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Cubic Equation Solver

Solve ax³ + bx² + cx + d = 0 for all three roots — real or complex — using the depressed cubic and Cardano's formula.

Discriminant
-0.037037
Nature
Three real roots
Root 1
1
Root 2
2
Root 3
3

How it works

This is the cubic sibling of this site's Quadratic Equation Solver. Dividing through by a and substituting x = t − B/3 (where B = b/a) eliminates the x² term, leaving a 'depressed' cubic t³ + pt + q = 0 with p = C − B²/3 and q = 2B³/27 − BC/3 + D. Its discriminant, Δ = (q/2)² + (p/3)³, plays the same role the quadratic formula's b² − 4ac does: it decides how many real roots there are.

When Δ ≥ 0, Cardano's formula gives the roots directly using real cube roots — one real root always, plus a complex conjugate pair when Δ > 0 (or a second repeated real root when Δ = 0). When Δ < 0, the cubic actually has three distinct real roots, but reaching them with Cardano's formula would require taking complex cube roots along the way (this is a genuine mathematical quirk called the 'casus irreducibilis') — so this case is instead solved with the standard trigonometric substitution, which stays entirely in real numbers and lands on the same three roots.

Every cubic with real coefficients has at least one real root (a real cubic's graph always crosses the x-axis at least once), which is why 'no real roots at all' never appears here the way it can for a quadratic.

FAQ

Why does a cubic always have at least one real root, unlike a quadratic?

A cubic's graph goes from −∞ to +∞ (or the reverse) as x moves from −∞ to +∞, since the x³ term dominates at the extremes. By the intermediate value theorem, a continuous curve that spans from negative to positive must cross zero somewhere — so at least one real root is guaranteed. A quadratic's graph can stay entirely above or below the x-axis (a parabola that never dips down to zero), which is why it can have zero real roots.

What does it mean for the discriminant to be exactly zero?

It means the cubic has a repeated root — two of its three roots coincide. Just like the quadratic case, this is the boundary between 'three distinct real roots' (Δ < 0) and 'one real root plus a complex pair' (Δ > 0), and all three roots come out real in this boundary case, with at least two of them equal.

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