Perfect Square Trinomial
A perfect square trinomial is a trinomial that equals the square of a binomial, following the patterns $a^2 + 2ab + b^2$ or $a^2 - 2ab + b^2$.
Formula
a^2 + 2ab + b^2 = (a + b)^2
Definition
A perfect square trinomial is a trinomial that factors as a binomial squared, following the pattern $a^2 + 2ab + b^2 = (a + b)^2$ (or $a^2 - 2ab + b^2 = (a-b)^2$ when the middle term is negative). To recognize one: check that the first and last terms are perfect squares and that the middle term is exactly twice the product of their square roots. Formally, a quadratic $f = ax^2 + bx + c$ is a perfect square exactly when its discriminant $b^2 - 4ac$ equals zero, meaning the quadratic has a double root, with factored form $a(x - r)^2$ where $r = -b/(2a)$, the degenerate case of the quadratic formula where the parabola just touches the x-axis at one point.
Example
$x^2 + 6x + 9$: first term $x^2$, last term $9 = 3^2$, and middle $6x = 2 \cdot x \cdot 3$ matches, so this is $(x + 3)^2$. $9x^2 - 24x + 16$: $\sqrt{9x^2} = 3x$, $\sqrt{16} = 4$, and $2 \cdot 3x \cdot 4 = 24x$ matches, giving $(3x - 4)^2$. $4x^2 - 20x + 25$ has discriminant $400 - 400 = 0$, root $x = 5/2$, and factors as $(2x - 5)^2$.
Key Insight
To check quickly: square-root the first term, square-root the last term, and see if twice their product equals the middle term. Recognizing this pattern saves time compared to the ac method or quadratic formula, and it arises naturally from completing the square and in vertex form. Perfect square trinomials correspond to double roots algebraically and to tangent points geometrically, and in algebraic geometry the multiplicity of a root has deep implications for a curve's local behavior.