Factoring Trinomials Calculator

Factoring a quadratic trinomial reverses polynomial multiplication. An expression in the form ax² + bx + c is rewritten as a product of two linear binomials when integer factors exist. For example, x² + 5x + 6 becomes (x + 2)(x + 3). Expanding the result gives x² + 3x + 2x + 6, which combines back to the original expression. For a monic trinomial, where a equals 1, look for two integers whose product is c and whose sum is b. Those integers become the constants in the two factors. Signs matter: x² - 5x + 6 requires two negative constants, producing (x - 2)(x - 3), because their product is positive 6 and their sum is negative 5. When the leading coefficient is not 1, the AC method provides a systematic path. Multiply a by c, find two integers whose product is ac and whose sum is b, split the middle term with those values, and factor by grouping. For 2x² + 7x + 3, the product ac is 6 and the useful pair is 6 and 1. Grouping 2x² + 6x + x + 3 produces (x + 3)(2x + 1). The discriminant b² - 4ac quickly indicates whether rational linear factors are possible. If it is negative, the quadratic has no real roots and cannot factor into real linear terms. If it is nonnegative but not a perfect square for integer coefficients, the roots are irrational and no integer binomial factorization exists. The calculator reports such expressions as not factorable over the integers rather than forcing an incorrect product. Special patterns can make factoring faster. A difference of squares such as x² - 9 factors as (x - 3)(x + 3). A perfect-square trinomial such as x² + 6x + 9 factors as (x + 3)(x + 3). If all coefficients share a greatest common factor, remove that content first and place it before the binomial product. Factored form reveals the zeros of a quadratic equation. If a product of two factors equals zero, each factor can be set equal to zero to find the roots. This is useful for graph intercepts, projectile-time equations, area models, and algebraic simplification. Always verify a factorization by multiplying the binomials and confirming that the x², x, and constant coefficients match the original trinomial exactly.