What does this calculator do?

This calculator factors quadratic trinomials of the form ax²+bx+c over the integers. When a valid factor pair exists, it rewrites the trinomial as a product of two binomials; when it does not, it states that the expression is not factorable over the integers.

The calculator follows the AC method: multiply a and c, find a product-sum pair, split the middle term, factor by grouping, and verify the result by FOIL.

What does factoring a trinomial mean?

Factoring a trinomial means rewriting a three-term quadratic expression as a multiplication of simpler expressions. For example, x²+5x+6 can be written as (x+2)(x+3).

This is the reverse direction of expanding binomials. If FOIL turns (x+2)(x+3) into x²+5x+6, factoring starts from x²+5x+6 and works back to (x+2)(x+3).

• Trinomial: in this tool, a quadratic expression in the form ax²+bx+c.
• Factoring: rewriting an expression as a product.
• AC method: using a×c to find the two numbers that split the middle term.
• Over the integers: the calculator looks for integer-coefficient factors in its stated scope.

How does the AC method work?

The AC method looks for two integers whose product is a×c and whose sum is b. Those two numbers are used to split the middle term, making factor-by-grouping possible.

1. Compute a×c.
2. Find two integers with product a×c and sum b.
3. Split bx into two terms using that pair.
4. Group the four terms into two pairs.
5. Factor out the common factor from each group.
6. Take out the shared binomial and check the answer by FOIL.

Worked example: factor 2x²+7x+3

For 2x²+7x+3, a=2, b=7, and c=3, so a×c=6. The two integers with product 6 and sum 7 are 1 and 6.

1. Split the middle term: 2x²+7x+3 = 2x²+x+6x+3.
2. Group: (2x²+x)+(6x+3).
3. Factor each group: x(2x+1)+3(2x+1).
4. Extract the shared binomial: (2x+1)(x+3).
5. FOIL check: (2x+1)(x+3)=2x²+6x+x+3=2x²+7x+3.

Perfect squares and special patterns

Some trinomials match familiar patterns. For example, x²+6x+9 factors as (x+3)² because the product-sum pair is 3 and 3.

A difference of squares can also appear inside this family when the middle coefficient is zero, such as x²+0x−9. That factors as (x−3)(x+3).

• Perfect square trinomial: x²+6x+9 → (x+3)².
• Negative perfect square pattern: x²−6x+9 → (x−3)².
• Difference of squares style case: x²−9 → (x−3)(x+3).

How is this different from Reverse FOIL?

The math engine is not different. Reverse FOIL describes the method as the opposite of expanding two binomials. This page describes the same operation using the more common search phrase “factoring trinomials.”

That distinction matters for users. Some users search by method name, while others search by the type of expression they need to factor. Both paths should lead to the same calculation behavior.

Related methods and when to use them

Several nearby tools solve different parts of the same algebra workflow. Choosing the right one prevents confusing expansion, factoring, and visual methods.

• Use polynomial multiplication when you want to expand two polynomial expressions.
• Use FOIL when you want to multiply two binomials.
• Use the box method when you want a visual layout for the same AC factoring idea.
• Use the diamond problem when you only need the product-sum pair.
• Use the perfect square trinomial calculator when the expression may be (px±q)².

How to use the calculator

Enter the integer coefficients a, b, and c from ax²+bx+c. The calculator then shows the AC product, the factor pair if one exists, the middle-term split, and the final factored form.

1. Enter a; it must not be zero.
2. Enter b, the middle coefficient.
3. Enter c, the constant term.
4. Read the product-sum pair and split step.
5. Check the final product with the FOIL verification.

Common mistakes

The most common mistake is looking only for two numbers that add to b. In the AC method, the same pair must also multiply to a×c.

• Using c instead of a×c when a is not 1.
• Losing the negative sign when c is negative.
• Splitting the middle term correctly but grouping incorrectly.
• Assuming every trinomial has integer binomial factors.
• Treating a not-factorable-over-integers result as a statement about all possible roots.

Limitations

This calculator is limited to integer-coefficient quadratic trinomials of the form ax²+bx+c. It does not perform general polynomial factorization, symbolic parameter solving, or factoring with arbitrary irrational coefficients.