Solving Cubic Equations: Cardano's Method in 2026
First published by Gerolamo Cardano in his 1545 masterpiece Ars Magna, the solution of the general cubic equation $ax^3 + bx^2 + cx + d = 0$ was the historical catalyst that necessitated the discovery of complex numbers.
Cubic Discriminant & Root Character Table
\Delta = \left(\frac{q}{2}\right)^2 + \left(\frac{p}{3}\right)^3
| Discriminant Sign | Nature of Cubic Roots | Geometric Shape of Cubic Curve |
|---|---|---|
| $\Delta < 0$ | 3 Distinct Real Roots | Crosses x-axis at 3 distinct points |
| $\Delta = 0$ | Real Roots with Multiplicity | Tangent to x-axis (at least 2 roots equal) |
| $\Delta > 0$ | 1 Real Root + 2 Complex Conjugates | Crosses x-axis once, 2 inflection turns off-axis |