11-5 Solving Radical Equations Warm Up Warm Up Lesson Presentation Lesson Presentation California...

11-5 Solving Radical Equations

Warm UpWarm Up

Lesson Presentation

California Standards

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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


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


radical equation

Vocabulary


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.


=


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


Additional Example 1B: Solving Simple Radical Equations


100 = 2x

50 = x

Square both sides.

Divide both sides by 2.

Check

10 10


Simplify.

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


Square both sides.

6 6

Check


Simplify.

Simplify.

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


81 = 27x

3 = x

Square both sides.



Simplify.

Check

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


3x = 1

Check

Square both sides.


Simplify.

Substitute for x in the original equation.

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


x = 12

Square both sides.

Multiply both sides by 3.


Check It Out! Example 1d continued


Check

Substitute 12 for x.

Simplify.

2 2


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


x = 81

Add 4 to both sides.

Square both sides.

Check

9 – 4 55 5


Additional Example 2B: Solving Simple Radical Equations


x = 46 Subtract 3 from both sides.

Square both sides.

Check

7 7


Additional Example 2C: Solving Simple Radical Equations


5x + 1 = 16

5x = 15

x = 3

Subtract 6 from both sides.

Square both sides.




Additional Example 2C Continued


4 + 6 10

10 10

Check

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


x = 9


Square both sides.

Check

1 1



x = 18 Subtract 7 from both sides.

Square both sides.

Check

5 5


Check It Out! Example 2c


3x = 9

x = 3


Square both sides.




Check It Out! Example 2c Continued


3 3

Check


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


Method 1

x = 64


Square both sides.


Additional Example 3A Continued


Method 2

x = 64

Square both sides.





32 32

Check


Simplify.

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

Multiplying or Dividing


Method 1

144 = x

Square both sides.



Additional Example 3B Continued


Method 2

Square both sides.


144 = x




6 6

Check


Simplify.


Check It Out! Example 3a


Method 1

Square both sides.



Check It Out! Example 3a Continued


Method 2

Square both sides.

Divide both sides by 4.x = 121





Simplify.

Check


Check It Out! Example 3b


Method 1

Square both sides.


64 = x

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


Method 2

Square both sides.



Check It Out! Example 3b Continued



Simplify.

Check



Method 1

Square both sides.


x = 100Divide both sides by 4.

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


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



Simplify.

Check

4 4

4

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

Square Roots on Both Sides


2x – 1 = x + 7

x = 8

Square both sides.

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

Check


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


5x – 4 = 6

5x = 10

x = 2

Add to both sides.

Square both sides.






Check

0 0




2x = 4

x = 2

Square both sides.

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





Check


Check It Out! Example 4b


2x – 5 = 6

2x = 11

Add to both sides.

Square both sides.



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


Check

0 0


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


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.


Additional Example 5A: Extraneous Solutions

Solve Check your answer.

Square both sides


Subtract 12 from each sides.

6x = 36

x = 6




Substitute 6 for x in the equation.

Check

6 does not check; Ø.

18 6


Additional Example 5B: Extraneous Solutions


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


Substitute –1 for x in the equation.

Check

–1 1


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




x = 5


Square both sides.

Simplify.

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



16 6

Check

The answer is extraneous.



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.


Solve for x.

x + 1 = 0 or x + 2 = 0


Check It Out! Example 5b Continued


Substitute –1 for x in the equation.

Check

–2 2Substitute –2 for x in the

equation.

Both answers are extraneous.



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.


Solve for x.

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

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




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.


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.


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


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.


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.


5

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


5


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

11-5 Solving Radical Equations Warm Up Warm Up Lesson Presentation Lesson Presentation California...

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