Factor Quadratics Calculator: AC Method, Step by Step
This factor quadratics calculator turns Ax² + Bx + C into brackets and shows every step of the AC method, so you can check homework or find where your own working went wrong. Type whole numbers for A, B and C, select Factor equation, and read the Factored form and the numbered Steps. It takes out a common factor, copes with a negative A, recognises perfect squares and a difference of squares, and says plainly when a quadratic is prime. It runs in your browser and nothing is uploaded.
How to factor a quadratic with this calculator
- Type the number in front of x² into the box labelled A (x² term). It starts at 1, and a blank A counts as 1.
- Type the number in front of x into B (x term) and the plain number into C (constant). A blank B or C counts as 0.
- For a negative number, type a minus sign or select ± under the box to flip its sign. Phone number pads often have no minus key, which is what the button is for.
- Select Factor equation.
- Read Factored form in large type, then work down the numbered Steps.
For 2x² + 2x − 24, enter A as 2, B as 2 and C as −24. Changing any box clears the result, so select Factor equation again after an edit. Only whole numbers between −1,000,000 and 1,000,000 are accepted, and A cannot be 0 because there would be no x² term left.
How the AC method works, step by step
The AC method factors Ax² + Bx + C even when A is not 1, which is where simple guess-and-check stops working. The calculator follows it in the same order every time: factor out the greatest common factor, multiply the first and last numbers, find two whole numbers that multiply to that product and add to B, split the middle term, group in pairs, then factor out the shared bracket. This is its working for 2x² + 5x + 3:
2. The three numbers have no common factor, so there is nothing to pull out first.
3. Multiply the first and last numbers: 2 × 3 = 6.
4. Find two whole numbers that multiply to 6 and add to 5. They are 2 and 3, because 2 × 3 = 6 and 2 + 3 = 5.
5. Split the middle term using those two numbers: 2x² + 2x + 3x + 3.
6. Group the terms in pairs and take the common factor out of each pair: (2x² + 2x) + (3x + 3) = 2x(x + 1) + 3(x + 1).
7. Both pairs now contain (x + 1), so factor it out: (x + 1)(2x + 3).
The brackets are listed with the smaller x coefficient first, so you may see (x + 1)(2x + 3) where a textbook writes (2x + 3)(x + 1). The product is the same.
Factoring with a common factor
When A, B and C share a factor, step 2 pulls it out first. For 2x² + 2x − 24 it reads Factor out the greatest common factor, 2: 2(x² + x − 12). The rest of the steps factor x² + x − 12 into (x − 3)(x + 4), and the last step reads Put the 2 back in front: 2(x − 3)(x + 4). If A is negative, the factor is signed so the x² term inside the brackets is positive: −x² + 5x − 6 becomes −(x − 3)(x − 2).
Perfect squares, difference of squares and other special cases
None of these needs a different mode. The same steps run, and the answer shows the pattern.
| A, B, C | Factored form | What happens |
|---|---|---|
| 1, 6, 9 | (x + 3)² | Perfect square: the two brackets match, so the last step writes them as a square. |
| 1, 0, −9 | (x − 3)(x + 3) | Difference of squares: the pair is −3 and 3, and the middle terms cancel. |
| 3, 6, 0 | 3x(x + 2) | C is 0, so x is a common factor along with 3. |
| 4, 0, 0 | 4x² | Only an x² term, so after taking out 4 nothing more can be factored. |
| −1, 5, −6 | −(x − 3)(x − 2) | Negative A: −1 is taken out first. |
| 1, 1, 1 | Prime | No pair of whole numbers works. |
| 2, 2, 2 | 2(x² + x + 1) | The 2 comes out, and the part in brackets is prime. |
When a quadratic is prime
The calculator shows Prime when no two whole numbers multiply to A × C and add to B (after any common factor is out). Then the quadratic cannot be written as brackets with whole numbers. If a common factor came out first, you get that factor in front of the prime part instead, with a note under the answer.
A quick check you can do yourself: whole-number brackets exist only when B² − 4AC is a perfect square (0, 1, 4, 9, 16 and so on). For x² + x + 1 it is −3, so the quadratic is prime. For x² − 2 it is 8, which is not a perfect square, so that one is prime too.
What this calculator does not do
It factors; it does not solve for x or graph the parabola. Decimals and fractions are rejected with a message, so scale them up to whole numbers first. When you need the roots, set each bracket to 0 by hand. To practise the multiplication facts the pair search relies on, try basic math flashcards, and to check arithmetic along the way use the expression calculator.
Frequently asked questions
How do I factor quadratics when a is not 1?
Use the AC method. Multiply A by C, find two whole numbers that multiply to that product and add to B, split the middle term into those two numbers, then group in pairs and factor out the shared bracket. For 2x² + 5x + 3, A × C is 6, the numbers are 2 and 3, and the result is (x + 1)(2x + 3). Type 2, 5 and 3 into A, B and C and the calculator shows each of those steps.
What is the AC method for factoring?
The AC method is a way to factor Ax² + Bx + C when A is not 1. You multiply A by C, look for a pair of whole numbers that multiply to AC and add to B, and use that pair to split the middle term into two. The four terms then factor by grouping. It also works when A is 1, which is why the calculator uses the same steps for every quadratic.
How do I factor a quadratic with a common factor?
Take the greatest common factor of A, B and C out first, then factor what is left and put the common factor back in front. For 2x² + 2x − 24 the Steps list starts with 2(x² + x − 12) and ends with 2(x − 3)(x + 4). If the three numbers share nothing, the calculator says there is nothing to pull out first.
How do I factor a quadratic with a negative leading coefficient?
Factor out −1 (or a negative common factor) first, so the x² term inside the brackets is positive. Select ± beside A to make it negative, then select Factor equation. For −x² + 5x − 6 the answer is −(x − 3)(x − 2), and the Steps list shows the −1 being taken out at the start and put back in front at the end.
Can all quadratics be factored?
Not into brackets with whole numbers. Quadratics such as x² + x + 1 or x² − 2 have no whole-number factors, and the calculator reports them as Prime. Their roots are irrational or complex, so they cannot be written as brackets with whole-number coefficients.
How can you tell if a quadratic is prime?
Take the common factor out, then look for two whole numbers that multiply to A × C and add to B. If none exist, the quadratic is prime. The same test can be run on the discriminant: B² − 4AC must be a perfect square for whole-number factors to exist. For x² + x + 1 no pair multiplies to 1 and adds to 1, and B² − 4AC is −3, so it is prime.
How do I factor a difference of squares?
A difference of squares such as x² − 9 factors as (x − 3)(x + 3). Enter A as 1, leave B blank and enter C as −9. The calculator reaches the answer through the usual steps: A × C is −9, the pair is −3 and 3, and the split is x² − 3x + 3x − 9. Quadratics like 4x² − 25 work the same way.
How do I factor a perfect square trinomial?
A perfect square trinomial such as x² + 6x + 9 factors into two identical brackets, written (x + 3)². Enter 1, 6 and 9. The calculator finds the pair 3 and 3, factors out the shared bracket to get (x + 3)(x + 3), then adds a final step that writes the matching brackets as a square.
Can this calculator solve a quadratic equation or find the roots?
No. It factors Ax² + Bx + C into brackets and shows the steps, but it does not solve for x or draw the graph. Once you have the brackets you can finish by hand: set each bracket to 0. From 2(x − 3)(x + 4) that gives x = 3 and x = −4.
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