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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