11-5 Solving Radical Equations

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Lesson Presentation

California Standards

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11-5 Solving Radical Equations

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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

11-5 Solving Radical Equations

Extension of 2.0 Students understand and use such operations as taking the opposite, finding the reciprocal, taking a root, and raising to a fractional power. They understand and use the rules of exponents.

California Standards

11-5 Solving Radical Equations

radical equation

Vocabulary

11-5 Solving Radical Equations

A radical equation is an equation that contains a variable within a radical. In this chapter, you will study radical equations that contain only 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, you can solve by squaring both sides of the equation.

11-5 Solving Radical Equations

=

11-5 Solving Radical Equations

Additional Example 1A: Solving Simple Radical Equations

Solve the equation. Check your answer.

x = 25

Square both sides.

Substitute 25 for x in the original equation.

55 5

Simplify.

Check

11-5 Solving Radical Equations

Additional Example 1B: Solving Simple Radical Equations

Solve the equation. Check your answer.

100 = 2x

50 = x

Square both sides.

Divide both sides by 2.

Check

10 10

Substitute 50 for x in the original equation.

Simplify.

11-5 Solving Radical EquationsCheck It Out! Example 1a

Solve the equation. Check your answer.

Square both sides.

6 6

Check

Substitute 36 for x in the original equation.

Simplify.

Simplify.

11-5 Solving Radical EquationsCheck It Out! Example 1b

Solve the equation. Check your answer.

81 = 27x

3 = x

Square both sides.

Divide both sides by 27.

Substitute 3 for x in the original equation.

Simplify.

Check

11-5 Solving Radical EquationsCheck It Out! Example 1c

Solve the equation. Check your answer.

3x = 1

Check

Square both sides.

Divide both sides by 3.

Simplify.

Substitute for x in the original equation.

11-5 Solving Radical EquationsCheck It Out! Example 1d

Solve the equation. Check your answer.

x = 12

Square both sides.

Multiply both sides by 3.

11-5 Solving Radical Equations

Check It Out! Example 1d continued

Solve the equation. Check your answer.

Check

Substitute 12 for x.

Simplify.

2 2

11-5 Solving Radical Equations

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.

11-5 Solving Radical EquationsAdditional Example 2A: Solving Simple Radical Equations

Solve the equation. Check your answer.

x = 81

Add 4 to both sides.

Square both sides.

Check

9 – 4 55 5

11-5 Solving Radical Equations

Additional Example 2B: Solving Simple Radical Equations

Solve the equation. Check your answer.

x = 46 Subtract 3 from both sides.

Square both sides.

Check

7 7

11-5 Solving Radical Equations

Additional Example 2C: Solving Simple Radical Equations

Solve the equation. Check your answer.

5x + 1 = 16

5x = 15

x = 3

Subtract 6 from both sides.

Square both sides.

Subtract 1 from both sides.

Divide both sides by 5.

11-5 Solving Radical Equations

Additional Example 2C Continued

Solve the equation. Check your answer.

4 + 6 10

10 10

Check

11-5 Solving Radical EquationsCheck It Out! Example 2a

Solve the equation. Check your answer.

x = 9

Add 2 to both sides.

Square both sides.

Check

1 1

11-5 Solving Radical EquationsCheck It Out! Example 2b

Solve the equation. Check your answer.

x = 18 Subtract 7 from both sides.

Square both sides.

Check

5 5

11-5 Solving Radical Equations

Check It Out! Example 2c

Solve the equation. Check your answer.

3x = 9

x = 3

Add 1 to both sides.

Square both sides.

Subtract 7 from both sides.

Divide both sides by 3.

11-5 Solving Radical Equations

Check It Out! Example 2c Continued

Solve the equation. Check your answer.

3 3

Check

11-5 Solving Radical Equations

Additional Example 3A: Solving Radical Equations by Multiplying or Dividing

Solve the equation. Check your answer.

Method 1

x = 64

Divide both sides by 4.

Square both sides.

11-5 Solving Radical Equations

Additional Example 3A Continued

Solve the equation. Check your answer.

Method 2

x = 64

Square both sides.

Divide both sides by 16.

