The quadratic formula
x = (−b ± √(b²−4ac)) / (2a). Solves any quadratic ax² + bx + c = 0.
The formula always works (when there are real roots). Discriminant b²−4ac tells you the type: positive = two solutions, zero = one repeated, negative = no real solutions. Higher tier requires fluency with surd answers.
Worked examples
x² − 5x + 6 = 0: x = (5 ± 1)/2 = 3 or 2.
2x² + 3x − 1 = 0: x = (−3 ± √17)/4.
x² + 2x + 5 = 0: discriminant = −16; no real solutions.
Frequently asked questions
When to use the formula vs factorising?
Try factorising first — quicker. If it doesn't factorise nicely, use the formula. Foundation tier is mostly given factorisable quadratics.
Calculator-friendly?
Yes — modern Casios have a quadratic-solver mode. But you must show working in the exam.