Factoring Trinomials Calculator (ax² + bx + c, ac method step by step)

Understanding the factoring trinomials calculator — concept and principle

This calculator factors a quadratic trinomial ax² + bx + c with the ac (grouping) method, step by step: greatest common factor, the two numbers whose product is ac and sum is b, splitting the middle term, and grouping. It also shows the discriminant, which tells whether the trinomial factors over the integers. Use it for algebra homework.

How it is calculated

1. Take out the greatest common factor of a, b and c.
2. For Ax² + Bx + C, find whole numbers m and n with m·n = A·C and m + n = B.
3. Split the middle term: Ax² + mx + nx + C.
4. Factor each pair and pull out the common bracket: (px + q)(rx + s).

• Such m and n exist exactly when the discriminant B² − 4AC is a perfect square; otherwise the trinomial is prime over the integers.

The principle

Multiplying (px + q)(rx + s) gives prx² + (ps + qr)x + qs. The two middle products ps and qr multiply to pr·qs, which is exactly A·C, and add up to B. So searching for two numbers with product AC and sum B recovers how the middle term was built. Splitting it and grouping then undoes the multiplication. The discriminant test works because the roots are rational exactly when √(B² − 4AC) is a whole number.

Worked example

Default inputs: a = 6, b = 7, c = −20.

1. GCF = 1, so the trinomial stays 6x² + 7x − 20.
2. a·c = 6 × (−20) = −120; two numbers with product −120 and sum 7: m = −8, n = 15.
3. Split the middle term: 6x² − 8x + 15x − 20.
4. Group in pairs: 2x(3x − 4) + 5(3x − 4).
5. Factored form: (2x + 5)(3x − 4); discriminant 7² − 4 × 6 × (−20) = 529 = 23².

How to read the result

Multiply back to check: (2x + 5)(3x − 4) = 6x² − 8x + 15x − 20 = 6x² + 7x − 20. The factors also give the roots of 6x² + 7x − 20 = 0: x = −5/2 and x = 4/3. If the discriminant is not a perfect square, use the quadratic formula instead.

Common mistakes

• Forgetting to take out a common factor first.
• Choosing a pair with the right product but the wrong sum.
• Losing a sign when factoring the second pair.
• Declaring a trinomial prime without checking the discriminant.

Frequently asked questions

What is the ac method?

Find two numbers that multiply to ac and add to b, split the middle term with them, and factor by grouping.

How do I know if a trinomial can be factored?

With integer coefficients, it factors over the integers exactly when b² − 4ac is a perfect square.

What if a is negative?

Take out −1 (with the GCF) first so the leading coefficient is positive.

Sources: OpenStax, Elementary Algebra 2e (2020), §7.3 Factor Trinomials of the Form ax² + bx + c