Reverse FOIL Calculator

The standard FOIL method multiplies two binomials—such as (x + a)(x + b)—into a quadratic trinomial of the form ax² + bx + c. The Reverse FOIL process does the exact opposite: it takes a quadratic expression in the form ax² + bx + c and finds the two binomials that produce it. When the leading coefficient is 1, the goal is to find two numbers that multiply to give the constant term (c) and add up to give the middle coefficient (b). Once these two numbers, say p and q, are identified, the quadratic can be written directly in factored form as (x + p)(x + q).

Let us walk through factoring the quadratic expression x² + 7x + 12 using the Reverse FOIL method. First, identify the coefficients: the first coefficient (a) is 1, the second coefficient (b) is 7, and the third coefficient (c) is 12. We need to find two numbers whose product is 12 and whose sum is 7. Testing the factor pairs of 12 (such as 1 and 12, or 2 and 6), we find that 3 and 4 satisfy both conditions since 3 times 4 equals 12, and 3 plus 4 equals 7. Therefore, the factored form of the expression is (x + 3)(x + 4).

Always start by factoring out the greatest common factor (GCF) from all terms if possible, which simplifies the remaining quadratic expression significantly. Pay close attention to negative signs; a negative constant term means your two factors must have opposite signs, while a positive constant term with a negative middle term means both factors must be negative. Double-check your final factored answer by using the standard FOIL method to ensure it multiplies back out to your original polynomial.