Math July 13, 2026 · 8 Min Read

Reverse FOIL Calculator – Guide & Formulas

Factor quadratic trinomials using reverse FOIL. Enter ax² + bx + c to see factored form (px + q)(rx + s) with step-by-step solutions.

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Put these formulas into practice with our instant, step-by-step Reverse FOIL Calculator.

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Use our free **Reverse FOIL Calculator** to **factor quadratic trinomials** with step-by-step results. This **trinomial factoring calculator** helps you **find binomial factors** of ax² + bx + c accurately. Whether you need to **factor by grouping** or **verify the factored form**, this tool provides complete breakdowns. Features include **coefficient analysis**, **factor pair testing**, and **verification by expansion**. 100% free — no signup required!

Key Takeaway

Use the free Reverse FOIL Calculator to factor quadratic trinomials using reverse foil. enter ax² + bx + c to see factored form (px + q)(rx + s) with step-by-step solutions. Get instant results with step-by-step explanations.

How to Use the Reverse FOIL Calculator

  1. Step 1: Enter the **coefficient a** (of x²) into the reverse FOIL calculator.
  2. Step 2: Enter the **coefficient b** (of x) to set up the trinomial.
  3. Step 3: Enter the **constant c** to complete the quadratic expression.
  4. Step 4: Click Factor to see the **factored form** (px + q)(rx + s).
  5. Step 5: Review the **step-by-step breakdown** showing how factors were found.
  6. Step 6: Follow the **factor pair analysis** for the constant and leading coefficient.
  7. Step 7: Verify the result by **expanding the factored form** using FOIL to confirm it matches the original.

The Formula

Reverse FOIL: Find two binomials (px + q) and (rx + s) whose product equals the original trinomial. The coefficients satisfy: p×r = a, q×s = c, and p×s + q×r = b.

Variable Definitions

  • a: Coefficient of the x² term
  • b: Coefficient of the x term
  • c: The constant term
  • p, r: Factors of a (leading coefficient)
  • q, s: Factors of c (constant term)
  • FOIL: First, Outer, Inner, Last — the method for multiplying binomials

Factoring x² + 5x + 6

Factor the trinomial x² + 5x + 6 using reverse FOIL.

  1. Step 1: Identify a = 1, b = 5, c = 6
  2. Step 2: Find factors of c = 6 that add to b = 5: (2, 3) since 2 + 3 = 5
  3. Step 3: Since a = 1, the factored form is (x + 2)(x + 3)
  4. Step 4: Verify by expanding: (x + 2)(x + 3) = x² + 3x + 2x + 6 = x² + 5x + 6 —

Frequently Asked Questions

What is the FOIL method?

FOIL stands for First, Outer, Inner, Last — the four products you multiply when expanding (a+b)(c+d) = ac + ad + bc + bd. Reverse FOIL undoes this process.

What if a ≠ 1?

When a ≠ 1, you need to find factors of a×c that add to b, then use factoring by grouping. For example, 2x² + 7x + 3 = (2x + 1)(x + 3).

Can all trinomials be factored?

No. Some trinomials are prime (irreducible) over the integers, meaning they cannot be factored into integer binomials. The discriminant b² - 4ac < 0 or not being a perfect square indicates this.

How do I know if a trinomial can be factored?

Check the discriminant: D = b² - 4ac. If D is a perfect square, the trinomial factors over integers. If D < 0, it has no real roots and cannot be factored over reals.

What is factoring by grouping?

When a ≠ 1, split the middle term bx into two terms whose coefficients multiply to a×c and add to b, then group and factor common terms from each pair.

How do I factor a trinomial where a = 1?

When a = 1, find two numbers that multiply to c and add to b. These become the constants in the factored form (x + p)(x + q). For x² + 5x + 6, find numbers that multiply to 6 and add to 5: 2 and 3, giving (x + 2)(x + 3).

What if the trinomial cannot be factored over integers?

If the discriminant b² - 4ac is not a perfect square, the trinomial cannot be factored over integers. It is called prime or irreducible over the integers. You can still find real roots using the quadratic formula.