Factorisation: Cross Method

1Summary 重點摘要

Learn a quick and neat method to factorise quadratic into two linear factors.

2Vocabulary cards Vocabulary

3Step by step Steps

1. Step 1: Understand what cross method doesThe cross method's goal: to factorise a quadratic such as x² + bx + c into the product of two linear factors (x + m)(x + n). Example: x² + 5x + 6 = (x + 2)(x + 3) — the two numbers sought are exactly m = 2 and n = 3. The name comes from the check: cross-multiplying the two brackets draws a cross. The method applies to quadratics with leading coefficient 1 and is a key S3 algebra tool.
2. Step 2: Find two numbers whose product equals constantThe first clue: find two numbers m and n whose product equals the constant term c. Example: in x² + 5x + 6 the constant is 6 — which integer pairs multiply to 6? 1 × 6 and 2 × 3, plus the negative pairs (−1) × (−6) and (−2) × (−3). List every possibility before choosing. A common error is skipping the negative pairs, especially when the constant is positive but the middle term is negative — then both numbers are in fact negative. List all pairs first; the next step screens them.
3. Step 3: Check if sum equals middle coefficientThe second clue: the two numbers must add up to the middle coefficient. Example: in x² + 5x + 6 the middle coefficient is 5 — test each pair: 1 + 6 = 7, no; 2 + 3 = 5, yes, so m = 2 and n = 3. With a negative constant, e.g. x² + 2x − 15, the pairs for product −15 include 3 × (−5) and (−3) × 5; 3 + (−5) = −2 fails, while (−3) + 5 = 2 works, giving (x − 3)(x + 5). Both clues must hold together before the answer stands.
4. Step 4: Write answerOnce the two numbers are found, the answer writes itself: x² + bx + c = (x + m)(x + n). Example: x² + 5x + 6 = (x + 2)(x + 3). The order may be swapped — (x + 3)(x + 2) is equally correct, by the commutative law of multiplication. A common error is losing the structure after finding the numbers, e.g. writing x² + 5x + 6 as (x + 2) + (x + 3) with a plus; a factorisation answer is always a product — nothing is written between the brackets.
5. Step 5: Remember signsSign patterns: a positive constant means both numbers share a sign — both positive or both negative, decided by the middle term; a negative constant means opposite signs, one plus and one minus. Example: in x² − 5x + 6 the constant 6 is positive and the middle term −5, so both numbers are negative: (x − 2)(x − 3), verified by (−2) + (−3) = −5. A common error is taking a negative middle term to mean a negative constant; the chant is 'constant positive, same signs; constant negative, opposite signs' — settle the signs before hunting the numbers.
6. Step 6: CheckAfter writing the answer, always multiply back. Example: (x + 2)(x + 3) = x² + 3x + 2x + 6 = x² + 5x + 6, matching the original — the factorisation is right. If the product differs, the two numbers were wrong; return to steps two and three and re-pair. The check takes ten seconds yet secures all the marks; under exam pressure, finish every question first, then verify them together. A careless check only matches the constant: (x + 1)(x + 6) also has constant 6 but middle term 7x — both terms must be compared.

4Diagrams Diagrams

5Worked example Example

6Common mistakes Mistakes

7Quick check Quick check

8Recap Recap

• Find two numbers: product equals constant
• Two numbers sum equals middle coefficient
• Constant positive → same signs, negative → opposite
• Write answer: (x + m)(x + n)
• Remember to check!