Factor quadratic expressions ax² + bx + c using the reverse FOIL (AC) method. See factor pair analysis, step-by-step grouping, and visual sum comparison for every factor pair.
The Reverse FOIL Calculator factors any quadratic trinomial ax² + bx + c back into two binomial factors using the AC method — the systematic reverse of FOIL multiplication. While FOIL expands (dx + e)(fx + g) into a trinomial, reverse FOIL does the opposite: starting from ax² + bx + c, it finds the two binomials whose product gives the original expression.
The AC method works by computing the product a × c, then searching for two numbers p and q such that p × q = ac and p + q = b. Once found, the middle term bx is split into px + qx and the expression is factored by grouping. This calculator automates the entire process, generating every factor pair of the AC product and checking which pair sums to b.
The interactive factor pair analysis table shows every candidate pair with a clear pass/fail indicator, while the visual bar chart compares each pair's sum against the target value b. Eight preset polynomials cover the most common textbook examples — simple monic trinomials, leading coefficients greater than 1, difference of squares, and expressions with negative terms. The step-by-step breakdown walks through the full AC method from identifying coefficients through the final factored form, making this an invaluable learning tool for algebra students mastering factorization.
Reverse FOIL Calculator helps you solve reverse foil problems quickly while keeping each step transparent. Instead of redoing long algebra by hand, you can enter Coefficient a (x² term), Coefficient b (x term), Coefficient c (constant) once and immediately inspect Expression, Factored Form, Discriminant (b²−4ac) to validate your work.
This is useful for homework checks, classroom examples, and practical what-if analysis. You keep the conceptual understanding while reducing arithmetic mistakes in multi-step calculations.
AC Method: Given ax² + bx + c, find p and q such that p × q = a × c and p + q = b. Rewrite as ax² + px + qx + c, then factor by grouping. Discriminant: Δ = b² − 4ac determines factorability over reals.
Result: Expression shown by the calculator
Using the preset "x²+5x+6", the calculator evaluates the reverse foil setup, applies the selected algebra rules, and reports Expression with supporting checks so you can verify each transformation.
This calculator takes Coefficient a (x² term), Coefficient b (x term), Coefficient c (constant), Show AC Method Steps and applies the relevant reverse foil relationships from your chosen method. It returns both final and intermediate values so you can audit the process instead of treating it as a black box.
Start with the primary output, then use Expression, Factored Form, Discriminant (b²−4ac), AC Product to confirm signs, magnitude, and internal consistency. If anything looks off, change one input and compare the updated outputs to isolate the issue quickly.
A strong workflow is manual solve first, calculator verify second. Repeating that loop improves speed and accuracy because you learn to spot common setup errors before they cost points on multi-step algebra problems.
FOIL multiplies two binomials into a trinomial. Reverse FOIL does the opposite — it takes a trinomial and finds the two binomials whose FOIL product gives that trinomial.
A systematic factoring technique: multiply a × c, find two numbers whose product is ac and sum is b, then rewrite the middle term and factor by grouping. Use this as a practical reminder before finalizing the result.
No — only those with rational roots. If the discriminant b² − 4ac is not a perfect square, the trinomial cannot be factored over integers.
The AC method handles any integer leading coefficient. You find factor pairs of a×c (not just c), which may produce more candidates to check.
Reverse FOIL gives exact factored form with integer or rational factors. The quadratic formula finds roots (which may be irrational) but does not directly give factored form.
If Δ > 0: two distinct real roots (factorable). If Δ = 0: one repeated root (perfect square). If Δ < 0: no real roots (not factorable over reals).