What Is a Perfect Square Trinomial?

A perfect square trinomial is the expanded product of two identical binomials. A perfect square trinomial is also the result that occurs when a binomial is squared. There are two general formulas for factoring a perfect square trinomial: x^2 + 2xy + y^2 = (x + y)^2, and x^2 – 2xy + y2 = (x – y)^2.

Consider the quadratic trinomial x^2 + 24x + 144. To be a perfect square trinomial, it must factor into two identical binomials, which means the first term and third term must both be perfect squares. The first term is the perfect square of x, and third term is the perfect square of 12. Before it can be positively determined that the trinomial is a perfect square, the middle term must be equal to two times the product of the first and third terms. In this case, 2 times 12x equals 24x, so x^2 + 24x + 144 is a perfect square trinomial equal to (x + 12)^2. The product of (x – 12)^2 has the same first and third terms as (x + 12)^2 because the square of -12 is also 144, but the middle term is negative: x^2 – 24x + 144. If (x + 12) is multiplied by (x – 12) the product is x^2 – 144, because the middle terms, 12x and -12x, cancel each other out.