Perfect Square Trinomials

The type form is a^2 \pm 2ab + b^2, and by the square-of-a-binomial rules, a^2 + 2ab + b^2 = (a + b)^2, \qquad a^2 - 2ab + b^2 = (a - b)^2. An expression that is the product of two equal factors is a perfect square.

A trinomial, arranged by powers of one letter, is a perfect square when its middle term (ignoring sign) is twice the product of the square roots of the other two terms.

For example 4x^2 - 20xy^2 + 25y^4 is a perfect square, since 20xy^2 = 2 \cdot 2x \cdot 5y^2.

Rule. Arrange the terms by descending powers of a letter. Take the square roots of the first and third terms and join them with the sign of the middle term.

So 4x^2 - 20xy^2 + 25y^4 = (2x - 5y^2)^2.

In real work nobody tells you which type you have, and answers often have three or more factors. Always pull out a common monomial first, then look again.