Beginning Algebra, 11th Edition

560 CHAPTER 9 Quadratic Equations

OBJECTIVES OBJECTIVE 1 Solve quadratic equations by completing the square when

the coefficient of the second-degree term is 1.The methods we have studied so

far are not enough to solve an equation such as

If we could write the equation in the form equals a constant, we could solve

it with the square root property discussed in Section 9.1.To do that, we need to have

a perfect square trinomial on one side of the equation.

Recall from Section 6.4that a perfect square trinomial has the form

or

where krepresents a number.

Creating Perfect Square Trinomials

Complete each trinomial so that it is a perfect square. Then factor the trinomial.

(a)

The perfect square trinomial will have the form Thus, the middle

term, 8x, must equal 2kx.

Solve this equation for k.

Divide each side by 2x.

Therefore, and The required perfect square trinomial is

which factors as

(b)

Here the perfect square trinomial will have the form The middle

term, must equal

Solve this equation for k.

Divide each side by

Thus, and The required perfect square trinomial is

which factors as NOW TRY

Rewriting an Equation to Use the Square Root Property

Solve

Subtract 7 from each side.

To solve this equation with the square root property, the quantity on the left side,

must be written as a perfect square trinomial in the form

x^2 + 6 x+

x^2 + 6 x, x^2 + 2 k x+k^2.

x^2 + 6 x=- 7

x^2 + 6 x+ 7 =0.

EXAMPLE 2

x^2 - 18 x+ 81 , 1 x- 922.

k= 9 k^2 = 92 = 81.

9 =k - 2 x.

• 18 x=- 2 kx


• 18 x, - 2 kx.

x^2 - 2 kx+k^2.

x^2 - 18 x+

x^2 + 8 x+ 16 , 1 x+ 422.

k= 4 k^2 = 42 = 16.

4 =k

8 x= 2 kx

x^2 + 2 kx+k^2.

x^2 + 8 x+

EXAMPLE 1

x^2 + 2 kx+k^2 x^2 - 2 kx+k^2 ,

1 x+ 322

x^2 + 6 x+ 7 =0.

Solving Quadratic Equations by Completing the Square

9.2

1 Solve quadratic

equations by

completing the

square when the

coefficient of the

second-degree term

is 1.

2 Solve quadratic

equations by

completing the

square when the

coefficient of the

second-degree term

is not 1.

3 Simplify the terms

of an equation

before solving.

4 Solve applied

problems that

require quadratic

equations.

NOW TRY
EXERCISE 1
Complete each trinomial so
that it is a perfect square.
Then factor the trinomial.

(a)

(b)x^2 - 22 x+

x^2 + 4 x+

NOW TRY ANSWERS

1. (a)4;
   (b)121; 1 x- 1122

1 x+ 222

A square must

go here.

Here, so and The required perfect square trinomial is

x^2 + 6 x+ 9 , which factors as 1 x+ 322.

2 k x= 6 x , k= 3 k^2 = 9.

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