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Reverse FOIL Calculator

Factor quadratic trinomials by reversing FOIL with GCF extraction, AC-method pairs, integer binomial candidates, and FOIL expansion checks.

Last Updated: May 2026

Factored form

(x + 3)(x + 4)

Original trinomial

x^2 + 7x + 12

AC product

12

Integer-factorable?

Yes

Quadratic Coefficients

Enter coefficients for ax^2 + bx + c. The calculator factors out any GCF, then searches for two binomials whose FOIL expansion returns the trinomial.

Factoring Summary

StepCalculationResult
Original trinomialx^2 + 7x + 12Input expression.
GCF stepNo shared coefficient factorPrimitive trinomial already.
AC product1 x 1212
Discriminantb^2 - 4ac1
Final factorization(x + 3)(x + 4)Integer binomial factors found.

AC Method Pairs

PairSumMatch
1 and 121 + 1213
-1 and -12-1 + -12-13
2 and 62 + 68
-2 and -6-2 + -6-8
3 and 43 + 4Middle-term match
-3 and -4-3 + -4-7
4 and 34 + 3Middle-term match
-4 and -3-4 + -3-7
6 and 26 + 28
-6 and -2-6 + -2-8
12 and 112 + 113
-12 and -1-12 + -1-13

Binomial Candidates

CandidateOuter + InnerMiddle
(x + 1)(x + 12)12 + 113
(x - 1)(x - 12)-12 + -1-13
(x + 2)(x + 6)6 + 28
(x - 2)(x - 6)-6 + -2-8
(x + 3)(x + 4)4 + 3Match

FOIL Check

FOIL partMultiplicationTerm
Firstx x xx^2
Outerx x 44x
Inner3 x x3x
Last3 x 412

Factoring Scope Notice

This calculator factors integer-coefficient quadratic trinomials over the integers. It does not handle decimals, higher-degree polynomials, nested expressions, or irrational factor forms.

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How to Use the Reverse FOIL Calculator

Enter the coefficients a, b, and c from the trinomial ax^2 + bx + c. Use negative signs when needed, and keep the coefficients as integers.

The calculator factors out any shared GCF, checks AC-method pairs, tests binomial candidates, and verifies the result with FOIL.

  1. Step 1: Enter a, b, and c

    Use the coefficients from ax^2 + bx + c, such as 1, 7, and 12.

  2. Step 2: Check the GCF

    A shared coefficient factor is pulled outside before reverse FOIL begins.

  3. Step 3: Review AC pairs

    Look for a factor pair of ac whose sum equals b.

  4. Step 4: Verify with FOIL

    The FOIL table multiplies the factors back out to confirm the trinomial.

How This Reverse FOIL Calculator Works

Reverse FOIL starts with the expanded form ax^2 + bx + c and searches for two binomials (mx + n)(px + q). The first product must equal ax^2, the last product must equal c, and the outer plus inner products must equal bx.

The calculator first removes any greatest common factor from the three coefficients. Then it checks integer factor pairs for the leading coefficient and constant term.

If no integer binomial pair works, the calculator leaves the trinomial marked as not factorable over integers and shows the discriminant for follow-up analysis.

Reverse FOIL Guide

Reverse FOIL Rules

ConceptFormulaMeaning
FOIL expansion(mx + n)(px + q)mpx^2 + (mq + np)x + nq
Reverse FOIL targetax^2 + bx + cFind binomials whose first, outer, inner, and last products match.
AC methodFind r and s with rs = ac and r + s = bUsed to split the middle term.
GCF first2x^2 + 14x + 20 = 2(x^2 + 7x + 10)Pull out shared coefficient factors before binomial factoring.
Integer factor testNo integer pair worksThe trinomial may need irrational or complex roots instead.

Worked Examples

TrinomialFactored FormReason
x^2 + 7x + 12(x + 3)(x + 4)3 and 4 multiply to 12 and add to 7.
6x^2 + 11x + 3(2x + 3)(3x + 1)Outer plus inner is 2x + 9x = 11x.
x^2 + 2x - 15(x + 5)(x - 3)5 and -3 multiply to -15 and add to 2.
2x^2 + 14x + 202(x + 2)(x + 5)GCF 2 comes out first.
9x^2 + 12x + 4(3x + 2)(3x + 2)Perfect-square trinomial.

Why FOIL Checks Matter

Factoring is easy to misread because signs and middle terms can be close. Multiplying the answer back out with FOIL confirms that the first, outer, inner, and last products rebuild the original trinomial exactly.

Keep the research moving with Distributive Property Calculator, Factor Calculator, Math Equation Solver, and GCF Calculator.

Frequently Asked Questions

Reverse FOIL means factoring an expanded quadratic trinomial back into two binomials.

It factors integer-coefficient trinomials in the form ax^2 + bx + c where a is nonzero.

The AC method looks for two numbers whose product is a times c and whose sum is b. Those numbers help split the middle term.

Yes. It pulls out the greatest common factor of a, b, and c before searching for binomial factors.

The calculator reports that no integer binomial factorization was found and shows the discriminant for context.

No. This calculator is scoped to integer coefficients for classroom-style reverse FOIL factoring.

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Sources & References

  1. 1.Khan Academy - Factoring quadratics intro(Accessed May 2026)
  2. 2.OpenStax Algebra and Trigonometry 2e - Factoring Polynomials(Accessed May 2026)
  3. 3.Khan Academy - FOIL method(Accessed May 2026)