Factoring Trinomials Calculator

A standard quadratic trinomial takes the algebraic form ax² + bx + c. To factor this expression, our calculator first extracts the greatest common factor (GCF) from coefficients a, b, and c to simplify the polynomial. It then computes the discriminant (delta = b² - 4ac) to determine whether the trinomial can be factored using integers. If factorable, the tool applies the ac method—multiplying the scaled leading coefficient a by the constant term c—to find two factors that multiply to ac and add up to the middle coefficient b. These values are used to split the middle term and group the expression completely.

Let us walk through factoring the quadratic expression 2x² + 7x + 3 step-by-step. First, identify the coefficients: a = 2, b = 7, and c = 3. The GCF of these numbers is 1, so the expression is already in its simplest form. Next, apply the ac method by multiplying a and c together, yielding 2 × 3 = 6. We look for two integers that multiply to 6 and add up to the middle term coefficient, 7. Those numbers are 6 and 1. We rewrite the middle term using these numbers: 2x² + 6x + 1x + 3. Grouping the terms gives 2x(x + 3) + 1(x + 3). Factoring out the common binomial (x + 3) yields the final factored form: (2x + 1)(x + 3).

Always check for a greatest common factor (GCF) before applying any advanced factoring techniques; pulling out the GCF early keeps numbers small and manageable. Pay close attention to the signs of your coefficients, as negative signs frequently cause arithmetic errors during the grouping phase. Finally, always verify your final factored binomials by using the FOIL method (First, Outer, Inner, Last) to multiply them back together and ensure they equal the original trinomial.