A quadratic equation graphs as a parabola and has at most two solutions (roots).
The Quadratic Formula
x = (−b ± √(b² − 4ac)) / 2a
The Discriminant (Δ = b² − 4ac)
- Δ > 0: Two distinct real roots
- Δ = 0: One repeated real root
- Δ < 0: Two complex (imaginary) roots
Example
2x² + 5x − 3 = 0 → x = (−5 ± √(25+24)) / 4 = (−5 ± 7) / 4 → x = 0.5 or x = −3
Three Ways to Solve a Quadratic
Beyond the quadratic formula, two other methods work in specific cases: factoring is fastest when the equation breaks into clean integer factors (x² − 5x + 6 = 0 → (x−2)(x−3) = 0). Completing the square is slower but always works and is the technique used to derive the quadratic formula itself.
Real-World Applications
Quadratic equations model projectile motion (a thrown ball's height over time), profit-maximization curves in economics, and satellite dish / parabolic reflector shapes. Any situation involving acceleration under constant force tends to produce a quadratic relationship between time and position.