Explanation:
To expand double brackets (x + a)(x + b), multiply every term in the first bracket by every term in the second bracket and simplify the middle x terms. To factorise x2 + bx + c, find two numbers that multiply to give c and add to give b.
Look out for the difference of two squares: a2 − b2 = (a + b)(a − b). For harder quadratics ax2 + bx + c (where a > 1), find two numbers that multiply to ac and add to b, then split the middle term and factorise in pairs.
Example 1: Factorise (a) x2 − 5x − 14 and (b) x2 − 49.
Solution:
(a) Multiply to −14 and add to −5 → −7 and +2
• x2 − 5x − 14 = (x − 7)(x + 2)
(b) Difference of two squares (x2 − 72):
• x2 − 49 = (x + 7)(x − 7)
Answer: (a) (x − 7)(x + 2), (b) (x + 7)(x − 7)
Example 2: Factorise 2x2 + 7x + 3.
Solution:
Multiply a × c = 2 × 3 = 6; factors adding to 7 are 6 and 1
Split middle term: 2x2 + 6x + x + 3
Factorise pairs: 2x(x + 3) + 1(x + 3)
Collect brackets: (2x + 1)(x + 3)
Answer: (2x + 1)(x + 3)
