Completing the Square

Procedure — To Complete the Square

To complete the square for a binomial of the form x² + bx:

Step 1 Calculate · b.

(That is, divide b by 2.)

Step 2 Square the result from Step 1.

That is, calculate .

Step 3 Add the result from Step 2 to x² + bx.

The new expression, x² + bx + , is a perfect square trinomial.

Example

Complete the square for each binomial. Then write each new expression as the square of a binomial.

a. x² + 12x

b. x² - 5x

Solution

a.	x2 + 12x has the form x2 + bx, where b = 12. Step 1 Calculate Step 2 Square the result from Step 1. Step 3 Add the result from Step 2 to x2 + bx.	62 = 36 x2 + 12x + 36
b.	Write the expression x2 + 12x + 36 as the square of a binomial. x2 - 5x has the form x2 + bx, where b = -5.	= (x + 6)2
	Step 1 Calculate Step 2 Square the result from Step 1. Step 3 Add the result from Step 2 to x2 + bx.

We have completed the square.

Write the expression as the square of a binomial.

Notes:

The expression x² + 12x + 36 is a perfect square trinomial.