Solve any cubic equation ax³ + bx² + cx + d = 0. Find all three roots (real and complex), discriminant, and Vieta's formulas with step-by-step analysis.
A cubic equation is a polynomial equation of degree three, written in the standard form ax³ + bx² + cx + d = 0, where a ≠ 0. Unlike quadratic equations — which always have a neat formula with a single square root — cubic equations require more sophisticated methods to solve. The general solution was first published in the 16th century by Italian mathematicians Cardano and Tartaglia and involves cube roots and a discriminant that determines the nature of the roots.
Every cubic equation has exactly three roots when counted with multiplicity in the complex numbers. The discriminant Δ tells you their nature: when Δ > 0 there are three distinct real roots, when Δ = 0 there is a repeated root, and when Δ < 0 there is one real root and a pair of complex conjugate roots. Our cubic equation solver computes all three roots using Cardano's method and trigonometric substitution. It displays the discriminant, verifies each root by substitution, and shows Vieta's formulas relating the roots to the coefficients. Presets for classic cubics let you explore the behavior instantly, and a visual number line shows where real roots are positioned. Whether you are solving homework problems, analyzing polynomial behavior, or building intuition for higher algebra, this tool gives you comprehensive results.
Cubic Equation Solver helps you solve cubic equation solver problems quickly while keeping each step transparent. Instead of redoing long algebra by hand, you can enter a (x³ coefficient), b (x² coefficient), c (x coefficient) once and immediately inspect Equation, Discriminant (Δ), Real Roots to validate your work.
This is useful for homework checks, classroom examples, and practical what-if analysis. You keep the conceptual understanding while reducing arithmetic mistakes in multi-step calculations.
Given ax³+bx²+cx+d=0, substitute x=t−b/(3a) to get depressed cubic t³+pt+q=0. Discriminant Δ=−4p³−27q². If Δ>0: three real roots via trigonometric method. If Δ<0: Cardano's formula with one real and two complex roots.
Result: Equation shown by the calculator
Using the preset "x³ − 6x² + 11x − 6", the calculator evaluates the cubic equation solver setup, applies the selected algebra rules, and reports Equation with supporting checks so you can verify each transformation.
This calculator takes a (x³ coefficient), b (x² coefficient), c (x coefficient), d (constant) and applies the relevant cubic equation solver relationships from your chosen method. It returns both final and intermediate values so you can audit the process instead of treating it as a black box.
Start with the primary output, then use Equation, Discriminant (Δ), Real Roots, Complex Roots to confirm signs, magnitude, and internal consistency. If anything looks off, change one input and compare the updated outputs to isolate the issue quickly.
A strong workflow is manual solve first, calculator verify second. Repeating that loop improves speed and accuracy because you learn to spot common setup errors before they cost points on multi-step algebra problems.
Every cubic equation has exactly three roots when counted with multiplicity in the complex numbers. It may have three real roots, or one real root and two complex conjugate roots.
The discriminant Δ = −4p³ − 27q² (after depressing the cubic). When Δ > 0, all roots are real and distinct. When Δ = 0, there is a repeated root. When Δ < 0, one root is real and two are complex conjugates.
A depressed cubic is one with no x² term: t³ + pt + q = 0. Any cubic can be converted to this form by substituting x = t − b/(3a), which eliminates the quadratic term.
For ax³+bx²+cx+d=0 with roots r₁, r₂, r₃: r₁+r₂+r₃ = −b/a, r₁r₂+r₁r₃+r₂r₃ = c/a, and r₁·r₂·r₃ = −d/a. Use this as a practical reminder before finalizing the result.
This calculator is designed for real coefficients. Complex-coefficient cubics require a more general treatment and may not have conjugate root pairs.
It uses Cardano's method for the case with one real and two complex roots, and the trigonometric method (which avoids complex intermediate values) when all three roots are real. Understanding this concept helps you apply the calculator correctly and interpret the results with confidence.