Perfect Square Trinomial
AlgebraA 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 can be factored as a binomial squared. It follows the pattern $a^2 + 2ab + b^2 = (a + b)^2$.
Example
$x^2 + 6x + 9$: the first term is $x^2 = x^2$, the last is $9 = 3^2$, and the middle is $6x = 2 \cdot x \cdot 3$. So this is $(x + 3)^2$.
Key Insight
To check: square root the first term, square root the last term. If twice their product equals the middle term, it is a perfect square trinomial.
Definition
A trinomial $a^2 + 2ab + b^2$ factors as $(a + b)^2$, and $a^2 - 2ab + b^2$ factors as $(a - b)^2$. The sign of the middle term determines whether the binomial uses $+$ or $-$. To identify: check that the first and last terms are perfect squares and the middle term is exactly twice the product of their square roots.
Example
$9x^2 - 24x + 16$: $\sqrt{9x^2} = 3x$, $\sqrt{16} = 4$, $2 \cdot 3x \cdot 4 = 24x$. Middle term matches. Factors as $(3x - 4)^2$.
Key Insight
Recognizing perfect square trinomials saves time compared to using the ac method or quadratic formula. They arise naturally from completing the square and in the vertex form of quadratic equations.
Definition
A perfect square trinomial is a trinomial $f(x)$ such that $f = g^2$ for some polynomial $g$. For a quadratic $f = ax^2 + bx + c$ to be a perfect square, the discriminant $b^2 - 4ac$ must equal zero, meaning the quadratic has a double root. The factored form is $a(x - r)^2$ where $r = -b/(2a)$. This is the degenerate case of the quadratic formula when the parabola touches the x-axis at exactly one point.
Example
$4x^2 - 20x + 25$: discriminant $= 400 - 400 = 0$. Root: $x = 20/8 = 5/2$. Factored: $4(x - 5/2)^2 = (2x - 5)^2$.
Key Insight
Perfect square trinomials correspond to double roots in the algebraic sense and tangent points in the geometric sense. In modular arithmetic and algebraic geometry, the multiplicity of a root has deep implications for the local behavior of a curve.