Quadratic Formula Calculator
Solve ax² + bx + c = 0 for any real coefficients. Get roots, discriminant, vertex, and more.
x = (−b ± √(b²−4ac)) / (2a). Discriminant Δ = b²−4ac: if Δ > 0 → 2 real roots; Δ = 0 → 1 repeated root; Δ < 0 → complex roots. Vertex at x = −b/(2a).
Quadratic Equations by Curriculum
| Curriculum | Methods taught | Local nickname |
|---|---|---|
| 🇺🇸 US (SAT / AP) | Factoring, completing the square, formula | — |
| 🇬🇧 UK (GCSE) | Factoring, formula (must memorise) | "Quadratic formula" |
| 🇩🇪 Germany (Gymnasium) | All methods, formula memorised | "Mitternachtsformel" (midnight formula) |
| 🇫🇷 France (Lycée) | Formula is standard method | "Formule du discriminant" |
| 🇯🇵 Japan (高校数学) | All methods | "解の公式" (Formula for roots) |
| 🇮🇳 India (CBSE) | Factoring, completing square, formula | "Shridharacharya method" (historical) |
| 🌍 IB Mathematics | All methods; GDC allowed in exams | — |
Frequently Asked Questions
What is the quadratic formula?
x = (−b ± √(b²−4ac)) / (2a). It solves any quadratic ax²+bx+c=0. The ± means there are two solutions. Enter any real values for a, b, c (a ≠ 0).
What does the discriminant tell you?
Δ = b²−4ac. If Δ > 0: two distinct real roots. If Δ = 0: one repeated real root (tangent to x-axis). If Δ < 0: two complex conjugate roots (no real x-intercepts, parabola doesn't cross x-axis).
What is the German "Mitternachtsformel"?
In Germany, the quadratic formula is nicknamed "Mitternachtsformel" (midnight formula) because students must be able to recite it even if woken at midnight. It reflects the cultural emphasis on formula memorisation in the Gymnasium curriculum.
How do you factor a quadratic instead of using the formula?
If ax²+bx+c factors nicely: find two numbers that multiply to ac and add to b. Example: x²−5x+6 → two numbers that multiply to 6, add to −5 → (−2)(−3) → (x−2)(x−3) = 0 → x = 2 or x = 3.