Perfect Square Trinomial Calculator

A perfect square trinomial is a three-term polynomial produced when a binomial is multiplied by itself. The familiar identities (p + q)² = p² + 2pq + q² and (p - q)² = p² - 2pq + q² explain its structure. In ax² + bx + c, the first and last terms must be squares and the middle coefficient must equal either positive or negative twice the product of their square roots. Recognizing this pattern turns a quadratic expression into a compact factored form. This calculator tests the equivalent discriminant condition b² - 4ac = 0 while also requiring a positive leading coefficient and a nonnegative constant. A zero discriminant means the associated quadratic equation has one repeated real root. That is exactly what happens when a binomial factor occurs twice. Once the test succeeds, the calculator reports the factor as (sqrt(a)x + sqrt(c))² when b is positive or (sqrt(a)x - sqrt(c))² when b is negative. For example, x² + 6x + 9 has square end terms x² and 9. Their roots are x and 3, and twice their product is 6x, matching the middle term. Therefore the expression factors as (x + 3)². Likewise, 4x² - 12x + 9 uses 2x and 3 as the square roots of its end terms. Twice their product is 12x, and the negative middle term selects subtraction, giving (2x - 3)². The coefficient test also works when roots are not integers. An expression can still be a perfect square over the real numbers even if its factored form contains decimal or irrational coefficients. The displayed factor is rounded for readability, while the decision uses the coefficients directly with a small numerical tolerance. This avoids rejecting a mathematically valid result merely because browser arithmetic cannot represent every decimal exactly. Perfect square trinomials appear in completing the square, solving quadratic equations, graphing parabolas, simplifying rational expressions, and deriving geometric formulas. Factoring one reveals the parabola's vertex and repeated x-intercept immediately. Use the calculator to verify your pattern recognition, then compare the reported discriminant and factor with your own algebra. It is designed as a quick checking aid while preserving the two key ideas: square outer terms and a middle term equal to twice their product.