Completing the Square
Procedure â€”
To Complete the Square
To complete the square for a binomial of the form x^{2} + bx:
Step 1 Calculate
Â· 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 x^{2} + bx.
The new expression, x^{2} + 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. x^{2} + 12x
b. x^{2}  5x
Solution
a. 
x^{2} + 12x has the form x^{2}
+ bx, where b = 12.
Step 1 Calculate
Step 2 Square the result from Step 1.
Step 3 Add the result from Step 2 to x^{2} + bx. 
6^{2} = 36
x^{2} + 12x + 36 
b. 
Write the expression x^{2} + 12x + 36 as the
square of a binomial. x^{2}  5x has the form x^{2} + 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
x^{2} + bx. 

We have completed the square.
Write the expression as the square of a binomial.
Notes:
The expression x^{2} + 12x + 36 is a perfect
square trinomial.
The expression
is a perfect
square trinomial.
