Completing the Square
The essential part of completing the square is to recognize a perfect square trinomial
when given its first two terms. For example, if we are given x^{2} + 6x, how do we recognize that these are the first two terms of the perfect square trinomial
x^{2} + 6x + 9? To answer this question, recall that
x^{2} + 6x + 9 is a perfect square
trinomial because it is the square of the binomial x + 3:
(x + 3)^{2} = x^{2} + 2 Â· 3x + 3^{2} = x^{2}
+ 6x + 9
Notice that the 6 comes frommultiplying 3 by 2 and the 9 comes from squaring the 3.
So to find the missing 9 in x^{2} + 6x, divide 6 by 2 to get 3, then square 3 to get 9. This
procedure can be used to find the last term in any perfect square trinomial in which
the coefficient of x^{2} is 1.
Rule for Finding the Last Term
The last term of a perfect square trinomial is the square of onehalf of the
coefficient of the middle term. In symbols, the perfect square trinomial whose
first two terms are
.
Helpful hint
Review the rule for squaring a
binomial: square the first term,
find twice the product of the
two terms, then square the
last term. If you are still using
FOIL to find the square of a binomial,
it is time to learn the
proper rule.
Example 1
Finding the last term
Find the perfect square trinomial whose first two terms are given.
Solution
a) Onehalf of 8 is 4, and 4 squared is 16. So the perfect square trinomial is
x^{2} + 8x + 16.
b) Onehalf of 5 is
, and
squared is
. So the perfect square trinomial is
.
c) Onehalf of
is
, and
squared is
. So the perfect square trinomial is
d) Onehalf of
is
, and
. So the perfect square trinomial is
Another essential step in completing the square is to write the perfect square
trinomial as the square of a binomial. Recall that
a^{2} + 2ab + b^{2} = (a + b)^{2} and
a^{2}  2ab + b^{2} = (a  b)^{2}.
Example 2
Factoring perfect square trinomials
Factor each trinomial.
Solution
a) The trinomial x^{2} + 12x + 36 is of the form a^{2} + 2ab
+ b^{2} with a = x and
b = 6. So
x^{2} + 12x + 36 = (x + 6)^{2}.
Check by squaring x + 6.
b) The trinomial
is of the form a^{2}
 2ab + b^{2} with a
=
y and
. So
Check by squaring
.
c) The trinomial
is of the form a^{2}
 2ab + b^{2} with a = z and
. So
