GCSE · Maths · AQA · Spec 8300 · Foundation

Factorising quadratics x^2+bx+c

Every quadratic like x² + 7x + 12 hides two numbers that multiply to 12 and add to 7 — find them, and the whole expression falls into two brackets.

The factorising method

Watch how one search — a number pair that multiplies to c and adds to b — drives every step.

  1. Read b and cFor x² + 7x + 12, compare it with x² + bx + c: b = 7 and c = 12. You need ONE pair of numbers that multiplies to 12 AND adds to 7 — both conditions, not just one.
  3. Keep the pair that sums to bCheck each pair against b = 7: 1 + 12 = 13, 2 + 6 = 8, 3 + 4 = 7. Only 3 and 4 add to 7 — that eliminates the other two candidates.
  4. Write the two bracketsUse the pair 3 and 4 as the two numbers inside the brackets: x² + 7x + 12 = (x + 3)(x + 4).
  5. Expand back to check(x + 3)(x + 4) = x² + 4x + 3x + 12 = x² + 7x + 12 — it matches the original expression exactly, so the factorisation is confirmed, not just guessed.

Why the same pair does two jobs

Splitting (x + 2)(x + 3) into an area model
x²3x2x6x3x2

Showing 1 layer: Rectangle

Explore

Select each part of the rectangle to see where x² + 5x + 6 comes from.

Select a part of the rectangle to see where each term of x² + 5x + 6 comes from.

Relationship matrix

Tap any cell to reveal it. Tap a column header to read one property down every item.

b is positiveb is negative
c is positivee.g. c = +12
c is negativee.g. c = −12

Each cell hides a short answer and the reason behind it. Predict before you tap.

Predict the middle term

Think about what +3x and −3x do when you add them, before you check.

Factorising x² − 9 means finding a pair that multiplies to −9 and adds to 0 (there's no x term to match). Using the pair +3 and −3, what happens to the two middle terms, +3x and −3x, when you add them together?

WHAT YOU'VE LEARNED

A quick recap of today's lesson.

x² + bx + c = (x + p)(x + q)

What you need to know

  • Factorising x² + bx + c means writing it as (x + p)(x + q), where p and q are the two numbers you're searching for.
  • The pair you need always multiplies to give c and adds to give b — check both conditions, not just one.
  • The difference of two squares, x² − a², is the same search with b = 0: the pair is +a and −a, which always cancel to give no x-term.
  • Always check your answer by expanding the brackets back out — if you don't get the original expression, the pair is wrong.

The big picture

Rewriting x² + bx + c as (x + p)(x + q) by finding the one pair of numbers that multiplies to c and adds to b — including the difference of two squares, which is the same search with b = 0.

Key points

1x² + bx + c factorises to (x + p)(x + q) when p × q = c and p + q = b.
2If c is positive, p and q have the same sign — matching the sign of b.
3If c is negative, p and q have opposite signs — the one with the larger size matches the sign of b.
4x² − a² = (x + a)(x − a): the difference of two squares is this method with b = 0.

Worked example

Problem

Factorise x² + 9x + 20.

⚠ Watch out

Stopping at the first pair that multiplies to c, without checking it also adds to b — both conditions must hold, not just one.

🧠

Memory hook

Multiply to c, add to b — same two numbers, two jobs.

✓

Check yourself

Factorise x² − 2x − 15 on paper, then check your answer by expanding your brackets back out. Did you get the original expression?

Flashcards

(6)
What does it mean to 'factorise' x² + bx + c?
Rewrite it as a product of two brackets, (x + p)(x + q), that multiply out to give the original expression.
What two conditions must the number pair p and q satisfy?
p × q = c and p + q = b — both must hold, not just one.
How do you factorise a difference of two squares, x² − a²?
(x + a)(x − a) — e.g. x² − 16 = (x + 4)(x − 4). It's the same pair-rule with b = 0.
If c is positive, what do you know about the signs of p and q?
They have the same sign as each other — both positive if b is positive, both negative if b is negative.
If c is negative, what do you know about the signs of p and q?
They have opposite signs — whichever one is larger in size takes the sign of b.
How do you check a factorisation is correct?
Expand the brackets back out — if you don't get the original expression, the pair is wrong.

Tap any card to flip it, or use Study as deck to go through them one at a time. In the full lesson these run as a spaced-repetition deck — you rate each card Hard, Good or Easy and the tricky ones keep coming back until they stick.

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How this lesson was checked. This AQA GCSE Maths (specification 8300)lesson was published through Lightbulb Learning's human-designed editorial process — the educational standards, accuracy rules and publication checks it must pass were authored and approved by Philip Halpin. It passed subject-specific assessment, automated educational checks and technical publication verification before going live (publication checks completed 23 September 2026). Published pages are monitored, human spot-checking is ongoing across the lesson library, and anything found wrong is corrected or withdrawn. How our lessons are made and checked. Spotted a mistake? Email hello@lightbulblearning.co and we'll review it.