Factorising Quadratic Expressions

1. 1 Expand (x+3)(x+4) = x²+4x+3x+12 = x²+7x+12. 'Where did the 7 come from?' (3+4). 'Where did the 12 come from?' (3×4). 'This is the key — to factorise, we reverse this process.'
2. 2 Factorise x²+7x+12. 'I need two numbers that multiply to 12 AND add to 7.' List factor pairs of 12: 1×12, 2×6, 3×4. Which pair adds to 7? (3 and 4.) So x²+7x+12 = (x+3)(x+4). Check by expanding.
3. 3 x²+x−12. Multiply to −12, add to 1. Factor pairs of −12 where one is negative: −3 and 4 (−3×4=−12, −3+4=1). x²+x−12 = (x−3)(x+4). Students practise four examples with negatives.
4. 4 x²−7x+12. Multiply to +12, add to −7. Both numbers must be negative: −3 and −4. (x−3)(x−4). Students explain why both negatives: 'negative × negative = positive, negative + negative = negative.'
5. 5 Spot the pattern: x²−9 = x²+0x−9. Need two numbers: multiply to −9, add to 0. (+3) and (−3). x²−9 = (x+3)(x−3). Shortcut: a²−b² = (a+b)(a−b). Students apply to x²−25, x²−49, 4x²−1.
6. 6 2x²+7x+3. Method: find two numbers that multiply to 2×3=6 and add to 7: 1 and 6. Split: 2x²+x+6x+3. Group: x(2x+1)+3(2x+1) = (2x+1)(x+3). Students apply to 3x²+10x+3.
7. 7 Display 8 expressions. Students categorise: simple (a=1), difference of squares, or a≠1, then factorise each using the appropriate method.

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