Intermediate Algebra (11th edition)

Using the Power Rule

Solve

Step 1 25 x- 1 =- 3

25 x- 1 + 3 = 0.

EXAMPLE 2

SECTION 8.6 Solving Equations with Radicals 469

Solving an Equation with Radicals

Step 1 Isolate the radical.Make sure that one radical term is alone on one side

of the equation.

Step 2 Apply the power rule.Raise each side of the equation to a power that is

the same as the index of the radical.

Step 3 Solvethe resulting equation. If it still contains a radical, repeat Steps 1

and 2.

Step 4 Checkall proposed solutions in the original equation.

NOW TRY

EXERCISE 2

Solve 23 x+ 4 + 5 =0.

Use the following steps to solve equations with radicals.

Step 2 Square each side.

Step 3 Apply the exponents.

Add 1.

Divide by 5.

Step 4 CHECK Original equation

Let.

3 + 3 = 0 False

25 # 2 - 1 + 3  0 x= 2

25 x- 1 + 3 = 0

x= 2

5 x= 10

5 x- 1 = 9

A 25 x- 1 B

2

= 1 - 322

To isolate the radical on one side,

subtract 3 from each side.

This false result shows that the proposedsolution 2 is nota solution of the original

equation. It is extraneous. The solution set is 0. NOW TRY

NOTE We could have determined after Step 1 that the equation in Example 2has no

solution because the expression on the left cannot be negative. (Why?)

OBJECTIVE 2 Solve radical equations that require additional steps.The

next examples involve finding the square of a binomial. Recall the rule from Section 5.4.

Using the Power Rule (Squaring a Binomial)

Solve

Step 1 The radical is alone on the left side of the equation.

Step 2 Square each side. The square of is

Twice the product of 2 and x

4 - x=x^2 + 4 x+ 4

A 24 - xB

2

= 1 x+ 222

x+ 2 1 x+ 222 =x^2 + 21 x 2122 + 4.

24 - x=x+ 2.

EXAMPLE 3

1 xy 22 x^2  2 xyy^2

NOW TRY ANSWER

2. 
   

Remember the

middle term.

Be sure to check the

proposed solution.