.
5x 10
Divide both sides by 2.
x 2
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Example 4B Continued
Solve the equation. Check your answer.
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0 0
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Check It Out! Example 4a
Solve the equation. Check your answer.
Square both sides.
Subtract x from both sides and subtract 2 from
both sides.
2x 4
x 2
Divide both sides by 2.
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Check It Out! Example 4a Continued
Solve the equation. Check your answer.
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Check It Out! Example 4b
Solve the equation. Check your answer.
Square both sides.
2x 5 6
2x 11
Add 5 to both sides.
Divide both sides by 2.
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Check It Out! Example 4b Continued
Solve the equation. Check your answer.
?
0 0
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Squaring both sides of an equation may result in
an extraneous solutiona number that is not a
solution of the original equation.
Suppose your original equation is x 3.
x 3
x2 9
Square both sides. Now you have a new equation.
Solve this new equation for x by taking the
square root of both sides.
x 3 or x 3
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Now there are two solutions. One (x 3) is the
original equation. The other (x 3) is
extraneousit is not a solution of the original
equation. Because of extraneous solutions, it is
important to check your answers.
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Example 5A Extraneous Solutions
Solve Check your answer.
Subtract 12 from each sides.
Square both sides
6x 36
Divide both sides by 6.
x 6
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Example 5A Continued
Solve Check your answer.
Substitute 6 for x in the equation.
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18 6
6 does not check. There is no solution.
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Example 5B Extraneous Solutions
Square both sides
x2 2x 3
x2 2x 3 0
Write in standard form.
(x 3)(x 1) 0
Factor.
x 3 0 or x 1 0
Zero-Product Property
x 3 or x 1
Solve for x.
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Example 5B Continued
Substitute 1 for x in the equation.
Substitute 3 for x in the equation.
1 does not check it is extraneous. The only
solution is 3.
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Check It Out! Example 5a
Solve the equation. Check your answer.
Subtract 11 from both sides.
Square both sides.
x 5
Simplify.
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Check It Out! Example 5a Continued
Solve the equation. Check your answer.
Substitute 5 for x in the equation.
No solution. The answer is extraneous.
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Check It Out! Example 5b
Solve the equation. Check your answer.
Square both sides
x2 3x 2
x2 3x 2 0
Write in standard form.
(x 1)(x 2) 0
Factor.
Zero-Product Property
x 1 0 or x 2 0
x 1 or x 2
Solve for x.
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Check It Out! Example 5b Continued
Solve the equation. Check your answer.
Substitute 1 for x in the equation.
Substitute 2 for x in the equation.
No solutions. Both answers are extraneous.
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Check It Out! Example 5c
Solve the equation. Check your answer.
Square both sides
x2 5x 4 0
Write in standard form.
(x 1)(x 4) 0
Factor.
X 1 0 or x 4 0
Zero-Product Property.
x 1 or x 4
Solve for x.
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Check It Out! Example 5c Continued
Solve the equation. Check your answer.
Substitute 1 for x in the equation.
Substitute 4 for x in the equation.
1 does not check it is extraneous. The only
solution is 4.
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Example 6 Geometry Application
Use the formula for area of a triangle.
Simplify.
Divide both sides by 4.
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Example 6 Continued
Square both sides.
81 x 1
82 x
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Example 6 Continued
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36 36
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Check It Out! Example 6
A rectangle has an area of 15 cm2. Its width is 5
cm, and its length is ( ) cm. What is
the value of x? What is the length of the
rectangle?
A lw
Use the formula for area of a rectangle.
Divide both sides by 5.
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Check It Out! Example 6 Continued
A rectangle has an area of 15 cm2. Its width is 5
cm, and its length is ( ) cm. What is
the value of x? What is the length of the
rectangle?
Square both sides.
8 x
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Check It Out! Example 6 Continued
A rectangle has an area of 15 cm2. Its width is 5
cm, and its length is ( ) cm. What is
the value of x? What is the length of the
rectangle?
Substitute 8 for x.
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15 15
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Lesson Quiz Part I
Solve each equation. Check your answer.
1.
2.
36
45
no solution
3.
4.
11
5.
4
6.
4
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Lesson Quiz Part II
253 16 ft