UNIT 6.5 SOLVING SQUARE UNIT 6.5 SOLVING SQUARE ROOT AND OTHER RADICALROOT AND OTHER RADICAL

EXPRESSIONSEXPRESSIONS

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

Solve radical equations.

Objective

radical equationextraneous solution

Vocabulary

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.

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

Example 1B: Solving Simple Radical Equations

Solve the equation. Check your answer.

100 = 2x50 = x

Square both sides.

Divide both sides by 2.

Check

10 10

Substitute 50 for x in the original equation.

Simplify.

Check 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.

Check It Out! Example 1b

Solve the equation. Check your answer.

81 = 27x3 = x

Square both sides.

Divide both sides by 27.

Substitute 3 for x in the original equation.

Simplify.

Check

Check 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.

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.

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

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

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.

Example 2C Continued

Solve the equation. Check your answer.

4 + 6 1010 10

Check

Check It Out! Example 2a

Solve the equation. Check your answer.

x = 9

Add 2 to both sides.

Square both sides.

Check

1 1

Check It Out! Example 2b

Solve the equation. Check your answer.

x = 18Subtract 7 from both sides.

Square both sides.

Check

5 5

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.

Check It Out! Example 2c

Solve the equation. Check your answer.

3 3

Check

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.

Example 3A Continued

Solve the equation. Check your answer.

Method 2

x = 64

Square both sides.

Divide both sides by 16.

Example 3A Continued

Solve the equation. Check your answer.

32 32

Check

Substitute 64 for x in the original equation.

Simplify.

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.

Example 3B Continued

Solve the equation. Check your answer.

Method 2

Square both sides.

Multiply both sides by 4.

144 = x

Example 3B Continued

Solve the equation. Check your answer.

6 6

Check

Substitute 144 for x in the original equation.

Simplify.

Check It Out! Example 3a

Solve the equation. Check your answer.

Method 1

Square both sides.

Divide both sides by 2.

Check It Out! Example 3a Continued

Solve the equation. Check your answer.

Method 2

Square both sides.

Divide both sides by 4.

x = 121

Check It Out! Example 3a Continued

Solve the equation. Check your answer.

Substitute 121 for x in the original equation.

Simplify.

Check

Check It Out! Example 3b

Solve the equation. Check your answer.

Method 1

Square both sides.

Multiply both sides by 4.

64 = x

Check It Out! Example 3b Continued

Solve the equation. Check your answer.

Method 2

Square both sides.

Multiply both sides by 16.

Check It Out! Example 3b Continued

Solve the equation. Check your answer.

Substitute 64 for x in the original equation.

Simplify.

Check

Check 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.

Check 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.

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

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

Solve the equation. Check your answer.

2x – 1 = x + 7x = 8

Square both sides.

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

Check

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

Solve the equation. Check your answer.

5x – 4 = 65x = 10x = 2

Add to both sides.

Square both sides.

Add 4 to both sides.

Divide both sides by 2.

Example 4B Continued

Solve the equation. Check your answer.

Check

0 0

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.

Check It Out! Example 4a Continued

Solve the equation. Check your answer.

Check

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.

Check It Out! Example 4b Continued

Solve the equation. Check your answer.

Check

0 0

Squaring both sides of an equation may result in an extraneous solution—a number that is not a solution of the original equation.

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

Now there are two solutions. 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 important to check your answers.

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

Example 5A Continued

Solve Check your answer.

Substitute 6 for x in the equation.

Check

6 does not check. There is no solution.

18 6

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

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.

Check It Out! Example 5a

Solve the equation. Check your answer.

x = 5

Subtract 11 from both sides.

Square both sides.

Simplify.

Check It Out! Example 5a Continued

Solve the equation. Check your answer.

Substitute 5 for x in the equation.

16 6

Check

No solution. The answer is extraneous.

Check 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

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.

No solutions. Both answers are extraneous.

Check 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

Write in standard form.

Factor.

Zero-Product Property.

Solve for x.

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.

Check

2 2

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.

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

8 ft

Check

36 36

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

Substitute 82 for x.

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?

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?

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.

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

8 = x

Square both sides.

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

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

Lesson Quiz: Part I

Solve each equation. Check your answer.

1.

3.

5.

2.

4.

6.

36 45

no solution 11

4 4

Lesson Quiz: Part II

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

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