Maths
Quadratic formula — practice & summary
GCSE MathsIGCSE MathsAP Algebra
The quadratic formula solves any equation of the form ax² + bx + c = 0 (a ≠ 0). The solutions are x = (-b ± √(b² − 4ac)) / 2a. The expression under the square root, b² − 4ac, is the discriminant — it tells you how many real roots exist before you even solve.
Key points
- Formula: x = (−b ± √(b² − 4ac)) / 2a
- Discriminant > 0 → two distinct real roots
- Discriminant = 0 → one repeated real root
- Discriminant < 0 → no real roots (two complex conjugate roots)
- Always check by substitution if the question asks for verification
Practice quiz
Click each question to reveal the answer.
1. Solve x² − 5x + 6 = 0
Answer: x = 2 or x = 3
Factorises to (x−2)(x−3) = 0, so x = 2 or 3.
2. What is the discriminant of 2x² + 3x − 5?
Answer: 49
b² − 4ac = 9 − 4(2)(−5) = 9 + 40 = 49.
3. How many real roots does x² + x + 1 = 0 have?
- 0
- 1
- 2
- Infinitely many
Answer: 0
Discriminant = 1 − 4 = −3 < 0, so no real roots.
Last reviewed: May 2026