Math
Factoring Trinomials Calculator
Enter a, b, and c and the factoring trinomials calculator runs the ac-method: it pulls out the GCF, lists every factor pair of a × c with its sum, splits the middle term, factors by grouping, and checks by expanding.
If no pair works, it says the trinomial is prime and gives the roots from the quadratic formula instead.
Whole number, not 0. Negative a is factored out as -1 first.
Whole number. Use 0 for a difference of squares such as x^2 - 9.
Whole number.
Try an example
Result
Factored form
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- Full statement
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- Pattern
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- GCF pulled out
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- a × c
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- Factor pairs checked
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- Discriminant
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More details (3 more)
- Roots (zeros)
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- Roots (decimal)
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- Check by expanding
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Study path
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What the factoring trinomials calculator solves
A trinomial ax^2 + bx + c factors over the integers when it can be written as a product of two binomials with whole-number The number multiplying a variable, like the 3 in 3x., such as 6x^2 + 11x + 4 = (2x + 1)(3x + 4). The ac-method turns the guess-and-check into a short search: find two integers whose product is a × c and whose sum is b.
Worked example: 6x^2 + 11x + 4
| Pair (m, n) | Product | Sum | Matches b = 11? |
|---|---|---|---|
| 1, 24 | 24 | 25 | no |
| 2, 12 | 24 | 14 | no |
| 3, 8 | 24 | 11 | yes |
| 4, 6 | 24 | 10 | no |
With m = 3 and n = 8: 6x^2 + 3x + 8x + 4 = 3x(2x + 1) + 4(2x + 1) = (2x + 1)(3x + 4).
Special patterns it recognizes
- Difference of squares: x^2 - 9 = (x - 3)(x + 3). Here b = 0 and c is a negative perfect square (times a square).
- Perfect square trinomial: 4x^2 - 12x + 9 = (2x - 3)^2. The The b² − 4ac part of the quadratic formula; its sign tells you how many real solutions exist. is 0.
- Common factor first: 2x^2 + 4x - 6 = 2(x - 1)(x + 3). The The largest number that divides evenly into every number in a set. is pulled out before the pair search.
- Missing constant: 3x^2 + 6x = 3x(x + 2). Factor out x.
How to use it
- Write the trinomial in the order ax^2 + bx + c and enter the three whole-number coefficients with their signs.
- Read the factored form at the top. The steps list every factor pair of a × c and mark the one whose sum is b.
- If the result says the trinomial does not factor, use the roots shown from the quadratic formula.
How to read the answer
The two binomials multiply back to the original trinomial; the check line expands them so you can verify. Setting each factor to 0 gives the roots, which are the x-A point where a graph crosses an axis. of the parabola.
Common mistakes and edge cases
- Forgetting the sign pattern: if a × c is negative, one of m and n is negative; if a × c is positive and b is negative, both are negative.
- Skipping the GCF. 2x^2 + 4x - 6 factors as 2(x - 1)(x + 3), and the pair search is easier on x^2 + 2x - 3.
- Stopping after splitting the middle term. Grouping must produce the same binomial in both groups.
- Trying to factor a A whole number above 1 whose only factors are 1 and itself. trinomial such as x^2 + x + 1. Check the discriminant first.
Worked examples
Leading coefficient 6
6x^2 + 11x + 4 = (2x + 1)(3x + 4)
Factored form
(2x + 1)(3x + 4)
Monic with negative constant
x^2 - 2x - 15 = (x - 5)(x + 3)
Factored form
(x + 3)(x - 5)
Common factor first
2x^2 + 4x - 6 = 2(x - 1)(x + 3)
Factored form
2(x - 1)(x + 3)
Difference of squares
x^2 - 9 = (x - 3)(x + 3)
Factored form
(x - 3)(x + 3)
Perfect square
4x^2 - 12x + 9 = (2x - 3)(2x - 3)
Factored form
(2x - 3)(2x - 3)
Does not factor
x^2 + x + 1 is prime; roots are complex
Factored form
Does not factor over the integers
Negative leading coefficient
-x^2 + 5x - 6 = -(x - 2)(x - 3)
Factored form
-(x - 2)(x - 3)
a = 0 is not a trinomial
0x^2 + 3x + 6
Factored form
Error
Frequently asked questions
What is the ac-method?+
Multiply a and c, then find two integers m and n whose product is a × c and whose sum is b. Rewrite bx as mx + nx and factor by grouping. It works for any trinomial that factors over the integers, including ones with a ≠ 1.
How do I know if a trinomial can be factored?+
Compute the discriminant b^2 - 4ac. If it is a perfect square (0, 1, 4, 9, 16, ...), the trinomial factors over the integers. Otherwise it is prime and you need the quadratic formula for the roots.
What does it mean when a trinomial is prime?+
It cannot be written as a product of two binomials with integer coefficients. It may still have irrational real roots, like x^2 - 3x - 2, or complex roots, like x^2 + x + 1.
Does the order of the factors matter?+
No. (2x + 1)(3x + 4) and (3x + 4)(2x + 1) are the same product. Your answer may list them in a different order from the calculator.
Why factor out the GCF first?+
It keeps the numbers small and guarantees the remaining trinomial is primitive, so the pair search is shorter and the final answer is fully factored.
How do I factor when a is negative?+
Factor out -1 first so the leading coefficient is positive, then use the ac-method on what is left. The calculator writes the answer as -(...)(...).
About this calculator
- Written by
- mathcheck editorial team
- Last reviewed
- September 4, 2026
Method
- Uses the values entered by the user and stable formulas documented on the page.
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Last updated: September 4, 2026