Math101 · Independent Ontario learning library
Math101Perfect Square Trinomials
A perfect square trinomial is produced by squaring a binomial: $a^2+2ab+b^2=(a+b)^2$ or $a^2-2ab+b^2=(a-b)^2$.
Recognizing perfect squares speeds factoring, equation solving, graphing, and completion of squares. The area model explains the otherwise easy-to-forget middle term.
Intuition and core definition
A perfect square trinomial is produced by squaring a binomial: $a^2+2ab+b^2=(a+b)^2$ or $a^2-2ab+b^2=(a-b)^2$. Its first and last terms are squares, and its middle term is exactly twice the product of their square roots, with the sign choosing the binomial.
Notation, language, and conditions
The square on $(a\pm b)^2$ applies to the entire binomial. The leading and constant terms should usually be positive squares over the reals. A GCF must be removed first because a scaled perfect-square pattern may be hidden.
Why this idea matters
A perfect-square trinomial records the expansion of a repeated binomial, with its middle term twice the product of the square roots of the outer terms.
A dependable method
- Factor out any overall GCF.
- Verify the first and last terms are perfect squares.
- Take their square roots as $a$ and $b$.
- Check whether the middle term equals $+2ab$ or $-2ab$.
- Write $(a+b)^2$ or $(a-b)^2$ and expand to verify.
