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Year 11 · Mathematics · Algebra

Factorising quadratic expressions

Factorise monic and simple non-monic quadratic expressions, including perfect squares and the difference of two squares.

Awaiting educator review. This material was drafted by our content team with AI assistance and is waiting for review by a qualified educator. Please check it suits your child before using it. Created 10 October 2026. How we review

These lessons were written around Australian Curriculum strands and matched to UK year groups by age (Year 1 is the year a child turns 6 in England). They are not mapped to the National Curriculum in England or the curricula of Scotland, Wales or Northern Ireland, so check them against your own plans.

Learning goals

Learners will

  • Take out a highest common factor before factorising further.
  • Factorise monic quadratics x^2 + bx + c, including perfect squares and the difference of two squares.
  • Factorise simple non-monic quadratics ax^2 + bx + c by splitting the middle term.

Success looks like

  • I can factorise an expression such as x^2 + 7x + 12 by finding two numbers that multiply to 12 and add to 7.
  • I can recognise and factorise a perfect square and a difference of two squares.
  • I can factorise an expression such as 3x^2 + 10x + 8 and check my answer by expanding.

The big idea

Factorising is the reverse of expanding. Expanding (x + 3)(x + 4) gives x^2 + 7x + 12; factorising x^2 + 7x + 12 takes us back to (x + 3)(x + 4). Factorising is the key skill for solving quadratic equations and simplifying algebraic fractions.

Step 1 – always look for a common factor first. For 3x^2 + 12x, both terms share 3x, so 3x^2 + 12x = 3x(x + 4). For 2x^2 − 18, take out 2 to get 2(x^2 − 9), then keep factorising. 'Factorise fully' means no bracket can be factorised any further.

Monic quadratics x^2 + bx + c. Find two numbers that multiply to c and add to b. For x^2 + 7x + 12: the numbers are 3 and 4 (3 × 4 = 12, 3 + 4 = 7), so x^2 + 7x + 12 = (x + 3)(x + 4).

Watch the signs. If c is positive, both numbers have the same sign as b. If c is negative, the numbers have opposite signs. For x^2 − 2x − 15: we need a product of −15 and a sum of −2, which gives −5 and 3, so x^2 − 2x − 15 = (x − 5)(x + 3).

Perfect squares follow the pattern a^2 + 2ab + b^2 = (a + b)^2 and a^2 − 2ab + b^2 = (a − b)^2. Spot them when the first and last terms are squares and the middle term is twice the product of their square roots: x^2 − 10x + 25 = (x − 5)^2.

Difference of two squares: a^2 − b^2 = (a − b)(a + b). It has only two terms, both perfect squares, with a minus between them: x^2 − 49 = (x − 7)(x + 7) and 4x^2 − 25 = (2x − 5)(2x + 5). Note that a sum of two squares such as x^2 + 9 does not factorise this way.

Non-monic quadratics ax^2 + bx + c – split the middle term. Multiply a × c, then find two numbers that multiply to ac and add to b. Rewrite the middle term using those numbers and factorise in pairs. For 2x^2 + 7x + 3: ac = 6, and 6 and 1 multiply to 6 and add to 7. So 2x^2 + 6x + x + 3 = 2x(x + 3) + 1(x + 3) = (2x + 1)(x + 3).

Always check by expanding. (2x + 1)(x + 3) = 2x^2 + 6x + x + 3 = 2x^2 + 7x + 3. If you get back to the original expression, your factorisation is correct.

Worked examples

A monic quadratic

Factorise x^2 + 2x − 24.

  1. We need two numbers that multiply to −24 and add to 2.
  2. Try factor pairs of 24 with opposite signs: 6 and −4 give 6 × (−4) = −24 and 6 + (−4) = 2.
  3. Write the factors: (x + 6)(x − 4).
  4. Check: (x + 6)(x − 4) = x^2 − 4x + 6x − 24 = x^2 + 2x − 24.

Answer: (x + 6)(x − 4)

Common factor, then difference of two squares

Factorise fully 3x^2 − 75.

  1. Take out the common factor 3: 3(x^2 − 25).
  2. Recognise x^2 − 25 as a difference of two squares: x^2 − 5^2.
  3. Factorise: 3(x − 5)(x + 5).
  4. Check: 3(x^2 − 25) = 3x^2 − 75.

Answer: 3(x − 5)(x + 5)

A non-monic quadratic

Factorise 3x^2 + 10x + 8.

