Cubic Equation Solver

Last updated: 2026-05-26

What is a cubic equation?

A cubic equation is a polynomial equation of degree three:

Every cubic with real coefficients has either three real roots (counting multiplicity) or one real root and two complex conjugates — it cannot have zero real roots, because a continuous function that goes from $-\infty$ to $+\infty$ must cross the axis at least once.

The depressed cubic substitution

The standard approach to solving a cubic begins by eliminating the $x^2$ term. Substituting $x = t - \tfrac{b}{3a}$ transforms the equation into a depressed cubic:

where the coefficients $p$ and $q$ depend only on $a$, $b$, $c$, and $d$:

Once the roots $t_1, t_2, t_3$ of the depressed cubic are found, the original roots are recovered by $x_k = t_k - \tfrac{b}{3a}$.

The discriminant

The key quantity that determines the root type is the discriminant:

• $\Delta > 0$: three distinct real roots — use the trigonometric method.
• $\Delta = 0$: at least two roots coincide (all real) — the trigonometric method still applies and yields the repeated value correctly.
• $\Delta < 0$: one real root and two complex conjugates — use Cardano's formula.

Cardano's formula ($\Delta < 0$)

When $\Delta < 0$ the quantity $C = \tfrac{q^2}{4} + \tfrac{p^3}{27}$ is positive, so the cube roots below are real:

The three roots of $t^3 + pt + q = 0$ are then:

The real root is $x_1 = t_1 - \tfrac{b}{3a}$; the complex pair is $\alpha \pm \beta i$ with $\alpha = -\tfrac{u+v}{2} - \tfrac{b}{3a}$ and $\beta = \tfrac{(u-v)\sqrt{3}}{2}$.

Worked example. Solve $x^3 + x - 2 = 0$.

Here $a = 1$, $b = 0$, $c = 1$, $d = -2$, giving $p = 1$, $q = -2$, and $\Delta = -112 < 0$.

Verification: $(x-1)(x^2+x+2) = x^3+x-2$ ✓.

The trigonometric method ($\Delta \geq 0$)

When $\Delta \geq 0$ the quantity $C \leq 0$, so $\sqrt{C}$ would be imaginary — Cardano's formula passes through complex numbers even when all three roots are real. This is casus irreducibilis ("the irreducible case"). The trigonometric method avoids it entirely.

Setting $r = \sqrt{-p/3}$ (which is real since $\Delta \geq 0$ implies $p \leq 0$) and $\theta = \tfrac{1}{3}\arccos\!\left(\tfrac{-q}{2r^3}\right)$, the three roots are:

The $\tfrac{2\pi}{3}$ angular spacing comes from the three cube roots of unity.

Worked example. Solve $x^3 - 6x^2 + 11x - 6 = 0$.

Here $a = 1$, $b = -6$, $c = 11$, $d = -6$. The depressed cubic has $p = -1$, $q = 0$, giving $\Delta = 4 > 0$.

Restoring the shift $h = -6/3 = -2$: $x = t - h = t + 2$, giving roots $x_1 = 3$, $x_2 = 2$, $x_3 = 1$.

Verification: $(x-1)(x-2)(x-3) = x^3-6x^2+11x-6$ ✓.

Historical context

The cubic formula was discovered by Scipione del Ferro around 1515 and later published by Gerolamo Cardano in his 1545 treatise Ars Magna. It was the first major algebraic result beyond the quadratic formula and sparked a century of competition — Cardano obtained the formula from Niccolò Tartaglia under an oath of secrecy and then published it anyway.

The discovery of casus irreducibilis in the same century forced mathematicians to take complex numbers seriously for the first time: even when the answer is a real number, the path to it requires complex arithmetic if you follow Cardano's route. Rafael Bombelli (1572) was the first to work consistently with complex intermediate values and verify that they cancel to give the correct real root.

Where cubic equations appear

Engineering — buckling loads, vibration frequencies, and deflection curves often produce cubic characteristic equations. Control system poles are solutions to cubic (or quartic) polynomials in the complex frequency variable $s$.

Computer graphics — cubic Bézier curves are parametrized by cubics in $t \in [0,1]$; finding where a Bézier curve crosses a given $y$-value requires solving a cubic.

Fluid dynamics — the van der Waals equation of state is cubic in the molar volume $V_m$: $(P + a/V_m^2)(V_m - b) = RT$. The three roots correspond to liquid, spinodal, and gas phases at temperatures below the critical point.

Finance — certain exotic option pricing models and internal rate of return calculations for three-period cash flows produce cubic equations.