Let us try to factor x² + 11x + 30 in a similar manner.

Step-by-step solution

Idea: “In a similar manner” means: compare with x² + (a + b)x + ab = (x + a)(x + b). Find two numbers with sum 11 and product 30, and split 11x using them.

1. x² + 11x + 30 = x² + (a + b)x + ab, so a + b = 11 and ab = 30.½ mark
2. List factor pairs of 30 and their sums: 1 × 30 (31), 2 × 15 (17), 3 × 10 (13), 5 × 6 (11) ✓. Pairs like 2 and 15 have the right product but the wrong sum.½ mark
3. So a = 5, b = 6, and 11x splits as 5x + 6x.½ mark
4. x² + 11x + 30 = x² + (5 + 6)x + 30 = (x + 5)(x + 6)½ mark

Check: Put x = 1: 1 + 11 + 30 = 42 and (6)(7) = 42 ✓.

Answer to write in the exam

Common mistakes that cost marks

• Stopping at the first factor pair of 30 (such as 2 × 15) without checking the sum.
• Writing (x + 5)(x − 6). Both numbers must be positive here, because their sum and product are both positive.
• Mixing up which number must be the sum: the sum is the middle coefficient 11, the product is the constant 30.

How this can come in the exam

Try one yourself

Factorise x² + 12x + 35.

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