Factorising Quadratics –

Factorising quadratics reverses the process of expanding double brackets. It is one of the most commonly tested algebra skills at GCSE and a prerequisite for solving quadratic equations, simplifying algebraic fractions, and sketching parabolas.

What Is Factorising Quadratics?

Factorising a quadratic expression means rewriting it as the product of two linear brackets. For a monic quadratic (where the coefficient of x² is 1) in the form x² + bx + c, the goal is to find two numbers p and q such that p + q = b and p times q = c, so that the expression becomes (x + p)(x + q).

The method works because expanding (x + p)(x + q) gives x² + (p + q)x + pq. By matching coefficients you can reverse-engineer the values of p and q from the original expression. Once factorised, you should always check your answer by expanding the brackets back out.

When the coefficient of x² is greater than 1, the expression is called a non-monic quadratic. At Foundation tier you will mostly meet monic quadratics; non-monic factorising appears at Higher level and uses the grouping (ac) method.

Key Formulas

Step-by-Step Method

1. Write down the quadratic in the form x² + bx + c and identify the values of b and c.
2. List all factor pairs of c (including negative pairs if c is negative).
3. Find the pair whose sum equals b.
4. Write the factorised form as (x + p)(x + q).
5. Expand the brackets to check that you recover the original expression.

Worked Example 1 — Foundation Level

Question: Factorise x² + 9x + 20.

Working:

Step 1 — Identify b = 9 and c = 20.

Step 2 — Factor pairs of 20: (1, 20), (2, 10), (4, 5).

Step 3 — Which pair sums to 9? 4 + 5 = 9.

Step 4 — Write: (x + 4)(x + 5).

Step 5 — Check: x² + 5x + 4x + 20 = x² + 9x + 20. Correct.

Answer: (x + 4)(x + 5)

Worked Example 2 — Higher Level

Question: Factorise x² - 3x - 28.

Working:

Step 1 — Here b = -3 and c = -28. Because c is negative, one number must be positive and one negative.

Step 2 — Factor pairs of -28: (1, -28), (-1, 28), (2, -14), (-2, 14), (4, -7), (-4, 7).

Step 3 — Which pair sums to -3? 4 + (-7) = -3.

Step 4 — Write: (x + 4)(x - 7).

Step 5 — Check: x² - 7x + 4x - 28 = x² - 3x - 28. Correct.

Answer: (x + 4)(x - 7)

Worked Example 3 — Exam Style

Question: Factorise x² - 11x + 24 and hence solve x² - 11x + 24 = 0. (3 marks)

Working:

b = -11 and c = 24. Both numbers must be negative (negative sum, positive product).

Factor pairs of 24 (negative): (-1, -24), (-2, -12), (-3, -8), (-4, -6).

-3 + (-8) = -11. This pair works.

Factorised form: (x - 3)(x - 8).

Setting each bracket to zero: x - 3 = 0 gives x = 3; x - 8 = 0 gives x = 8.

Check: 3² - 11(3) + 24 = 9 - 33 + 24 = 0. Correct.

Answer: (x - 3)(x - 8); solutions are x = 3 or x = 8.

Common Mistakes

• Getting signs wrong when c is negative. If c < 0, one factor must be positive and one negative. List pairs systematically to avoid missing the correct combination.
• Choosing factors that multiply correctly but do not add correctly. Always check both conditions: the pair must multiply to c and add to b.
• Forgetting to check by expanding. A quick expansion confirms your answer and prevents dropped marks. It takes only a few seconds.

Exam Tips

• Write out all factor pairs neatly before choosing. This avoids guesswork and shows the examiner your method.
• If both b and c are positive, both numbers in the brackets are positive. If b is negative and c is positive, both numbers are negative.
• When a question says "factorise and hence solve," factorise first, then set each bracket equal to zero.

Practice Questions

Q1 (Foundation): Factorise x² + 7x + 12.

Q2 (Foundation): Factorise x² - 2x - 15.

Q3 (Higher): Factorise x² - 10x + 25.

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Related Topics

• Factorising Expressions
• Solving Quadratic Equations by Factorising
• Expanding Double Brackets

Summary

• Factorising a quadratic reverses expanding double brackets.
• For x² + bx + c, find two numbers that multiply to c and add to b.
• Write the expression as (x + p)(x + q) using those two numbers.
• Always expand your answer to verify it matches the original expression.
• Sign rules: if c is positive, both numbers share the same sign (both positive if b > 0, both negative if b < 0). If c is negative, the numbers have opposite signs.