Cubic Equation Calculator

Solve cubic equations ax³ + bx² + cx + d = 0 with step-by-step solutions. Find real and complex roots using Cardano's method and advanced algorithms.

General Cubic Equation: ax³ + bx² + cx + d = 0

Where a ≠ 0 (otherwise it's a quadratic equation)

Coefficient a
1
Coefficient b
-6
Coefficient c
11
Coefficient d
-6
x³ - 6x² + 11x - 6 = 0
x³ - 1 = 0
x³ - 6x² + 11x - 6 = 0
x³ - 3x² - 3x - 1 = 0
x³ - 2x = 0
2x³ - 3x² + 5 = 0
x³ - 7x + 6 = 0
x³ + 8 = 0
x³ - x² - x + 1 = 0
x³ - 3x² + 3x - 1 = 0
x³ - 3x = 0
Choose solution method for cubic equation
Range of x-values to display on the graph
Calculating roots...

Understanding Cubic Equations

A cubic equation is a polynomial equation of degree three. The general form is ax³ + bx² + cx + d = 0, where a, b, c, d are coefficients and a ≠ 0. Cubic equations always have three roots (solutions), which may be real or complex numbers.

Fundamental Theorem of Algebra:

Every cubic equation with real or complex coefficients has exactly three solutions (roots) in the complex numbers, counting multiplicities.

Types of Cubic Equations

Type General Form Characteristics
Depressed Cubic t³ + pt + q = 0 No x² term (obtained by substitution x = t - b/(3a))
Monic Cubic x³ + bx² + cx + d = 0 Leading coefficient a = 1
Biquadratic Cubic ax³ + cx + d = 0 Missing x² term (b = 0)
Simple Cubic x³ + d = 0 Only constant term besides x³
Factorable Cubic (x - r₁)(x - r₂)(x - r₃) = 0 Can be expressed as product of linear factors

Solution Methods

1

Cardano's Formula: General algebraic solution for cubic equations, discovered in the 16th century. Works for all cubic equations but can involve complex numbers even when final roots are real.

2

Trigonometric Method: Alternative approach for cases with three real roots (Δ < 0). Uses trigonometric functions to avoid complex number calculations in intermediate steps.

3

Numerical Methods: For equations that don't factor nicely, numerical methods like Newton-Raphson can approximate real roots to desired precision.

4

Factorization: When one rational root can be found (using Rational Root Theorem), the cubic can be reduced to a quadratic for finding remaining roots.

Discriminant of a Cubic Equation

The discriminant Δ of a cubic equation ax³ + bx² + cx + d = 0 determines the nature of its roots:

Discriminant Formula:

Δ = 18abcd - 4b³d + b²c² - 4ac³ - 27a²d²

Or for depressed cubic t³ + pt + q = 0:

Δ = -4p³ - 27q²

Δ > 0
1 Real
2 Complex Conjugate
Δ = 0
3 Real
At least 2 equal
Δ < 0
3 Real
All distinct

Applications of Cubic Equations

  • Physics: Equations of state (Van der Waals equation), motion under constant jerk
  • Engineering: Cubic splines in computer graphics, solving equilibrium equations
  • Economics: Cubic cost functions, optimal pricing models
  • Geometry: Volume calculations, intersection of curves
  • Computer Science: Bézier curves, interpolation algorithms

Calculator Features:

  • Solves any cubic equation with real coefficients
  • Uses Cardano's formula, trigonometric method, and numeric approximation
  • Calculates discriminant to determine root nature
  • Provides factorization when possible
  • Graphs the cubic function with roots highlighted
  • Step-by-step solution showing derivation process

Frequently Asked Questions

Cardano's formula is an algebraic method for solving cubic equations. For a depressed cubic t³ + pt + q = 0, the solution is: t = ∛(-q/2 + √((q/2)² + (p/3)³)) + ∛(-q/2 - √((q/2)² + (p/3)³)). This formula works for all cases but may involve complex numbers in intermediate steps even when final roots are real.

Yes, when the discriminant Δ > 0, the cubic equation has one real root and two complex conjugate roots. For example, x³ - 3x + 2 = 0 has one real root (x = 1) and two complex roots.

Use the Rational Root Theorem: If a polynomial has integer coefficients, then any rational root p/q (in lowest terms) satisfies that p is a factor of the constant term d, and q is a factor of the leading coefficient a. Test these possible rational roots by direct substitution or synthetic division.

The trigonometric method is used when the cubic has three real roots (Δ < 0). For depressed cubic t³ + pt + q = 0 with p < 0, the roots are: t_k = 2√(-p/3) · cos(⅓ arccos(3q/(2p)√(-3/p)) - 2πk/3) for k = 0,1,2. This method avoids complex numbers in intermediate calculations.

For a cubic ax³ + bx² + cx + d = 0 with roots r₁, r₂, r₃:
• Sum of roots: r₁ + r₂ + r₃ = -b/a
• Sum of products of roots taken two at a time: r₁r₂ + r₁r₃ + r₂r₃ = c/a
• Product of roots: r₁r₂r₃ = -d/a
These are known as Vieta's formulas for cubic equations.