Beginning Algebra, 11th Edition

SECTION 8.6 Solving Equations with Radicals 533

Solving a Radical Equation

Step 1 Isolate a radical.Arrange the terms so that a radical is isolated on

one side of the equation.

Step 2 Square each side.

Step 3 Combine like terms.

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

Step 5 Solve the equation.Find all proposed solutions.

Step 6 Check all proposed solutionsin the original equation.

Using the Squaring Property with a Quadratic Expression

Solve

Step 1 The radical is already isolated on the right side of the equation.

Step 2 Square each side.

Squaring property

Step 3 Subtract.

Step 4 This step is not needed.

Step 5 Subtract 10.

Proposed solution

Step 6 CHECK Original equation

Let

Multiply.

• 2 = 2 False

- 2  24 - 10 + 10

• 2  21 - 222 + 51 - 22 + 10 x=-2.

x= 2 x^2 + 5 x+ 10

• 2 =x


• 10 = 5 x

0 = 5 x+ 10 x^2

x^2 =x^2 + 5 x+ (^10) A 2 x^2 + 5 x+ (^10) B^2 =x^2 + 5 x+ 10
x^2 = A 2 x^2 + 5 x+ (^10) B
2
x= 2 x^2 + 5 x+ 10.
NOW TRY EXAMPLE 4
EXERCISE 4
Solve t= 2 t^2 + 3 t+ 9.
NOW TRY ANSWER

Since substituting for xleads to a false result, the equation has no solution, and

the solution set is 0. NOW TRY

- 2

Using the Squaring Property when One Side Has Two Terms

Solve

Square each side.

2 x- 3 =x^2 - 6 x+ 9 1 x-y 22 =x^2 - 2 xy+y^2

(^) A 22 x- (^3) B
2
= 1 x- 322
22 x- 3 =x- 3
22 x- 3 =x-3.
EXAMPLE 5
OBJECTIVE 3 Solve equations by squaring a binomial.Recall the rules for
squaring binomials from Section 5.6.
and
We apply the second pattern in Example 5 when finding.
=x^2 - 6 x+ 9
=x^2 - 2 x 132 + 32
1 x- 322
1 x- 322
1 xy 22 x^2  2 xyy^21 xy 22 x^2  2 xyy^2
Remember the middle
term when squaring.
The principal square
root of a quantity
cannotbe negative.