Therefore, if we add 9 to each side of the equation will have a perfect

square trinomial on the left side, as needed.

Add 9.

1 x+ 322 = 2 Factor. Add.

x^2 + 6 x+ 9 =- 7 + 9

x^2 + 6 x=- 7

x^2 + 6 x=-7,

SECTION 9.2 Solving Quadratic Equations by Completing the Square 561

This is a

key step.

Now use the square root property to complete the solution.

or

or

Checkby substituting and for xin the original equation. The

solution set is E- 3  (^22) F. NOW TRY

- 3 + 22 - 3 - 22

x=- 3 + 22 x = - 3 - 22

x+ 3 = 22 x + 3 =- 22

The process of changing the form of the equation in Example 2from

to

is called completing the square.Completing the square changes only the form of the

equation. To see this, multiply out the left side of and combine like

terms. Then subtract 2 from each side to see that the result is.

Look again at the original equation in Example 2.

If we take half the coefficient of x, which is 6 here, and square it, we get 9.

and

Coefficient of x Quantity added to each side

To complete the square in Example 2,we added 9 to each side.

32 = 9

1

2

6 = 3

x^2 + 6 x+ 7 = 0

x^2 + 6 x+ 7 = 0

1 x+ 322 = 2

x^2 + 6 x+ 7 = 0 1 x+ 322 = 2

Completing the Square to Solve a Quadratic Equation

Solve

To complete the square on take half the coefficient of xand square it.

and

Coefficient of x

Add the result, 16 , to each side of the equation.

Given equation

Add 16.

Factor on the left. Add on the right.

Square root property

Add 4.

A check indicates that the solution set is E 4  221 F. NOW TRY

x= 4  221

x- 4 = 221

1 x- 422 = 21

x^2 - 8 x+ 16 = 5 + 16

x^2 - 8 x= 5

1 - 422 = 16

1

2

1 - 82 =- 4

x^2 - 8 x,

x^2 - 8 x=5.

NOW TRY EXAMPLE 3
EXERCISE 3
Solve x^2 - 6 x=9.

3. 
   

   E 3  (^322) F
   NOW TRY
   EXERCISE 2
   Solve x^2 + 10 x+ 8 =0.
   NOW TRY ANSWERS

   
   


4. 
   

   E- 5  217 F