Chapter 8 Section 6 - Ligon NC Math Grade 8, Math 2 and 3danielsroar.weebly.com/uploads/5/3/1/4/5314494/... · Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Chapter 8

Section 6

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.

1

4

3

2

8.6


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 1

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


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 2

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.

4x2 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:


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 3

Slide 8.6 - 12

Solve equations by squaring a

binomial.


EXAMPLE 5

Solve

Solution:


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 0x2 1 0x

4x1

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 x2 2

25 6x x

225 12 325 256x x xx x20 13 36x x

0 4 9x x

0 9x0 4x

9x4x

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 x2 2

1 1 4x x

1 1 2 4 4x x x2

24 2 4x

16 4 16x

32

4 4

4x

8xThe solution set is {8}.


Objective 4

Slide 8.6 - 17


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 8 Solving Equations with

Cube Root Radicals

Slide 8.6 - 19

Solve each equation.

Solution:

3 37 4 2x x 3 2 3 26 27x x3 3

3 2 3 26 27x x

2 26 27x x20 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

327,1

or

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Chapter 8 Section 6 - Ligon NC Math Grade 8, Math 2 and 3danielsroar.weebly.com/uploads/5/3/1/4/5314494/... · Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley