🔢 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.
Discriminant Δ
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
Worked examples
x³ − 6x² + 11x − 6 = 0 has the three distinct real roots 1, 2 and 3, and a positive discriminant.
x³ + x = 0 factors as x(x² + 1), giving one real root at 0 plus ±i — exactly what a negative discriminant signals.
What the roots assume
Every cubic has at least one real root, because the curve runs from −∞ to +∞ and must cross the axis, so an empty real solution set is impossible. The discriminant sign separates the cases: positive for three distinct real roots, zero when two coincide, negative for one real plus a complex pair. Coefficients are read as real numbers and roots are rounded for display.
Frequently asked questions
- What does the sign of the discriminant mean?
- Δ > 0 gives three distinct real roots, Δ = 0 gives a repeated real root, and Δ < 0 gives one real root with a complex conjugate pair.
- Why must a be nonzero?
- Without the x³ term the degree drops to two and a different method applies; a = 0 makes the equation quadratic rather than cubic.
- How does the graph help?
- The real roots are where the curve crosses the x-axis. A cubic crosses once or three times and just touches the axis when a root repeats.