Cubic Equation Calculator

Third-Degree Polynomial Solver

Solve cubic equations of the form ax³ + bx² + cx + d = 0 using Cardano's formula. Find all real and complex roots.

Equation Input

ax³ + bx² + cx + d = 0
Standard Form:
ax³ + bx² + cx + d = 0
where a ≠ 0

About Cubic Equations

A cubic equation is a polynomial equation of degree 3.

General Form:

ax³ + bx² + cx + d = 0

Solution Method:

Cardano's formula (16th century)

Features

  • Cardano's formula solver
  • Real and complex roots
  • Discriminant calculation
  • Root nature identification
  • High precision results
  • Verification values
  • Quick examples
  • Copy results

Discriminant

Δ > 0:

Three distinct real roots

Δ = 0:

At least two equal roots

Δ < 0:

One real, two complex conjugate roots

Key Properties

Degree:

3 (cubic/third-degree)

Number of Roots:

Always 3 (counting multiplicity)

Complex Roots:

Always appear in conjugate pairs

Sum of Roots:

-b/a

Applications

  • Physics (motion equations)
  • Engineering (structural analysis)
  • Economics (cost functions)
  • Computer graphics (Bézier curves)
  • Signal processing
  • Optimization problems
  • Chemistry (reaction rates)
  • Mathematics education

Historical Note

Cardano's Formula:

Discovered: 16th century by Italian mathematicians

Gerolamo Cardano: Published the solution in 1545 in "Ars Magna"

Tartaglia & del Ferro: Earlier contributors to the solution

Impact: Led to the discovery of complex numbers

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