Pythagorist
Algebra

Factorising Calculator

Factorise quadratic expressions ax² + bx + c, find common factors, recognise difference of two squares, and more — with full step-by-step working. The calculator shows exactly how to find the factor pairs.

Factorise ax² + bx + c

Factorising: 1x² -5x +6

Factorised form

(x -3)(x -2)

How it works

  1. Write the expression

    1x2−5x+61x^2 -5x +6

    Factorise 1x² -5x +6.

  2. Find two numbers

    We need two numbers that multiply to c (6) and add to b (-5).

  3. Found: -3 and -2

    −3×−2=6and−3+−2=−5-3 \times -2 = 6 \quad \text{and} \quad -3 + -2 = -5

    -3 × -2 = 6 ✓ and -3 + -2 = -5 ✓

  4. Write the factorised form

    (x−3)(x−2)(x -3)(x -2)

    The factorised form is (x -3)(x -2).

    Result:(x -3)(x -2)

The formula

x2+bx+c=(x+p)(x+q)where p+q=b and pq=cx^2 + bx + c = (x + p)(x + q) \quad \text{where } p + q = b \text{ and } pq = c
p,qp, qTwo numbers that multiply to c and add to b

Example substitution

x2−5x+6:  pq=6,  p+q=−5  ⟹  p=−2,q=−3x^2 - 5x + 6: \; pq = 6, \; p + q = -5 \implies p = -2, q = -3

Find two numbers that multiply to 6 and add to −5: they are −2 and −3. So x² − 5x + 6 = (x − 2)(x − 3).

Worked examples

Factorise x² + 7x + 12

  1. 1
    Find p and q: pq = 12, p + q = 7
  2. 2
    p = 3, q = 4: 3 × 4 = 12, 3 + 4 = 7 ✓
Answer: (x + 3)(x + 4)

Factorise x² − x − 6

  1. 1
    Find p and q: pq = −6, p + q = −1
  2. 2
    p = 2, q = −3: 2 × (−3) = −6, 2 + (−3) = −1 ✓
Answer: (x + 2)(x − 3)

Factorise x² − 9

  1. 1
    Difference of squares: b = 0, c = −9
  2. 2
    Recognise: x² − 3²
Answer: (x + 3)(x − 3)

Frequently asked questions

How do you factorise a quadratic?+

Find two numbers that multiply to give ac and add to give b. Use these to split the middle term and factor by grouping.

How do you factorise x² + 5x + 6?+

Find two numbers that multiply to 6 and add to 5: 2 and 3. So x² + 5x + 6 = (x+2)(x+3).

How do you factorise using a common factor?+

Find the highest common factor of all terms, then divide each term by it and write it outside a bracket. E.g., 6x² + 4x = 2x(3x + 2).

What is the difference of two squares?+

a² − b² = (a+b)(a−b). Recognise it by two perfect squares with a minus sign between them.

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