perfect-square trinomial

A perfect-square trinomial is a trinomial that is secretly the square of a single binomial. Just as 25 is the square of 5, the expression x^2 + 6x + 9 is the square of (x + 3). Spotting these saves you the work of the full factoring search: you recognize the shape and write the answer immediately.

The patterns to memorize are a^2 + 2ab + b^2 = (a + b)^2 and a^2 - 2ab + b^2 = (a - b)^2. The tell-tale signs: the first and last terms are perfect squares (a^2 and b^2), and the middle term is exactly twice the product of their square roots (2ab). For x^2 + 6x + 9, the ends are x^2 and 3^2, and the middle 6x equals 2·x·3 — so it is (x + 3)^2.

Always verify the middle term, because that is where the pattern is faked. x^2 + 5x + 9 has perfect-square ends but its middle term should be 6x to be a perfect square, so it is not one. Perfect-square trinomials are also the engine behind completing the square, the technique used to derive the quadratic formula.