11-5 Solving Radical Equations

Additional Example 3A Continued

Solve the equation. Check your answer.

32 32

Check

Substitute 64 for x in the original equation.

Simplify.

11-5 Solving Radical EquationsAdditional Example 3B: Solving Radical Equations by

Multiplying or Dividing

Solve the equation. Check your answer.

Method 1

144 = x

Square both sides.

Multiply both sides by 2.

11-5 Solving Radical Equations

Additional Example 3B Continued

Solve the equation. Check your answer.

Method 2

Square both sides.

Multiply both sides by 4.

144 = x

11-5 Solving Radical Equations

Additional Example 3B Continued

Solve the equation. Check your answer.

6 6

Check

Substitute 144 for x in the original equation.

Simplify.

11-5 Solving Radical Equations

Check It Out! Example 3a

Solve the equation. Check your answer.

Method 1

Square both sides.

Divide both sides by 2.

11-5 Solving Radical Equations

Check It Out! Example 3a Continued

Solve the equation. Check your answer.

Method 2

Square both sides.

Divide both sides by 4.x = 121

11-5 Solving Radical Equations

Check It Out! Example 3a Continued

Solve the equation. Check your answer.

Substitute 121 for x in the original equation.

Simplify.

Check

11-5 Solving Radical Equations

Check It Out! Example 3b

Solve the equation. Check your answer.

Method 1

Square both sides.

Multiply both sides by 4.

64 = x

11-5 Solving Radical EquationsCheck It Out! Example 3b Continued

Solve the equation. Check your answer.

Method 2

Square both sides.

Multiply both sides by 16.

11-5 Solving Radical Equations

Check It Out! Example 3b Continued

Solve the equation. Check your answer.

Substitute 64 for x in the original equation.

Simplify.

Check

11-5 Solving Radical EquationsCheck It Out! Example 3c

Solve the equation. Check your answer.

Method 1

Square both sides.

Multiply both sides by 5.

x = 100Divide both sides by 4.

11-5 Solving Radical EquationsCheck It Out! Example 3c Continued

Solve the equation. Check your answer.

Method 2

Square both sides.

Multiply both sides by 25.4x = 400

x = 100 Divide both sides by 4.

11-5 Solving Radical Equations

Check It Out! Example 3c Continued

Solve the equation. Check your answer.

Substitute 100 for x in the original equation.

Simplify.

Check

4 4

4

11-5 Solving Radical EquationsAdditional Example 4A: Solving Radical Equations with

Square Roots on Both Sides

Solve the equation. Check your answer.

2x – 1 = x + 7

x = 8

Square both sides.

Add 1 to both sides and subtract x from both sides.

Check

11-5 Solving Radical Equations

Additional Example 4B: Solving Radical Equations with Square Roots on Both Sides

Solve the equation. Check your answer.

5x – 4 = 6

5x = 10

x = 2

Add to both sides.

Square both sides.

Add 4 to both sides.

Divide both sides by 5.

11-5 Solving Radical Equations

Additional Example 4B Continued

Solve the equation. Check your answer.

Check

0 0

11-5 Solving Radical Equations

Check It Out! Example 4a

Solve the equation. Check your answer.

2x = 4

x = 2

Square both sides.

Subtract x from both sides and subtract 2 from both sides.

Divide both sides by 2.

11-5 Solving Radical Equations

Check It Out! Example 4a Continued

Solve the equation. Check your answer.

Check

11-5 Solving Radical Equations

Check It Out! Example 4b

Solve the equation. Check your answer.

2x – 5 = 6

2x = 11

Add to both sides.

Square both sides.

Add 5 to both sides.

Divide both sides by 2.

11-5 Solving Radical EquationsCheck It Out! Example 4b Continued

Solve the equation. Check your answer.

Check

0 0

11-5 Solving Radical Equations

Squaring both sides of an equation may result in an extraneous solution.

Suppose your original equation is x = 3.

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

x2 = 9

x = 3 or x = –3

11-5 Solving Radical Equations

Now there are two solutions of the new equation. One (x = 3) is the original equation. The other (x = –3) is extraneous–it is not a solution of the original equation. Because of extraneous solutions, it is especially important to check your answers to radical equations.