  1. Multiply a × c: 3 × 8 = 24. Find two numbers that multiply to 24 and add to 10: 6 and 4.
  2. Split the middle term: 3x^2 + 6x + 4x + 8.
  3. Factorise in pairs: 3x(x + 2) + 4(x + 2).
  4. Take out the common bracket: (3x + 4)(x + 2).
  5. Check: (3x + 4)(x + 2) = 3x^2 + 6x + 4x + 8 = 3x^2 + 10x + 8.

Answer: (3x + 4)(x + 2)

Practice check

Have a go, then check your answers. Each answer comes with an explanation.

  1. 1.Factorise x^2 + 8x + 15.
  2. 2.Factorise x^2 − 49.
  3. 3.Factorise x^2 − 10x + 25.
  4. 4.Factorise x^2 + 2x − 15.
  5. 5.Factorise 3x^2 + 12x.
  6. 6.Factorise 2x^2 + 5x + 3.
  7. 7.Factorise fully 2x^2 − 18.
  8. 8.Which expression is a perfect square?
Answer guide for parents

  1. Factorise x^2 + 8x + 15.

    (x + 3)(x + 5) — 3 × 5 = 15 and 3 + 5 = 8, so x^2 + 8x + 15 = (x + 3)(x + 5).

  2. Factorise x^2 − 49.

    (x − 7)(x + 7) — x^2 − 49 is a difference of two squares: x^2 − 7^2 = (x − 7)(x + 7).

  3. Factorise x^2 − 10x + 25.

    (x − 5)^2 — −5 × −5 = 25 and −5 + −5 = −10, so it is a perfect square: (x − 5)^2.

  4. Factorise x^2 + 2x − 15.

    (x + 5)(x − 3) — 5 × (−3) = −15 and 5 + (−3) = 2, so x^2 + 2x − 15 = (x + 5)(x − 3).

  5. Factorise 3x^2 + 12x.

    3x(x + 4) — The highest common factor of 3x^2 and 12x is 3x, leaving x + 4 in the bracket.

  6. Factorise 2x^2 + 5x + 3.

    (2x + 3)(x + 1) — ac = 6; 3 and 2 multiply to 6 and add to 5. 2x^2 + 2x + 3x + 3 = 2x(x + 1) + 3(x + 1) = (2x + 3)(x + 1).

  7. Factorise fully 2x^2 − 18.

    2(x − 3)(x + 3) — Take out 2 to get 2(x^2 − 9), then factorise the difference of two squares: 2(x − 3)(x + 3).

  8. Which expression is a perfect square?

    x^2 + 6x + 9 — x^2 + 6x + 9 = (x + 3)^2, because 3 × 3 = 9 and 3 + 3 = 6.

Watch out for

  • Forgetting to take out a common factor first, which makes the remaining factorisation harder or leaves it incomplete.
  • Mixing up the signs, for example writing x^2 − 2x − 15 as (x + 5)(x − 3). Check that the two numbers add to b, including its sign.
  • Trying to factorise a sum of two squares, such as x^2 + 9, as (x + 3)(x − 3). That expands to x^2 − 9; x^2 + 9 does not factorise using real numbers.

Tips for parents

  • Ask your learner to expand their answer every time. If it does not give the original expression, they know to try again.
  • Practise quick-fire 'multiply to, add to' puzzles out loud: 'Two numbers multiply to 18 and add to 9?' (3 and 6).
  • If non-monic quadratics cause frustration, return to monic ones until those feel automatic, then add the splitting step.

Go further

  • Use algebra tiles or cut-out paper shapes to build x^2 + 5x + 6 as a rectangle. What are the rectangle's side lengths?
  • Use the difference of two squares to calculate 51 × 49 mentally, by writing it as (50 + 1)(50 − 1) = 2500 − 1 = 2499.
  • Investigate why x^2 + bx + c can only be factorised over the integers when b^2 − 4c is a perfect square.

About 55 minutes

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You'll need

  • Paper and pencil
  • A calculator (optional, for checking)
  • Algebra tiles or cut-out paper squares and rectangles (optional)

Key words

Factorise
Write an expression as a product of its factors; the reverse of expanding.
Quadratic expression
An expression of the form ax^2 + bx + c, where a is not zero.
Monic quadratic
A quadratic whose x^2 coefficient is 1, such as x^2 + 5x + 6.
Non-monic quadratic
A quadratic whose x^2 coefficient is not 1, such as 2x^2 + 7x + 3.
Perfect square
An expression such as x^2 + 6x + 9 that factorises to a bracket squared, (x + 3)^2.
Difference of two squares
An expression such as x^2 − 25 that factorises to (x − 5)(x + 5).
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