Solving Quadratics: Factorisation & Quadratic Formula
Quadratic equations at GCSE Higher tier appear in the form ax² + bx + c = 0. Factorisation: find two numbers that multiply to ac and add to b. For x² - 7x + 10 = 0: find numbers that multiply to 10 and add to -7: those are -2 and -5. Factorise to (x - 2)(x - 5) = 0, giving x = 2 or x = 5. When a ≠ 1, factorise carefully: for 2x² + 5x - 3 = 0, multiply a × c = -6, find numbers that multiply to -6 and add to 5: 6 and -1. Rewrite: 2x² + 6x - x - 3 = 0, factor by grouping: 2x(x + 3) - 1(x + 3) = 0, so (2x - 1)(x + 3) = 0, giving x = ½ or x = -3. The quadratic formula x = (-b ± √(b² - 4ac)) / 2a works for all quadratics. Substitute carefully and simplify. At GCSE, answers are often required to be given to 2 decimal places (2 d.p.). The discriminant b² - 4ac tells you: if > 0, two solutions; if = 0, one solution; if < 0, no real solutions.