Cubic Equation Solver

Solve any cubic equation ax³ + bx² + cx + d = 0 and find all three roots (real and complex).

Enter Coefficients

1x³ + (-6)x² + 11x + (-6) = 0

Roots

Solutions
x = 1, 2, 3
3 real roots
Root x&sub1; 1
Root x&sub2; 2
Root x&sub3; 3
Discriminant 0
Nature of Roots 3 distinct real roots

Step-by-Step Solution

x³ - 6x² + 11x - 6 = 0

Understanding Cubic Equations

A cubic equation is a polynomial equation of degree three, written in the general form ax³ + bx² + cx + d = 0, where a is not zero. Every cubic equation with real coefficients has at least one real root, and it has exactly three roots when counted with multiplicity (including complex roots).

Cubic equations arise in many areas of mathematics and science, including geometry (volumes and intersections), physics (motion and equilibrium), engineering (signal processing), and economics (supply-demand models).

Methods for Solving Cubic Equations

Cardano's Formula

The classical algebraic method published in 1545 by Gerolamo Cardano.

Uses depressed cubic t³ + pt + q = 0

Trigonometric Method

When all three roots are real (casus irreducibilis).

Uses cos(arccos(...)/3)

Rational Root Theorem

Test factors of d/a to find rational roots first.

Possible roots: ±(factors of d)/(factors of a)

Factoring by Grouping

Rearrange terms to identify common factors.

ax³ + bx² + cx + d = (x - r)(ax² + ...)

The Discriminant of a Cubic

The discriminant of a cubic equation determines the nature of its roots:

  • Δ > 0: The equation has three distinct real roots.
  • Δ = 0: The equation has a repeated root (all roots are real).
  • Δ < 0: The equation has one real root and two complex conjugate roots.

The discriminant is calculated as Δ = 18abcd - 4b³d + b²c² - 4ac³ - 27a²d².

Cardano's Method Explained

Cardano's method first reduces the general cubic to a "depressed" cubic (without an x² term) using the substitution x = t - b/(3a). The resulting equation t³ + pt + q = 0 is then solved using the formulas:

  1. Substitute x = t - b/(3a) to eliminate the x² term.
  2. Compute p and q for the depressed cubic t³ + pt + q = 0.
  3. Calculate the discriminant: Δ = -(4p³ + 27q²).
  4. Apply Cardano's formula or trigonometric method based on the discriminant.
  5. Convert back to x using x = t - b/(3a).

Complex Roots

When the discriminant is negative, the cubic has one real root and a pair of complex conjugate roots. Complex roots always come in conjugate pairs for polynomials with real coefficients: if a + bi is a root, then a - bi is also a root. This calculator displays complex roots in the standard form a + bi.

Historical Significance

The solution of the cubic equation was one of the great mathematical achievements of the Renaissance. It was first discovered by Scipione del Ferro around 1515, independently by Niccolo Tartaglia in 1535, and published by Gerolamo Cardano in his 1545 masterwork "Ars Magna." The cubic formula led to the discovery of complex numbers, as Cardano encountered square roots of negative numbers even when all three roots were real (the "casus irreducibilis"), forcing mathematicians to take these "imaginary" quantities seriously.