Beginning Algebra, 11th Edition

OBJECTIVE 2 Solve quadratic equations by completing the square when the coefficient of the second-degree term is not 1.If a quadratic equation has the form

where

we obtain 1 as the coefficient of by dividing each side of the equation by a. The steps used to solve a quadratic equation by completing the square are summarized here.

ax^2 +bx+c= 0

x^2

ax^2 +bx+c=0, aZ 1 ,

562 CHAPTER 9 Quadratic Equations

Solving a Quadratic Equation by Completing the Square

Step 1 Be sure the second-degree term has coefficient 1.If the coeffi-

cient of the second-degree term is 1, go to Step 2. If it is not 1, but

some other nonzero number a, divide each side of the equation by a.

Step 2 Write in correct form.Make sure that all variable terms are on one

side of the equation and that all constant terms are on the other side.

Step 3 Complete the square.Take half the coefficient of the first-degree

term, and square it. Add the square to each side of the equation. Fac-

tor the variable side, and combine like terms on the other side.

Step 4 Solvethe equation by using the square root property.

Solving a Quadratic Equation by Completing the Square Solve Step 1 Before completing the square, the coefficient of must be 1,not 4. We get 1 as the coefficient of here by dividing each side by 4.

Given equation

x^2 + 4 x- Divide by 4.

9

4

= 0

4 x^2 + 16 x- 9 = 0

x^2

4 x^2 + 16 x- 9 =0.

EXAMPLE 4

The coefficient of x^2 must be 1.

Step 2 Write the equation so that all variable terms are on one side of the equation and all constant terms are on the other side.

Add

Step 3 Complete the square by taking half the coefficient of x, and squaring it.

and

We add the result, 4 , to each side of the equation.

Add 4.

1 x+ 222 = Factor;^94 + 4 = 49 +^164 =^254.

25

4

x^2 + 4 x+ 4 =

9

4

+ 4

22 = 4

1

2

142 = 2

9 x 4.

(^2) + 4 x=^9
4
NOW TRY
EXERCISE 4
Solve 4t^2 - 4 t- 3 =0.
NOW TRY ANSWER

E-^12 ,^32 F

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