Quadratic Equations
The exam tests recognising quadratic equations, solving by factorisation and the quadratic formula, using the discriminant to describe roots, and forming equations from given roots.
Core rules
- Standard form ax² + bx + c = 0 with a ≠ 0.
- Discriminant D = b² − 4ac. D > 0 gives two distinct real roots; D = 0 gives two equal real roots; D < 0 gives no real roots.
- Roots: x = (−b ± √D)/(2a).
- Sum of roots = −b/a, product = c/a. An equation from roots α, β is x² − (α + β)x + αβ = 0.
- Factorisation works when two numbers multiply to ac and add to b.
Worked example
Solve 2x² − 5x + 3 = 0. Here a = 2, b = −5, c = 3, so D = 25 − 24 = 1. Then x = (5 ± 1)/4, giving x = 6/4 = 3/2 or x = 4/4 = 1. Check the product: (3/2)(1) = 3/2 = c/a, as required.
Traps
Forgetting a ≠ 0 before applying the formula; writing D = b² + 4ac; counting D = 0 as two distinct roots; losing the ± sign; retaining a negative root for a distance or time in a word problem; not verifying roots in the original equation.