Reverse Foiling Calculator: Factor Quadratics Into Binomials
This reverse foiling calculator factors a quadratic trinomial into two binomials and shows each step. Enter x² + 5x + 6, and it returns (x + 2)(x + 3).
Use the Reverse Foiling Calculator
Type a quadratic in the form ax² + bx + c and select Calculate. You'll see a, b, c, the two numbers that split the middle term, the grouping, and the factored answer. If no integer factors exist, it labels the trinomial prime.
Input tips
- Use whole-number coefficients and ^2 or ² for the square.
- Any single letter works as the variable, such as a² + 7a + 10.
- Factor out any common factor first, e.g. 4x² + 10x + 6 → 2(2x² + 5x + 3).
- Select Clear to start a new problem.
What Is Reverse FOILing?
Reverse FOILing is factoring a quadratic trinomial back into the two binomials that produced it. The FOIL method turns (x + 2)(x + 3) into x² + 5x + 6. Reverse FOILing goes the other way. It is how you solve quadratic equations by factoring, a skill in the Common Core standard HSA-SSE.B.3a.
How the Reverse Foiling Calculator Works
The calculator uses the ac method, also called factoring by grouping. Here it is for 6x² − x − 2:
- Find a, b, c: a = 6, b = −1, c = −2.
- Multiply a · c: 6 · (−2) = −12.
- Find two numbers that multiply to −12 and add to −1: 3 and −4.
- Split the middle term: 6x² + 3x − 4x − 2.
- Group: 3x(2x + 1) − 2(2x + 1) = (3x − 2)(2x + 1).
- Check with FOIL: 6x² + 3x − 4x − 2 = 6x² − x − 2 ✓
When a = 1, skip grouping. Just find two numbers that multiply to c and add to b.
Reverse FOILing Examples
| Trinomial | Factors | Why |
|---|---|---|
| x² + 5x + 6 | (x + 2)(x + 3) | 2 · 3 = 6 and 2 + 3 = 5 |
| x² − 2x − 15 | (x + 3)(x − 5) | Negative c, so the signs are opposite |
| 2x² + 7x + 3 | (2x + 1)(x + 3) | ac = 6, so the pair is 1 and 6 |
| x² − 9 | (x + 3)(x − 3) | Difference of squares |
| x² + 6x + 9 | (x + 3)² | Perfect square trinomial |
| x² + x + 1 | Prime | No integer pair works |
Sign Rules and Quick Checks
- ac positive: both numbers take the sign of b.
- ac negative: the numbers have opposite signs, and the larger one takes the sign of b.
- Can it be factored? If b² − 4ac is a perfect square, yes. If not, the trinomial is prime, so use the quadratic formula.
- Common factor first: the calculator factors exactly what you type, so 4x² + 10x + 6 returns (4x + 6)(x + 1). Factor out the 2 first to get 2(2x + 3)(x + 1).
Factor Your Next Trinomial
Find two numbers that multiply to ac and add to b, split the middle term, group, and check with FOIL. Try the reverse foiling calculator above with a homework problem. To go the other way, use our FOIL calculator or the box method calculator.
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