Cubic Equations

The idea

A cubic equation in one variable has the general form:

Its graph is an S-shaped curve with up to two turning points, so it can cross the x-axis once, twice (with a repeated root), or three times — meaning a cubic has up to three real roots.

Unlike linear or quadratic equations, most cubics don't factor neatly by inspection. The most practical approach in a classroom is the Rational Root Theorem: try small integer divisors of the constant term d as candidate roots, confirm one by substitution, then divide it out (see the Polynomials page for synthetic division) to reduce the cubic to a quadratic.

A true general formula does exist — Cardano's formula — but it's considerably more involved than the quadratic formula, which is part of why it took mathematicians until the 16th century to find it.

Switch to Interactive Grapher to change a, b, c, d and watch the curve — and its real roots — update live.

Worked example

Method 1 — Factorization (via trial root + synthetic division)

By the Rational Root Theorem, try divisors of −6: ±1, ±2, ±3, ±6. Testing x = 1:

So (x − 1) is a factor. Dividing the cubic by (x − 1) using synthetic division gives a quotient of x² − 5x + 6, so:

Setting each factor to zero: x = 1, 2, 3.

Method 2 — Using Cardano's formula

First, depress the cubic by substituting x = t − b/3a, which removes the x² term. For ax³+bx²+cx+d, this gives an equation of the form t³ + pt + q = 0, where:

The real root of the depressed cubic is then given by Cardano's formula:

Substituting a = 1, b = −6, c = 11, d = −6 through this formula produces t = 0, and reversing the substitution x = t − b/3a = t + 2 gives x = 2 — one of the same three roots found by factorization, confirming the two methods agree. In practice, once one root is found, factorization is almost always the faster route for a cubic with rational roots; Cardano's formula matters most when no rational root exists at all.