Perfect Square Trinomials: Definition and Factoring Rules
A perfect square trinomial is a three-term polynomial that results from squaring a binomial:
(px + q)^2 = p^2 x^2 + 2pq\,x + q^2
(px - q)^2 = p^2 x^2 - 2pq\,x + q^2
The Three Identification Criteria
To confirm whether $ax^2 + bx + c$ is a perfect square trinomial, check these three non-negotiable mathematical conditions:
- 1The leading coefficient $a$ must be positive, with a real square root $\sqrt{a}$.
- 2The constant term $c$ must be positive, with a real square root $\sqrt{c}$.
- 3The middle term coefficient $b$ must equal exactly $\pm 2\sqrt{a}\sqrt{c}$.
Equivalently, the algebraic discriminant must be zero:
\Delta = b^2 - 4ac = 0
Common Perfect Square Patterns
| Standard Trinomial | Factored Binomial Square | Square Root Components |
|---|---|---|
| $x^2 + 6x + 9$ | $(x + 3)^2$ | $p = 1, q = 3$ |
| $x^2 - 10x + 25$ | $(x - 5)^2$ | $p = 1, q = 5$ |
| $4x^2 + 12x + 9$ | $(2x + 3)^2$ | $p = 2, q = 3$ |
| $9x^2 - 30x + 25$ | $(3x - 5)^2$ | $p = 3, q = 5$ |
| $16x^2 + 56x + 49$ | $(4x + 7)^2$ | $p = 4, q = 7$ |