About

A cubic equation ax³ + bx² + cx + d = 0 always has three roots in the complex plane, counted with multiplicity. The nature of those roots - three distinct reals, a repeated real, or one real with two complex conjugates - depends entirely on the sign of the discriminant Δ. Misclassifying root type leads to incorrect factorizations in engineering filter design, control system pole placement, and structural eigenvalue analysis. This calculator applies Cardano's analytical method with Vieta's trigonometric refinement when Δ > 0, avoiding the numerical instability that plagues naive implementations near the casus irreducibilis.

The tool assumes a ≠ 0. If a = 0, the equation degenerates to a quadratic, which is handled separately via the standard quadratic formula. All roots are refined with a Newton-Raphson polishing pass. Results are accurate to 10 significant digits for coefficients within ±10¹⁵. Note: for coefficients exceeding IEEE 754 double-precision range, rounding artifacts may appear.

Formulas

The general cubic equation is first converted to the depressed cubic by substituting x = t − b3a, yielding:

t³ + pt + q = 0

where the intermediate coefficients are:

p = 3ac − b²3a²

q = 2b³ − 9abc + 27a²d27a³

The discriminant of the original cubic is:

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

For the depressed cubic, define:

Δ₀ = b² − 3ac

Δ₁ = 2b³ − 9abc + 27a²d

When Δ > 0 (three real roots), the trigonometric method is used:

x_k = 2√−p3 cos(13 arccos(3q2p √−3p) − 2πk3) − b3a

for k = 0, 1, 2.

When Δ ≤ 0, Cardano's direct formula computes:

C = ∛Δ₁ ± √Δ₁² − 4Δ₀³2

where a = leading coefficient, b = quadratic coefficient, c = linear coefficient, d = constant term, p = depressed cubic linear coefficient, q = depressed cubic constant, Δ = discriminant, k = root index.

Reference Data

Discriminant Condition	Root Classification	Example Equation	Roots
Δ > 0	Three distinct real roots	x³ − 6x² + 11x − 6 = 0	1, 2, 3
Δ = 0 (with Δ₀ ≠ 0)	One simple root + one double root	x³ − 3x + 2 = 0	1 (double), −2
Δ = 0 (with Δ₀ = 0)	One triple root	x³ − 3x² + 3x − 1 = 0	1 (triple)
Δ < 0	One real root + two complex conjugates	x³ + x + 1 = 0	−0.6824, 0.3412 ± 1.1615i
Notable Cubic Identities & Special Forms
Sum of cubes	a³ + b³	(a + b)(a² − ab + b²)
Difference of cubes	a³ − b³	(a − b)(a² + ab + b²)
Perfect cube	(a + b)³	a³ + 3a²b + 3ab² + b³
Depressed cubic	t³ + pt + q = 0	Obtained by substituting x = t − b3a
Vieta's Formulas for x³ + px² + qx + r = 0
x₁ + x₂ + x₃	= −p	x₁x₂ + x₁x₃ + x₂x₃	= q
x₁x₂x₃	= −r	(Product of roots equals negative constant term)
Historical Milestones
1545	Cardano publishes Ars Magna	First general algebraic solution of the cubic
1572	Bombelli introduces complex numbers	Resolved casus irreducibilis paradox
1637	Descartes' La Géométrie	Modern algebraic notation standardized
1799	Fundamental Theorem of Algebra	Every degree-n polynomial has exactly n roots in C

Frequently Asked Questions

When a = 0, the equation is no longer cubic. The calculator automatically falls back to the quadratic formula x = −b ± √b² − 4ac2a (using the remaining non-zero coefficients as the quadratic parameters). If both a and b are zero, it reduces further to a linear equation.

The casus irreducibilis occurs when Δ > 0 (three distinct real roots) but Cardano's formula produces intermediate complex cube roots. Naively computing real cube roots of complex numbers introduces floating-point errors. This calculator avoids the problem entirely by switching to Vieta's trigonometric method in that case, using arccos and cos which are numerically stable in IEEE 754 arithmetic.

Δ > 0 means three distinct real roots. Δ = 0 means at least two roots coincide (a repeated root). Δ < 0 means one real root and two complex conjugate roots. The discriminant formula Δ = 18abcd − 4b³d + b²c² − 4ac³ − 27a²d² generalizes the quadratic discriminant b² − 4ac to the third degree.

JavaScript uses IEEE 754 double-precision floats with approximately 15-17 significant decimal digits. Coefficients within ±10¹⁵ yield reliable results. Beyond that range, the discriminant computation may overflow or lose precision. For coefficients exceeding 10¹⁰⁰, specialized arbitrary-precision arithmetic libraries would be required.

Complex roots are displayed in the standard form a ± bi. The calculator verifies each root by substituting it back into the original polynomial and checking that the residual |f(x)| < 10⁻⁸. Complex conjugate pairs always share the same real part and have imaginary parts of equal magnitude but opposite sign, which is enforced by the algorithm.

Cardano's formula involves nested cube roots and square roots, each introducing rounding error of order ε ≈ 2.2 × 10⁻¹⁶. One or two Newton-Raphson iterations (x_n+1 = x_n − f(x_n)f′(x_n)) polish the root to near machine epsilon, quadratically converging since the analytical solution is already close.