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
| Step | Calculation | Result |
|---|---|---|
| Original trinomial | x^2 + 7x + 12 | Input expression. |
| GCF step | No shared coefficient factor | Primitive trinomial already. |
| AC product | 1 x 12 | 12 |
| Discriminant | b^2 - 4ac | 1 |
| Final factorization | (x + 3)(x + 4) | Integer binomial factors found. |
AC Method Pairs
| Pair | Sum | Match |
|---|---|---|
| 1 and 12 | 1 + 12 | 13 |
| -1 and -12 | -1 + -12 | -13 |
| 2 and 6 | 2 + 6 | 8 |
| -2 and -6 | -2 + -6 | -8 |
| 3 and 4 | 3 + 4 | Middle-term match |
| -3 and -4 | -3 + -4 | -7 |
| 4 and 3 | 4 + 3 | Middle-term match |
| -4 and -3 | -4 + -3 | -7 |
| 6 and 2 | 6 + 2 | 8 |
| -6 and -2 | -6 + -2 | -8 |
| 12 and 1 | 12 + 1 | 13 |
| -12 and -1 | -12 + -1 | -13 |
Binomial Candidates
| Candidate | Outer + Inner | Middle |
|---|---|---|
| (x + 1)(x + 12) | 12 + 1 | 13 |
| (x - 1)(x - 12) | -12 + -1 | -13 |
| (x + 2)(x + 6) | 6 + 2 | 8 |
| (x - 2)(x - 6) | -6 + -2 | -8 |
| (x + 3)(x + 4) | 4 + 3 | Match |
FOIL Check
| FOIL part | Multiplication | Term |
|---|---|---|
| First | x x x | x^2 |
| Outer | x x 4 | 4x |
| Inner | 3 x x | 3x |
| Last | 3 x 4 | 12 |
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.
Checked by Jitendra Kumar
Reverse FOIL Calculator is checked for formula labels, source links, and result limits.
Jitendra Kumar, Founder & Editorial Standards Lead. Updated May 2026. Scope: math calculators.
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.
Step 1: Enter a, b, and c
Use the coefficients from ax^2 + bx + c, such as 1, 7, and 12.
Step 2: Check the GCF
A shared coefficient factor is pulled outside before reverse FOIL begins.
Step 3: Review AC pairs
Look for a factor pair of ac whose sum equals b.
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
| Concept | Formula | Meaning |
|---|---|---|
| FOIL expansion | (mx + n)(px + q) | mpx^2 + (mq + np)x + nq |
| Reverse FOIL target | ax^2 + bx + c | Find binomials whose first, outer, inner, and last products match. |
| AC method | Find r and s with rs = ac and r + s = b | Used to split the middle term. |
| GCF first | 2x^2 + 14x + 20 = 2(x^2 + 7x + 10) | Pull out shared coefficient factors before binomial factoring. |
| Integer factor test | No integer pair works | The trinomial may need irrational or complex roots instead. |
Worked Examples
| Trinomial | Factored Form | Reason |
|---|---|---|
| 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 + 20 | 2(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.
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- 1.Khan Academy - Factoring quadratics intro(Accessed May 2026)
- 2.OpenStax Algebra and Trigonometry 2e - Factoring Polynomials(Accessed May 2026)
- 3.Khan Academy - FOIL method(Accessed May 2026)