Factoring Trinomials Calculator
Everyone who has factored a trinomial knows the pause. You look at 6x² + 7x − 3, you know you need two numbers, and then you sit there trying pairs in your head: 1 and 18, 2 and 9, 3 and 6… one of them adds to 7 and you cannot remember which.
That pause is the whole topic, and it is the one thing calculators refuse to show you. They print (2x + 3)(3x − 1) and leave the search invisible. This factoring trinomials calculator shows the search: every factor pair of a×c, each one tested against b, with the one that works marked in green.
The Pair Is the Whole Method
Every trinomial ax² + bx + c comes down to one question: which two numbers multiply to a×c and add to b?
Find them and everything else is bookkeeping. For 6x² + 7x − 3, a×c is −18 and you need a sum of 7:
1 × −18 = −18 sum −17
−1 × 18 = −18 sum 17
2 × −9 = −18 sum −7
−2 × 9 = −18 sum 7 ✓
Four attempts, and the fourth one is it. Nothing clever happened — it is a list, worked through in order. Seeing it as a list is what turns this from guessing into a method.
When a = 1, and When It Is Not
With a leading coefficient of 1 the job is smaller, because a×c is just c. For x² + 5x + 6 you want two numbers multiplying to 6 and adding to 5, which is 2 and 3, so it is (x + 2)(x + 3). Most people can do these by eye after a while.
When a is not 1, eyeballing stops working and you need the AC method. Multiply a by c first, find the pair for that product, then split the middle term with it. The multiplication is the whole trick: it converts an ax² problem into the same search you already know how to do.
Splitting the Middle Term
Once you have the pair, write the middle term as those two terms instead of one. For 6x² + 7x − 3 with the pair −2 and 9:
6x² − 2x + 9x − 3
= 2x(3x − 1) + 3(3x − 1)
= (3x − 1)(2x + 3)
Four terms now, worth exactly the same as three — because −2 and 9 add back to 7. Take a common factor out of the first pair and out of the second, and the same bracket appears twice. That bracket lifts out, and what is left of the two front factors becomes the other one.
The order does not matter. Write it 6x² + 9x − 2x − 3 instead and you get the same two brackets with the front and back swapped.
Take Out the Common Factor First
2x² + 4x − 6 shares a 2 across all three terms. Pull it out and you are left with 2(x² + 2x − 3), and that trinomial has a leading coefficient of 1 — a much easier search than the one you started with.
Skipping this does not make your answer wrong, but it makes the numbers bigger than they needed to be, and it is the commonest reason a factorisation is marked incomplete. If you want the full ladder of techniques rather than trinomials alone, the factoring polynomials calculator works through all of them in order.
When There Is No Pair
x² + x + 1 has no pair. Neither does x² − 2. That is not a failure of technique — the numbers genuinely do not exist.
You can settle it before you start searching. The discriminant b² − 4ac tells you: a trinomial factors over the whole numbers only when that is a perfect square. For x² + x + 1 it is −3, so the trinomial never touches the x-axis and there is nothing to find. For x² − 2 it is 8, which is positive but not a square, so the zeros are irrational and no whole-number pair exists.
The calculator says which of those two it is, rather than just refusing.
How to Use This Factoring Trinomials Calculator
Type the trinomial and press =. Terms can be in any order, brackets are expanded first, and a common factor is taken out for you. The keypad has x, powers, brackets and a fraction bar; the key icon switches to your own keyboard.
The band gives the factored form and the pair that produced it. Tap show how the pair was found and the whole search opens — every factor pair tested, the winner in green. Underneath, the working: the product a×c, the search, the middle term split, the grouping, and a final line that multiplies the brackets back out to check.
Worked Examples
| Trinomial | a×c | The pair | Factored |
|---|---|---|---|
| x² + 5x + 6 | 6 | 2 and 3 | (x + 2)(x + 3) |
| 6x² + 7x − 3 | −18 | −2 and 9 | (2x + 3)(3x − 1) |
| 12x² + 7x − 10 | −120 | −8 and 15 | (4x + 5)(3x − 2) |
| 4x² − 12x + 9 | 36 | −6 and −6 | (2x − 3)² |
| 2x² + 4x − 6 | −3 | −1 and 3 | 2(x + 3)(x − 1) |
| x² + x + 1 | 1 | none exists | prime |
The fourth row is worth a look. Both numbers came out the same, which is what a perfect square trinomial is — and it is also why its discriminant is exactly zero.
Frequently Asked Questions
How do you factor a trinomial?
What is the AC method?
How do you factor a trinomial when a is 1?
What if the trinomial will not factor?
What is a perfect square trinomial?
Do you take out a common factor before factoring a trinomial?
Is this factoring trinomials calculator free?
Factoring trinomials is a search, not a trick. Take the common factor out, multiply a by c, work through the factor pairs until one adds to b, split the middle term and group. The only part that takes practice is the search itself — so use the calculator above to watch it happen a few dozen times, and it stops being a pause.