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Zuletzt aktualisiert: 19. August 2026

Umgekehrtes FOIL Rechner

Quick Answer

Reverse FOIL factors a monic quadratic x² + bx + c by finding integers m and n such that m + n = b and mn = c. This calculator returns the factored form, the individual factors, and the discriminant so users can confirm whether clean integer factoring is available.

To reverse FOIL a monic quadratic, find two integers whose sum is the x coefficient and whose product is the constant term.

Wichtige Erkenntnisse

  • Reverse FOIL works with monic trinomials x² + bx + c.
  • You need two integers whose sum is b and product is c.
  • A negative constant means opposite signs.
  • A repeated pair creates a perfect square.
  • Expanding the factors is the best final check.
Dharmendra Singh - Creator
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Formel

For x² + bx + c, find integers m and n such that m + n = b and mn = c; then x² + bx + c = (x + m)(x + n)

Wobei:

  • b=Middle coefficient
  • c=Constant term
  • m,n=Numbers whose sum is b and product is c
Reverse FOIL CalculatorDiagram showing the sum and product search for x squared plus 7x plus 12.Reverse FOILx² + 7x + 12Sum = 7Product = 123 and 43 + 4 = 73 × 4 = 12Factored form(x + 3)(x + 4)
Reverse FOIL works by matching a required sum and product to build two binomial factors.

Rechenbeispiele

x² + 7x + 12

The pair 3 and 4 has the needed sum and product.

  1. 1List factor pairs of 12: (1, 12), (2, 6), (3, 4).
  2. 2Only 3 + 4 = 7.
  3. 3So x² + 7x + 12 = (x + 3)(x + 4).
Endergebnis: factorization = (x + 3)(x + 4)

x² − x − 12

A negative constant means the two numbers have opposite signs.

  1. 1Find factors of -12 whose sum is -1.
  2. 23 and -4 multiply to -12 and add to -1.
  3. 3Therefore x² − x − 12 = (x + 3)(x − 4).
Endergebnis: factorization = (x + 3)(x - 4)

x² + 10x + 25

A perfect-square trinomial uses a repeated factor.

  1. 1Look for two numbers whose product is 25 and sum is 10.
  2. 2The pair 5 and 5 works.
  3. 3So x² + 10x + 25 = (x + 5)(x + 5).
Endergebnis: factorization = (x + 5)(x + 5)

Einführung

The Reverse FOIL Calculator factors monic quadratic trinomials of the form x² + bx + c by reversing the usual binomial multiplication pattern. Instead of expanding two factors, you work backward to find the two integers whose sum matches the middle coefficient and whose product matches the constant term. This page keeps the method intentionally simple, explains when it works cleanly over the integers, and shows how the discriminant can support the factoring decision when you want an additional algebra check.

What reverse FOIL means

FOIL is a mnemonic for multiplying two binomials. Reverse FOIL asks the inverse question: if a trinomial is already expanded, which two binomials would multiply to create it?

  • This calculator focuses on monic quadratics.

  • You look for a sum and a product at the same time.

  • The target form is x² + bx + c.

  • Integer factoring is the main goal here.

How the method works

To factor x² + bx + c, search for integers m and n such that m + n = b and mn = c. If such integers exist, then the trinomial factors as (x + m)(x + n).

  • Sum must equal the middle coefficient.

  • Product must equal the constant term.

  • Negative c means opposite signs.

  • Positive c means the two signs match.

Worked example

For x² + 7x + 12, the factor pairs of 12 are 1 and 12, 2 and 6, and 3 and 4. Only 3 and 4 add to 7, so the factorization is (x + 3)(x + 4).

  • List factor pairs systematically.

  • Check the sum after checking the product.

  • Write the binomial factors carefully.

  • Expand mentally to verify the answer.

Inputs and outputs

You enter the middle coefficient and constant term of a monic trinomial. The calculator then returns the factored form, each factor separately, a yes-or-no indicator for integer factorization, and the discriminant.

  • The leading coefficient is fixed at 1.

  • Both inputs must be integers.

  • The factored form is the main output.

  • The discriminant helps cross-check the algebra.

Common mistakes

The most common mistake is finding numbers with the right product but the wrong sum. Another is forgetting that a negative constant requires one positive and one negative factor.

  • Check the sign of the constant first.

  • Do not stop after matching the product alone.

  • Watch for perfect-square trinomials.

  • Verify the factorization by multiplying back out.

When this method is useful

Students use reverse FOIL in algebra classes to solve equations, identify intercepts, and simplify expressions. It is also useful whenever a quick factorization reveals structure faster than a longer symbolic method.

  • Solving quadratic equations by zero-product reasoning.

  • Recognizing perfect-square patterns.

  • Finding roots quickly when factors are integers.

  • Checking homework and worksheet answers efficiently.

Manual method versus calculator use

Reverse FOIL is easy to practice by hand for small constants, but a calculator reduces sign errors and confirms borderline cases quickly. The fastest habit is to estimate likely factor pairs before reading the exact output.

  • Start with factor pairs closest together.

  • Use the sign pattern to narrow the search.

  • Multiply the factors back out to verify.

  • Switch to the quadratic formula when integer factoring is impossible.

Kurzreferenzkarte

Reverse FOIL quick reference

Kurzreferenz • Umgekehrtes FOIL Rechner

Find m and n with m + n = b and mn = c

Gültiger Bereich: Use integer values for b and c in monic trinomials x² + bx + c.

Häufige Werte

b = 7, c = 12(x + 3)(x + 4)
b = -1, c = -12(x + 3)(x - 4)
b = 10, c = 25(x + 5)(x + 5)
No matching pairNot factorable over integers

⚠ Achtung

  • •Do not use this simplified method when the leading coefficient is not 1.
  • •Do not match the product without checking the sum.
  • •Do not forget sign patterns.
  • •Do not skip expansion as a final verification step.

Profi-Tipps

  • →Start with factor pairs near √|c|.
  • →Use the sign of c to predict the signs.
  • →Recognize repeated factors quickly.
  • →Switch methods immediately if no integer pair works.

FAQ

What does reverse FOIL do?

It factors a monic trinomial x² + bx + c by finding two integers whose sum is b and whose product is c.

Why does the calculator ask only for b and c?

This version is intentionally limited to monic quadratics, where the leading coefficient of x² is 1.

How do I know whether integer factoring is possible?

If no integer pair satisfies both the required sum and product, the trinomial is not factorable over the integers.

What does a negative constant term tell me?

It tells you the two numbers must have opposite signs because their product is negative.

What if the same number works twice?

Then the trinomial is a perfect square, such as x² + 10x + 25 = (x + 5)².

Why is the discriminant shown?

The discriminant helps you connect factoring with root behavior and provides a second algebra check.

Can I use this for ax² + bx + c when a is not 1?

Not this simplified reverse-FOIL version. For non-monic quadratics, use a broader trinomial factoring method.

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