Definition: A perfect square trinomial is a trinomial that equals a binomial squared: a^2 + 2ab + b^2 = (a + b)^2, or a^2 – 2ab + b^2 = (a – b)^2.
Why is it called perfect?
It is what you get when a binomial is multiplied by itself. The first and last terms are perfect squares, and the middle term is twice the product of their square roots.
How do you test for one?
Take the square roots of the first and last terms, multiply them, double the result, and compare with the middle term. If it matches in size, the sign of the middle term tells you the sign inside the binomial.
Example. Is 25x2 – 60x + 36 a perfect square? The roots are 5x and 6. Twice their product is 2(5x)(6) = 60x. The middle term is -60x, so the trinomial equals (5x – 6)2. Now try 9x2 + 20x + 16. The roots are 3x and 4, and 2(3x)(4) = 24x, not 20x, so it is not a perfect square.
What mistake is typical?
Believing (a + b)2 = a2 + b2. The middle term 2ab is always present. This pattern also speeds up completing the square.
Where to learn it
- Factoring Polynomials with Patterns and Grouping – lesson
- Solving Quadratic Equations Four Ways – lesson