EXAMPLE 1 Solve a quadratic equation by finding square roots

Solve x2 – 8x + 16 = 25.

x2 – 8x + 16 = 25 Write original equation.

(x – 4)2 = 25 Write left side as a binomial squared.

x – 4 = +5 Take square roots of each side.

x = 4 + 5 Solve for x.

The solutions are 4 + 5 = 9 and 4 –5 = – 1.

ANSWER

EXAMPLE 2 Make a perfect square trinomial

Find the value of c that makes x2 + 16x + c a perfect square trinomial. Then write the expression as the square of a binomial.

SOLUTION

STEP 1

Find half the coefficient of x.

STEP 2

162 = 8

Square the result of Step 1. 82 = 64

STEP 3

Replace c with the result of Step 2. x2 + 16x + 64

EXAMPLE 2 Make a perfect square trinomial

The trinomial x2 + 16x + c is a perfect square when c = 64. Then x2 + 16x + 64 = (x + 8)(x + 8) = (x + 8)2.

ANSWER

GUIDED PRACTICE for Examples 1 and 2

1. x2 + 6x + 9 = 36.

x2 + 6x + 9 = 36 Write original equation.

(x + 3)2 = 36 Write left side as a binomial squared.

x + 3 = + 6 Take square roots of each side.

x = – 3 + 6 Solve for x.

The solutions are – 3 + 6 = 3 and – 3 – 6 = – 9.

ANSWER

Solve the equation by finding square roots.

GUIDED PRACTICE for Examples 1 and 2

2. x2 – 10x + 25 = 1.

x2 – 10x + 25 = 1 Write original equation.

(x – 5)2 = 1 Write left side as a binomial squared.

x – 5 = + 1 Take square roots of each side.

x = 5 + 1 Solve for x.

The solutions are 5 – 1 = 4 and 5 + 1 = 6.

ANSWER

GUIDED PRACTICE for Examples 1 and 2

3. x2 – 24x + 144 = 100.

x2 – 24x + 144 = 100 Write original equation.

(x – 12)2 = 100 Write left side as a binomial squared.

x – 12 = + 10 Take square roots of each side.

x = 12 + 10 Solve for x.

The solutions are 12 – 10 = 2 and 12 + 10 = 22.

ANSWER

GUIDED PRACTICE for Examples 1 and 2

4.

x2 + 14x + c

x2 7x

7x 49

x

x

7

7

SOLUTION

STEP 1

Find half the coefficient of x.

STEP 2

142 = 7

Square the result of Step 1. 72 = 49

STEP 3

Replace c with the result of Step 2. x2 + 14x + 49

Find the value of c that makes the expression a perfect square trinomial.Then write the expression as the square of a binomial.

GUIDED PRACTICE for Examples 1 and 2

The trinomial x2 + 14x + c is a perfect square when c = 49.

Then x2 + 14x + 49 = (x + 7)(x + 7) = (x + 7)2.

ANSWER

GUIDED PRACTICE for Examples 1 and 2

5.

x2 + 22x + c

x2 11x

11x 121

x

x

11

11

SOLUTION

STEP 1

Find half the coefficient of x.

STEP 2

222 = 22

Square the result of Step 1. 112 = 121

STEP 3

Replace c with the result of Step 2. x2 + 22x + 121

GUIDED PRACTICE for Examples 1 and 2

The trinomial x2 + 22x + c is a perfect square when c = 144.

Then x2 + 22x + 144 = (x + 11)(x + 11) = (x + 11)2.

ANSWER

GUIDED PRACTICE for Examples 1 and 2

6.

x2 – 9x + c

x2x

x

SOLUTION

STEP 1

Find half the coefficient of x.

STEP 2

STEP 3

Replace c with the result of Step 2.

92

– x

92

– x

814

92

–

92

–

Square the result of Step 1.

x2 – 9x + 814

92

92=

92( ) =

814

GUIDED PRACTICE for Examples 1 and 2

ANSWER

The trinomial x2 – 9x + c is a perfect square when c = .

Then x2 – 9x + = (x – )(x – ) = (x – )2.

814 81

492

92

92