🔢 Cubic Equation Calculator
Solve the cubic equation ax³ + bx² + cx + d = 0.
f(x) = ax³ + bx² + cx + d
What is a Cubic Equation?
An equation where the highest-degree term has degree 3, in the form ax³ + bx² + cx + d = 0 (a≠0). It has at least 1 and at most 3 real roots.
Cardano's Formula
Substitute to depressed cubic t³ + pt + q = 0 then apply the following.
x = t − b/(3a)
p = (3ac − b²) / (3a²)
q = (2b³ − 9abc + 27a²d) / (27a³)
Discriminant Δ
Δ = −(4p³ + 27q²)
Δ > 0: Δ > 0: Three distinct real roots (trigonometric method)
Δ = 0: Δ = 0: Real roots including double root
Δ < 0: Δ < 0: One real root + two complex roots (conjugate pair)
Vieta's Formulas
x₁+x₂+x₃ = −b/a
x₁x₂+x₁x₃+x₂x₃ = c/a
x₁·x₂·x₃ = −d/a