Chapter 8 Section 6 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley.

Chapter Chapter 88Section Section 66

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley


Solving Equations with Radicals

Solve radical equations having square root radicals.Identify equations with no solutions.Solve equations by squaring a binomial.Solve radical equations having cube root radicals.

11

44

33

22

8.68.68.68.6


Solving Equations with Radicals.

A radical equation is an equation having a variable in the radicand, such as

Slide 8.6 - 3

1 3x or 3 8 9x x


Objective 11

Slide 8.6 - 4

Solve radical equations having square root radicals.


To solve radical equations having square root radicals, we need a new property, called the squaring property of equality.

Be very careful with the squaring property: Using this property can give a new equation with more solutions than the original equation has. Because of this possibility, checking is an essential part of the process. All proposed solutions

from the squared equation must be checked in the original equation.

Slide 8.6 - 5

Solve radical equations having square root radicals.

If each side of a given equation is squared, then all solutions of the original equation are among the solutions of the squared equation.


EXAMPLE 1

Solve.

Solution:

Using the Squaring Property of Equality

Slide 8.6 - 6

It is important to note that even though the algebraic work may be done perfectly, the answer produced may not make the original equation true.

9 4x

229 4x

9 16x 9 169 9x

7x 7x 7


Solve.

EXAMPLE 2 Using the Squaring Property with a Radical on Each Side

Slide 8.6 - 7

Solution:

3 9 2x x

2 2

3 9 2x x

3 9 4x x

3 33 9 4xx x x

9x

9


Objective 22

Identify equations with no solutions.

Slide 8.6 - 8


EXAMPLE 3

Solution:

Using the Squaring Property when One Side Is Negative

Slide 8.6 - 9

Solve.4x

2 2

4x

16x 16 4

4 4

4x

False

Because represents the principal or nonnegative square root of x in Example 3, we might have seen immediately that there is no solution.

x

Check:


Use the following steps when solving an equation with radicals.

Step 1 Isolate a radical. Arrange the terms so that a radical is isolated on one side of the equation.

Solving a Radical Equation.

Slide 8.6 - 10

Step 6 Check all proposed solutions in the original equation.

Step 5 Solve the equation. Find all proposed solutions.

Step 4 Repeat Steps 1-3 if there is still a term with a radical.

Step 3 Combine like terms.

Step 2 Square both sides.


EXAMPLE 4

Solution:

Using the Squaring Property with a Quadratic Expression

Slide 8.6 - 11

Solve 2 4 16.x x x

22 2 4 16x x x

2 22 24 16x xx x x

44 40 16xx x 4 1

4 4

6x

4x Since x must be a positive number the solution set is Ø.


Objective 33

Slide 8.6 - 12

Solve equations by squaring a binomial.


EXAMPLE 5

Solve

Solution:

Using the Squaring Property when One Side Has Two Terms

Slide 8.6 - 13

2 1 10 9.x x

222 1 10 9x x

2 10 94 4 1 10 99 10x x xx x 24 14 8 0x x

2 1 2 8 0x x

2 8 0x 2 1 0x 4x 1

2x

Since x must be positive the solution set is {4}.

or


Solve.

EXAMPLE 6 Rewriting an Equation before using the Squaring Property

Slide 8.6 - 14

Solution:

25 6x x

625 66x x

2 2

25 6x x 225 12 325 256x x xx x 20 13 36x x

0 4 9x x 0 9x 0 4x

9x 4x

The solution set is {4,9}.

or


Solve equations by squaring a binomial.

Errors often occur when both sides of an equation are squared. For instance, when both sides of

are squared, the entire binomial 2x + 1 must be squared to get 4x2 + 4x + 1. It is incorrect to square the 2x and the 1 separately to get 4x2 + 1.

Slide 8.6 - 15

9 2 1x x


EXAMPLE 7 Using the Squaring Property Twice

Slide 8.6 - 16

Solve.

Solution:

1 4 1x x

1 1 4x x

2 2

1 1 4x x

1 1 2 4 4x x x

224 2 4x

16 4 16x 32

4 4

4x

8x The solution set is {8}.


Objective 44

Slide 8.6 - 17

Solve radical equations having cube root radicals.


Solve radical equations having cube root radicals.

Slide 8.6 - 18

We can extend the concept of raising both sides of an equation to a power in order to solve radical equations with cube roots.


EXAMPLE 8Solving Equations with Cube Root Radicals

Slide 8.6 - 19

Solve each equation.

Solution:

3 37 4 2x x 3 2 3 26 27x x

3 3

3 2 3 26 27x x 2 26 27x x

20 26 27x 0 27 1x x

0 27x 0 1x 27x 1x

3 33 37 4 2x x

7 4 2x x 3 2

3 3

x

2

3x

2

3

27,1

or

Chapter 8 Section 6 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley.

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