# Is  a perfect square trinomial and how do you factor it?

May 31, 2015

You seem to have lost your trinomial from the question, so let me address the general question.

How do you recognise a perfect square trinomial and how do you factor it?

A perfect square trinomial is the square of a binomial. A general binomial is of the form $a + b$, so we need to look at:

${\left(a + b\right)}^{2} = {a}^{2} + 2 a b + {b}^{2}$

So a trinomial is a perfect square trinomial if the following three conditions hold:

(1) The first term is a square.
(2) The last term is a square.
(3) The middle term is twice the product of square roots (positive or negative) of the first and last term.

For example, $16 {x}^{2} - 24 x + 9$ is a perfect square trinomial:

(1) $16 {x}^{2} = {\left(4 x\right)}^{2}$
(2) $9 = {3}^{2}$
(3) $- 24 x = 2 \times \left(4 x\right) \times - 3$

To factor such a trinomial:
(1) Take the positive square root of the first term (e.g. $4 x$)
(2) Take the square root of the second term whose sign matches that of the middle term. (e.g. $- 3$)
(3) Add them together to get your repeated factor (e.g. $\left(4 x - 3\right)$)