Quadratic Equations
The exam tests solving quadratic equations by factorisation and the quadratic formula, interpreting the discriminant, and translating word problems into quadratics.
Core rules
- Standard form: ax² + bx + c = 0, a ≠ 0.
- Quadratic formula: x = [−b ± √(b² − 4ac)]/(2a).
- Discriminant D = b² − 4ac: D > 0 gives two distinct real roots; D = 0 gives two equal real roots; D < 0 gives no real roots.
- Sum of roots = −b/a; product of roots = c/a.
- In word problems, define the variable, form the equation, solve it, and reject values that are impossible in context.
Worked example
The sum of the squares of two consecutive positive integers is 365. Find them. Let the integers be n and n + 1. Then n² + (n + 1)² = 365. 2n² + 2n + 1 = 365 → n² + n − 182 = 0. Factorise: (n + 14)(n − 13) = 0, so n = −14 or n = 13. Since the integers are positive, n = 13 and the integers are 13 and 14. Check: 169 + 196 = 365.
Traps
- Forgetting to reject a negative root in a length, age or speed problem.
- Substituting into the formula without writing the equation in standard form.
- Treating D = 0 as no roots rather than two equal real roots.
- Losing the ± and reporting only one root.