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 (x + 3)(x + 3) = (x + 3)^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 = (x + 3)^2.

In the last section you expanded (a + b)^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 (a + b)^2.