Cubic Equation Solver
Solve any cubic equation ax³ + bx² + cx + d = 0 online for free. Our cubic equation solver applies Cardano's method and the trigonometric method to find all three roots — real or complex — with the discriminant, depressed cubic coefficients, Vieta's formula verification, and a full step-by-step solution. All calculations run locally in your browser. No signup required.
Enter the four coefficients a, b, c, d for the equation ax³ + bx² + cx + d = 0. The cubic equation solver applies Cardano's method and the trigonometric method to find all three roots — real or complex — with a full step-by-step breakdown. All calculations run locally in your browser.
Cannot be 0
- Substitute x = t − b/(3a) to eliminate the x² term → depressed cubic t³ + pt + q = 0
- Compute discriminant Δ = −4p³ − 27q²
- Δ > 0: three distinct real roots (trigonometric method)
- Δ = 0: repeated roots (double root + simple root)
- Δ < 0: one real root + two complex conjugate roots (Cardano's formula)
Why Use Our Cubic Equation Solver?
Instant Cubic Equation Solving
Solve any cubic equation ax³ + bx² + cx + d = 0 instantly in your browser. Our cubic equation solver applies Cardano's method and the trigonometric method to find all three roots — real or complex — in milliseconds.
Secure Cubic Equation Solver Online
All cubic equation calculations happen locally in your browser. Your coefficients and results never leave your device, ensuring 100% privacy when you solve cubic equations online.
Cubic Equation Solver Online — No Installation
Use our cubic equation solver directly in any browser with no downloads, plugins, or software required. Solve cubic equations from any device — desktop, tablet, or mobile — instantly.
Full Cardano's Method Breakdown
Get all three roots (real or complex), the depressed cubic coefficients, the discriminant, Vieta's formula verification, and a complete step-by-step Cardano's method solution — all in one place.
Common Use Cases for Cubic Equation Solver
Algebra & Calculus Coursework
Students solving cubic equations in algebra, precalculus, and calculus courses use our cubic equation solver to check answers and understand Cardano's method step by step. The full breakdown makes it easy to follow the derivation.
Physics & Engineering Problems
Cubic equations appear in mechanics (equilibrium of forces), thermodynamics (van der Waals equation of state), and structural engineering (beam deflection). Our cubic equation solver handles all three root cases accurately.
Control Systems & Signal Processing
Control engineers find the poles of third-order transfer functions by solving cubic characteristic equations. Use our cubic equation solver to find all three poles — real or complex — and determine system stability.
Optimization & Economics
Cubic cost, revenue, and profit functions in economics and operations research require finding critical points by solving cubic derivative equations. Our solver handles decimal and negative coefficients with full precision.
Computer Graphics & Geometry
Cubic Bézier curves, ray-sphere intersections, and spline computations in computer graphics require solving cubic equations. Our cubic equation solver provides the analytical roots needed for exact geometric calculations.
Number Theory & Pure Mathematics
Mathematicians studying algebraic number theory, Galois theory, and polynomial roots use cubic equations as fundamental examples. Our solver shows the full Cardano's method derivation including the depressed cubic substitution.
Understanding Cubic Equation Solving
What is a Cubic Equation?
A cubic equation is a polynomial equation of degree 3 in the form ax³ + bx² + cx + d = 0, where a ≠ 0. Every cubic equation with real coefficients has exactly three roots (counting multiplicity) in the complex numbers — guaranteed by the Fundamental Theorem of Algebra. These roots are either three real numbers, or one real number and two complex conjugates. Unlike quadratic equations, cubic equations always have at least one real root. Our cubic equation solver finds all three roots analytically using Cardano's method (published by Gerolamo Cardano in 1545) and the trigonometric method for the three-real-roots case.
Historical Note: Cardano's Method: Cardano's method was published in Ars Magna (1545) by Gerolamo Cardano, though the original discovery is attributed to Scipione del Ferro and Niccolò Tartaglia. It was the first general algebraic solution to a polynomial equation of degree higher than 2. The method introduces the substitution x = t − b/(3a) to eliminate the quadratic term, then uses the identity (u + v)³ = u³ + v³ + 3uv(u + v) to reduce the depressed cubic to a quadratic in u³. The three-real-roots case (casus irreducibilis) requires complex intermediate values even when all roots are real — which is why the trigonometric method is preferred for that case.
How Our Cubic Equation Solver Works
- 1. Enter your coefficients: Type the four coefficients a, b, c, d for ax³ + bx² + cx + d = 0. All processing happens locally in your browser — your data never leaves your device.
- 2. Depressed cubic substitution: The solver applies the Tschirnhaus–Vieta substitution x = t − b/(3a) to eliminate the x² term, reducing the equation to the depressed cubic t³ + pt + q = 0.
- 3. Discriminant and root type: The discriminant Δ = −4p³ − 27q² determines the root nature. Δ > 0 gives three distinct real roots (solved via the trigonometric method). Δ = 0 gives repeated roots. Δ < 0 gives one real root and two complex conjugates (solved via Cardano's formula).
What the Solver Computes
The Three Root Cases
| Discriminant Δ | Root Nature |
|---|---|
| Δ > 0 | Three distinct real roots |
| Δ = 0 | Repeated roots (one double + one simple, or triple) |
| Δ < 0 | One real root + two complex conjugate roots |
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Frequently Asked Questions About Cubic Equation Solver
A cubic equation solver finds all three roots of any equation in the form ax³ + bx² + cx + d = 0. Our cubic equation solver applies Cardano's method and the trigonometric method to compute real and complex roots analytically, with a full step-by-step breakdown — all running instantly in your browser with no signup required.
Every cubic equation with real coefficients has exactly three roots in the complex numbers (counting multiplicity), guaranteed by the Fundamental Theorem of Algebra. These are either three real roots, or one real root and two complex conjugate roots. Unlike quadratic equations, a cubic always has at least one real root.
Cardano's method is the analytical technique for solving cubic equations, published by Gerolamo Cardano in 1545. It works by substituting x = t − b/(3a) to eliminate the x² term (giving the depressed cubic t³ + pt + q = 0), then using the discriminant Δ = −4p³ − 27q² to determine the root type and apply the appropriate formula.
For the depressed cubic t³ + pt + q = 0, the discriminant Δ = −4p³ − 27q² determines the root nature. If Δ > 0, there are three distinct real roots. If Δ = 0, there are repeated roots (a double root and a simple root, or a triple root). If Δ < 0, there is one real root and two complex conjugate roots.
The trigonometric method (also called the casus irreducibilis solution) is used when Δ > 0 — the three-distinct-real-roots case. Cardano's formula would require taking cube roots of complex numbers in this case, which is cumbersome. The trigonometric method uses the substitution t = m·cos(θ) to express all three roots cleanly using cosine.
Yes. When Δ < 0, the cubic equation solver computes the one real root and the two complex conjugate roots using Cardano's formula. The complex roots are displayed in standard a + bi form with copy buttons.
If a = 0, the equation is not cubic — it becomes quadratic or lower degree. The solver displays an error asking you to enter a non-zero value for a. For quadratic equations, use our Quadratic Equation Solver instead.
Yes, completely. All calculations happen locally in your browser using JavaScript. Your coefficients and results are never sent to any server. Your data never leaves your device, ensuring 100% privacy when you solve cubic equations online.
Yes. Our cubic equation solver is 100% free with no signup, no account, and no usage limits. Solve as many cubic equations as you need — with full Cardano's method steps — at no cost, forever.