Quick Answer

The expression x^2 + 5x + 6 factors into (x + 2)(x + 3), with roots x = -2 and x = -3.

Enter integer coefficients for ax² + bx + c

a (x² coefficient)

b (x coefficient)

c (constant)

Common Examples

Input	Result
a=1, b=5, c=6	(x + 2)(x + 3)
a=1, b=-7, c=12	(x - 3)(x - 4)
a=2, b=7, c=3	(2x + 1)(x + 3)
a=1, b=-1, c=-12	(x + 3)(x - 4)
a=3, b=12, c=12	3(x + 2)(x + 2)

How It Works

The AC Method

The AC method (also called factoring by grouping) is the standard technique for factoring quadratic trinomials of the form ax^2 + bx + c:

1. Compute the product a * c.
2. Find two integers m and n such that m * n = a * c and m + n = b.
3. Rewrite the middle term: ax^2 + mx + nx + c.
4. Factor by grouping the first two terms and the last two terms.
5. Extract the common binomial factor to get the final result.

If no such integer pair m, n exists, the quadratic is not factorable over the integers. It may still have irrational or complex roots, which can be found using the quadratic formula.

Special Cases

Difference of squares: a^2 - b^2 = (a + b)(a - b). For example, x^2 - 9 = (x + 3)(x - 3).

Perfect square trinomial: a^2 + 2ab + b^2 = (a + b)^2. For example, x^2 + 6x + 9 = (x + 3)^2.

GCF extraction: Always factor out the greatest common factor first. For example, 6x^2 + 12x + 6 = 6(x^2 + 2x + 1) = 6(x + 1)^2.

Worked Example

Factor 2x^2 + 7x + 3.

Compute a * c = 2 * 3 = 6. Find m and n where m * n = 6 and m + n = 7. The pair is m = 6, n = 1.

Rewrite: 2x^2 + 6x + 1x + 3. Group: (2x^2 + 6x) + (x + 3) = 2x(x + 3) + 1(x + 3) = (2x + 1)(x + 3).

Verification: (2x + 1)(x + 3) = 2x^2 + 6x + x + 3 = 2x^2 + 7x + 3. The factorization is correct.

The roots are x = -1/2 and x = -3. The discriminant is 7^2 - 4(2)(3) = 49 - 24 = 25, which is a perfect square, confirming rational roots.

Related Calculators

Quadratic Formula Calculator

Solve quadratic equations (ax^2 + bx + c = 0), find roots, vertex, and graph properties.

Prime Factorization Calculator

Break any integer into its prime factors and find all positive divisors.

Frequently Asked Questions

What does it mean to factor a quadratic expression?

Factoring a quadratic means rewriting ax^2 + bx + c as a product of simpler expressions, typically two binomials like (px + q)(rx + s). The factored form reveals the roots (zeros) of the expression and simplifies solving equations.

What if the expression cannot be factored?

Not all quadratic expressions factor neatly over the integers. If the discriminant (b^2 - 4ac) is not a perfect square, the roots are irrational and the expression does not factor into binomials with integer coefficients. The quadratic formula can still find the exact roots.

How does factoring relate to the quadratic formula?

Both methods find the roots of ax^2 + bx + c = 0. Factoring works when the roots are rational numbers. The quadratic formula x = (-b +/- sqrt(b^2 - 4ac)) / (2a) works for all quadratics, including those with irrational or complex roots.

What is the GCF and why extract it first?

The GCF (greatest common factor) is the largest integer that divides all three coefficients a, b, and c. Extracting it first simplifies the remaining expression. For example, 4x^2 + 8x + 4 becomes 4(x^2 + 2x + 1) = 4(x + 1)^2, which is much easier to factor.

Can this calculator handle negative coefficients?

Yes. Enter any integer values for a, b, and c, including negative numbers. For example, x^2 - 5x + 6 (a=1, b=-5, c=6) factors into (x - 2)(x - 3).