In order to factor x² − 5x + 6, we first note that the coefficient of x is negative.

Step-by-step solution

Idea: The product 6 is positive, so both numbers have the same sign. Their sum −5 is negative, so both must be negative.

1. Compare x² − 5x + 6 with x² + (a + b)x + ab: a + b = −5 and ab = 6.½ mark
2. ab = 6 is positive → same signs. a + b = −5 is negative → both negative.½ mark
3. Negative pairs with product 6: (−1)(−6), sum −7; (−2)(−3), sum −5 ✓. So a = −2, b = −3 (or the other way round).½ mark
4. x² − 5x + 6 = x² − 2x − 3x + 6 = x(x − 2) − 3(x − 2) = (x − 2)(x − 3)½ mark

Check: (x − 2)(x − 3) = x² − 3x − 2x + 6 = x² − 5x + 6 ✓.

Answer to write in the exam

Common mistakes that cost marks

• Using +2 and +3: their sum is +5, not −5.
• Writing (x − 2)(x + 3): the product would be −6, not +6.
• In the grouping step, writing −3(x + 2) instead of −3(x − 2): taking out −3 changes the sign inside.

How this can come in the exam

Try one yourself

Factorise x² − 9x + 20.

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