Cubic Equations

Calculator to determine all three roots of cubic equations using Cardano's formula

Cubic Equation Calculator

What is calculated?

This function solves cubic equations of the form ax³ + bx² + cx + d = 0 using the Cardano formula and returns all three roots (real or complex).

Coefficients of the cubic equation
ax³ + bx² + cx + d = 0
a ≠ 0 required
Quadratic term
Linear term
Absolute term
Three roots of the cubic equation
Root x₁
Root x₂
Root x₃
Cubic equations always have at least one real root

Cubic Equations Info

Properties

Cubic equations:

  • Degree 3 polynomial
  • Always at least 1 real root
  • At most 3 real roots
  • Solvable with Cardano's formula

Cardano's formula: Closed-form solution for cubic equations, developed in the 16th century.

Root types
3 real roots 1 real + 2 complex
Examples
x³ - 1 = 0:
x₁ = 1, x₂,₃ = complex cube roots of unity
x³ - 6x² + 11x - 6 = 0:
x₁ = 1, x₂ = 2, x₃ = 3
Related topics

→ Quadratic Equations
→ Polynomial arithmetic
→ Complex Numbers

Cardano's formula for cubic equations

Standard form
\[ax^3 + bx^2 + cx + d = 0\] Cubic equation
Depressed form
\[t^3 + pt + q = 0\] After substitution x = t - b/(3a)
Cardano solution
\[x_1 = S + T - \frac{b}{3a}\] Real principal root
Complex roots
\[x_{2,3} = -\frac{S+T}{2} \pm i\frac{\sqrt{3}}{2}(S-T) - \frac{b}{3a}\] Complex conjugates
Auxiliary quantities S and T
\[S = \sqrt[3]{R + \sqrt{Q^3 + R^2}}\] \[T = \sqrt[3]{R - \sqrt{Q^3 + R^2}}\]
Parameters Q and R
\[Q = \frac{3ac - b^2}{9a^2}\] \[R = \frac{9abc - 27a^2d - 2b^3}{54a^3}\]
Discriminant
\[\Delta = Q^3 + R^2\] Determines root type
Vieta's formulas
\[x_1 + x_2 + x_3 = -\frac{b}{a}\] \[x_1 x_2 x_3 = -\frac{d}{a}\]

Detailed worked example

Example: -2x³ + 3x² + 5x + 8 = 0

Given:

  • a = -2, b = 3, c = 5, d = 8

Step 1 - Compute parameters:

\[Q = \frac{3(-2)(5) - 3^2}{9(-2)^2} = \frac{-39}{36}\] \[R = \frac{9(-2)(3)(5) - 27(-2)^2(8) - 2(3)^3}{54(-2)^3}\]

Step 2 - Discriminant:

\[\Delta = Q^3 + R^2\]

Δ > 0: One real, two complex roots
Δ = 0: Three real roots (at least one repeated)
Δ < 0: Three distinct real roots

Step 3 - Compute S and T:

\[S = \sqrt[3]{R + \sqrt{\Delta}}\] \[T = \sqrt[3]{R - \sqrt{\Delta}}\]

Step 4 - Determine roots:

\[x_1 = S + T - \frac{b}{3a}\] \[x_{2,3} = \text{complex}\]

Cardano's formula systematically yields all three roots

Geometric interpretation

Cubic curve and x-axis intersections
3 real roots

Cubic intersects x-axis three times

1 real root

One real, two complex roots

Double root

Tangent and intersection

Discriminant analysis

Δ < 0 (Casus irreducibilis)
  • 3 distinct real roots
  • Trigonometric representation required
  • No closed radical form
  • Geometrically: 3 intersections
Δ = 0
  • At least one repeated root
  • 2 or 3 real roots
  • Tangent to the x-axis
  • Special case of Cardano's formula
Δ > 0
  • 1 real, 2 complex roots
  • Standard case of Cardano's formula
  • Complex roots are conjugate
  • Single intersection with x-axis

Practical applications

Engineering
  • Bending problems in beams
  • Stability analysis
  • Optimization problems
  • Material stresses
Physics
  • Quantum mechanics
  • Thermodynamics
  • Vibration analysis
  • Wave functions
Economics
  • Cost functions
  • Profit maximization
  • Market analysis
  • Production optimization

Historical context

Gerolamo Cardano (1501-1576)
  • Ars Magna (1545): First publication
  • Tartaglia's formula: Actual discoverer
  • Renaissance mathematics: Breakthrough of algebra
  • Complex numbers: Early use
Mathematical significance
  • Galois theory: Solvability in radicals
  • Algebraic equations: General theory
  • Complex analysis: Foundational aspects
  • Numerical methods: Modern algorithms
Summary

Cardano's formula for cubic equations was a milestone of Renaissance mathematics. It led to the development of complex numbers and laid the foundation for modern algebra. Despite its complexity it remains a fundamental mathematical tool with wide applications in science and engineering.

Mathematical Equations

Associative Property  •  Linear equation  •  Quadratic equation  •  Cubic equation  •  Discriminant  •  Gauss elimination method  •