Functions
AI Assistant

Cubic Equation Solver

Graph any cubic — find all real roots with sliders and AI

A cubic equation has the form ax³ + bx² + cx + d = 0. Unlike quadratics, every cubic has at least one real root — but can have one, two (with a repeated root), or three distinct real roots depending on the coefficients.

This tool pre-draws the cubic with sliders for a, b, c, and d. Move the sliders and ask the AI to find and mark all roots — it reads the x-intercepts directly from the graph and labels them for you.

Unlike the quadratic formula, there's no simple closed-form expression for cubic roots that survives all cases, so this solver finds them numerically — which is actually how most software does it too.

Graph

FAQ

How many real roots can a cubic have?
A cubic always has exactly three roots in the complex numbers. In the real numbers, it can have one real root (two complex conjugate roots) or three real roots (which may include repeated roots). It always has at least one real root because odd-degree polynomials must cross the x-axis.
Why doesn't the cubic formula work like the quadratic formula?
Cardano's cubic formula exists but it's extremely complex — it involves cube roots of complex numbers even when the roots are real (the "casus irreducibilis"). For practical purposes, numerical methods like Newton-Raphson find roots faster and more reliably than the formula.
How do I solve x³ − 6x + 4 = 0 using this tool?
Set a=1, b=0, c=−6, d=4. The cubic crosses the x-axis at approximately x ≈ −2.73, x ≈ 0.68, and x ≈ 2.05. Ask the AI to mark all three roots on the graph.
What does the sign of the leading coefficient a do?
When a > 0, the cubic goes from bottom-left to top-right (like x³). When a < 0, it goes from top-left to bottom-right. Changing the sign of a flips the curve vertically — but the number of real roots stays the same.
What is the difference between a root and a factor?
If r is a root of the cubic, then (x − r) is a factor. A repeated root r means the graph touches the x-axis without crossing it, and (x − r)² is a factor. Three distinct real roots means the cubic factors completely as a(x − r₁)(x − r₂)(x − r₃).

Related Topics

Quadratic Formula CalculatorPolynomial FunctionsRoot Finder