Factorising quadratics is a core algebra skill for GCSE Maths (AQA, Edexcel, OCR). It's the process of rewriting a quadratic expression, like (x² + 5x + 6), as a product of two brackets, ((x + 2)(x + 3)). Mastering this unlocks solving quadratic equations and simplifying algebraic fractions. This guide walks you through every type you'll meet at Higher and Foundation Tier.

What is a Quadratic Expression?

A quadratic expression is any in the form (ax² + bx + c), where (a), (b), and (c) are numbers (constants) and (a ≠ 0). The highest power of the variable (usually (x)) is 2.

Method 1: Factorising Simple Quadratics ((a = 1))

These are in the form (x² + bx + c).

Step-by-Step:

Find two numbers that multiply to make (c) (the constant term) and add to make (b) (the coefficient of (x)).

Write the brackets: ((x + first number)(x + second number)).

Example: Factorise (x² + 7x + 12).

We need two numbers that multiply to (12) and add to (7).

The pairs that multiply to 12 are: (1, 12), (2, 6), (3, 4).

Which pair adds to 7? (3 + 4 = 7). ✔

Therefore: (x² + 7x + 12 = (x + 3)(x + 4)).

Watch out for negatives:

(x² - 5x + 6): Need two numbers that multiply to (+6) and add to (-5). That's (-2) and (-3). So, ((x - 2)(x - 3)).

(x² + 2x - 8): Multiply to (-8), add to (+2). That's (+4) and (-2). So, ((x + 4)(x - 2)).

Method 2: The Difference of Two Squares

This is a special case: (x² - k²), where (k) is any number. It factorises to ((x + k)(x - k)).

Why it works: The (x) terms cancel out: ((x + k)(x - k) = x² - kx + kx - k² = x² - k²).

Examples:

(x² - 25 = x² - 5² = (x + 5)(x - 5))

(4x² - 9 = (2x)² - 3² = (2x + 3)(2x - 3)) (Note: both terms must be perfect squares).

Method 3: Factorising Harder Quadratics ((a > 1))

These are in the form (ax² + bx + c) where (a) is not 1 (e.g., (2x² + 5x + 3)). Many students find this trickier, but a systematic approach works.

The 'AC' Method (Splitting the Middle Term):

Step-by-Step:

Multiply (a) and (c) (the coefficient of (x²) and the constant).

Find two numbers that multiply to make this product ((a × c)) and add to make (b).

Split the middle term ((bx)) into two terms using these numbers.

Factorise in pairs the first two terms and the last two terms separately.

You should find a common bracket, which you can then factorise out.

Example: Factorise (6x² + 11x + 4).

(a × c = 6 × 4 = 24).

Need two numbers that multiply to 24 and add to 11. That's 8 and 3 ((8 × 3 = 24, 8 + 3 = 11)).

Split the middle term: (6x² + 8x + 3x + 4).

Factorise in pairs:

First pair: (6x² + 8x = 2x(3x + 4))

Second pair: (3x + 4 = 1(3x + 4))

Both parts now contain the common bracket ((3x + 4)). Factorise this out:
(2x(3x +
4) + 1(3x +
4) = (3x + 4)(2x + 1)).

So, (6x² + 11x + 4 = (3x + 4)(2x + 1)).

Which Method Should I Use? A Decision Flowchart

Use this flowchart to decide your approach every time.

flowchart TD
    A["Start: Quadratic Expression
 ax² + bx + c"] --> B["Is it Difference of Two Squares?, x² - k² or (px)² - (q)²?"];
    B -- Yes --> C["Factorise as (x + k)(x - k), or (px + q)(px - q)"];
    B -- No --> D["Is a = 1?"];
    D -- Yes --> E["Simple Case:, Find two numbers that multiply to c, and add to b"];
    D -- No --> F["Harder Case (a > 1):, Use the AC Method"];
    E --> G["Write as (x + m)(x + n)"];
    F --> H["Find factors of a×c that add to b,, split the middle term,, factorise in pairs"];
    C --> I["Finished! Check by expanding brackets."]
    G --> I
    H --> I

Why is Factorising Important?

Factorising isn't just an exercise; it's a key tool for:

Solving Quadratic Equations: Once factorised to ((x+p)(x+q)=0), the solutions are (x = -p) and (x = -q) (using the principle that if the product is zero, one of the factors must be zero).

Simplifying Algebraic Fractions: You can cancel common factors from the numerator and denominator only after factorising.

Sketching Quadratic Graphs: The factorised form reveals the roots (where the graph crosses the x-axis).

Common Pitfalls and Top Tips

Pitfall 1: Forgetting the HCF. Always check for a Highest Common Factor (HCF) first. E.g., (2x² + 10x + 12 = 2(x² + 5x + 6) = 2(x+2)(x+3)).

Pitfall 2: Incorrect signs. Be meticulous with positive and negative numbers when finding your number pairs.

Tip: Always expand your final brackets as a quick check. You should get back to the original expression.

Tip: For harder quadratics, if you struggle to find the splitting numbers, list all factor pairs of (a × c) systematically.

Practice Makes Perfect

Try these, mixing all types:

(x² + 9x + 20)

(x² - 81)

(2x² + 7x + 3)

(3x² - 10x - 8)

(4x² - 25)

Answers:

((x+4)(x+5))

((x+9)(x-9))

((2x+1)(x+3))

((3x+2)(x-4))

((2x+5)(2x-5))
Master these steps, use the flowchart, and you'll confidently tackle any factorising question in your GCSE Maths exam. Good luck!

How to Factorise Quadratics: A Step-by-Step GCSE Maths Guide