The Perfect Square

To factor $x^{2}+6x+9$, you need a pair with product 9 and sum 6. The pair is 3 and 3. The two factors are the same, so $\left(x+3\right)\left(x+3\right) = \left(x+3\right)^{2}$. A trinomial that factors into a square like this is called a perfect square trinomial.

There is a quick test for a monic trinomial. Take half of the middle coefficient and square it. If the result equals the constant, the trinomial is a perfect square. Here half of 6 is 3, and $3^{2} = 9$, so $x^{2}+6x+9 = \left(x+3\right)^{2}$.

In the last section you expanded $\left(a+b\right)^{2} = a^{2}+2ab+b^{2}$. Factoring a perfect square trinomial uses the same rule backwards. The middle term is $2ab$, so half of it is $ab$. The constant is $b^{2}$. Look for this pattern: a square, then twice a product, then a square. When you see it, the trinomial is $\left(a+b\right)^{2}$.