11-5 Solving Radical Equations

Additional Example 5A: Extraneous Solutions

Solve Check your answer.

Square both sides

Divide both sides by 6.

Subtract 12 from each sides.

6x = 36

x = 6

11-5 Solving Radical Equations

Additional Example 5A Continued

Solve Check your answer.

Substitute 6 for x in the equation.

Check

6 does not check; Ø.

18 6

11-5 Solving Radical Equations

Additional Example 5B: Extraneous Solutions

Solve Check your answer.

x2 – 2x – 3 = 0

(x – 3)(x + 1) = 0

x – 3 = 0 or x + 1 = 0

x = 3 or x = –1

Square both sides

Write in standard form.

Factor.

Zero-Product Property

Solve for x.

x2 = 2x + 3

11-5 Solving Radical EquationsAdditional Example 5B Continued

Solve Check your answer.

Substitute –1 for x in the equation.

Check

–1 1

Substitute 3 for x in the equation.

3 3–1 does not check; it is extraneous. The only solution is 3.

11-5 Solving Radical Equations

Check It Out! Example 5a

Solve the equation. Check your answer.

x = 5

Subtract 11 from both sides.

Square both sides.

Simplify.

11-5 Solving Radical EquationsCheck It Out! Example 5a Continued

Solve the equation. Check your answer.

Substitute 5 for x in the equation.

16 6

Check

The answer is extraneous.

11-5 Solving Radical EquationsCheck It Out! Example 5b

Solve the equation. Check your answer.

x2 = –3x – 2

x2 + 3x + 2 = 0

(x + 1)(x + 2) = 0

x = –1 or x = –2

Square both sides

Write in standard form.

Factor.

Zero-Product Property

Solve for x.

x + 1 = 0 or x + 2 = 0

11-5 Solving Radical Equations

Check It Out! Example 5b Continued

Solve the equation. Check your answer.

Substitute –1 for x in the equation.

Check

–2 2Substitute –2 for x in the

equation.

Both answers are extraneous.

11-5 Solving Radical EquationsCheck It Out! Example 5c

Solve the equation. Check your answer.

x2 – 5x + 4 = 0

(x – 1)(x – 4) = 0

x = 1 or x = 4

x – 1 = 0 or x – 4 = 0

Square both sides.

Factor.

Zero-Product Property

Solve for x.

x2 – 4x + 4 = x Subtract x from both sides.

11-5 Solving Radical EquationsCheck 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.

Check

2 2

11-5 Solving Radical EquationsAdditional Example 6: Geometry Application

8 ft

Use the formula for area of a triangle.

Substitute 8 for b, 36 for A, and for h.

Divide both sides by 4.

A triangle has an area of 36 square feet, its base is 8 feet, and its height is feet. What is the value of x? What is the height of the triangle?

Simplify.

11-5 Solving Radical Equations

Additional Example 6 Continued

8 ft

82 = x

Square both sides.

A triangle has an area of 36 square feet, its base is 8 feet, and its height is feet. What is the value of x? What is the height of the triangle?

81 = x – 1

11-5 Solving Radical Equations

8 ft

Check

36 36The value of x is 82. The height of the triangle is 9 feet.

Substitute 82 for x.

Additional Example 6 ContinuedA triangle has an area of 36 square feet, its base is 8 feet, and its height is feet. What is the value of x? What is the height of the triangle?

11-5 Solving Radical EquationsCheck 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?

5

A = lw Use the formula for area of a rectangle.

Divide both sides by 5.

Substitute 5 for w, 15 for A, and for l.

11-5 Solving Radical EquationsCheck It Out! Example 6 Continued

8 = x

Square both sides.

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?

5

The value of x is 8. The length of the rectangle is cm.

11-5 Solving Radical Equations

Check A = lw

15 15

Substitute 8 for x.

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?

5

11-5 Solving Radical Equations

Lesson Quiz

Solve each equation. Check your answer.

1.

3.

5.

2.

4.

6.

36 45

ø 11

4 4

7. A triangle has an area of 48 square feet, its base is 6 feet, and its height is feet. What is the value of x? What is the height of the triangle?

253; 16 ft