Reverse FOIL Calculator
Factor any quadratic trinomial ax^2 + bx + c into binomials using the reverse FOIL method. Free factoring calculator with step-by-step solutions and root finding.
About This Calculator
The Reverse FOIL Calculator helps you factor second-degree trinomials of the form ax^2 + bx + c into the product of two binomials (px + q)(rx + s). This process -- also known as factoring trinomials -- is the reverse of the FOIL multiplication method. Students, teachers, and anyone working with quadratic equations can use this tool to quickly find factor pairs and roots.
How the Reverse FOIL Method Works
The FOIL acronym stands for First, Outer, Inner, Last -- the four multiplications required to expand (px + q)(rx + s): First (p·r)x^2, Outer (p·s)x, Inner (q·r)x, and Last (q·s). To reverse this process, the calculator searches all integer factor pairs of a and c and checks whether any combination produces a matching middle coefficient b. When a match is found, it returns the factored binomials and the roots x = -q/p and x = -s/r.
This tool works for any quadratic trinomial with integer coefficients, whether the leading coefficient is 1 (monic) or greater than 1. If no integer-factor pair satisfies the condition, the trinomial does not factor over the integers and the calculator informs you accordingly.
Frequently Asked Questions
What is the reverse FOIL method?
The reverse FOIL method is a factorization algorithm for second-degree trinomials in one variable. It finds two first-degree binomials whose product equals the original trinomial by matching factor pairs of the first and last coefficients. The name FOIL stands for First, Outer, Inner, Last -- referring to the four multiplication steps when expanding binomials.
How does the Reverse FOIL calculator work?
Enter the coefficients a, b, and c from your trinomial ax^2 + bx + c. The calculator tries all possible integer factor pairs of a and c to find a combination where the sum of the outer and inner products equals b. If found, it displays the factored form (px + q)(rx + s) and both roots. If factoring over integers fails, you will be informed.
When would a trinomial not factor over integers?
A trinomial ax^2 + bx + c does not factor over the integers when no integer factor pairs of a and c satisfy the reverse FOIL condition p·s + q·r = b. In such cases the quadratic is irreducible over the rational numbers, and you would need to use the quadratic formula or complete the square to find its roots.
Can the Reverse FOIL calculator handle negative coefficients?
Yes. The calculator accepts any integer values for coefficients a, b, and c, including negative numbers. It searches all possible factor pair combinations including negative factors and correctly formats the resulting binomials with proper signs. For example, 6x^2 - 7x - 5 factors as (2x + 1)(3x - 5).
What are the roots from reverse FOIL factoring?
Once a trinomial ax^2 + bx + c is factored as (px + q)(rx + s), the roots are found by setting each binomial to zero: x = -q/p and x = -s/r. These roots are the x-intercepts of the quadratic function and the solutions to the equation ax^2 + bx + c = 0.