How to Solve a Cubic Equation

Finds the real roots of ax³ + bx² + cx + d = 0. Shifting x by b/(3a) removes the squared term, and the remaining t³ + pt + q = 0 is solved according to the sign of the discriminant. There is always at least one real root.

This finds the real roots of $ax^3 + bx^2 + cx + d = 0$ . A cubic always has at least one real root: its graph runs from the bottom left to the top right, so it has to cross the $x$ axis somewhere.

x = t - \dfrac{b}{3a}

Shifting $x$ this way removes the squared term and leaves $t^3 + pt + q = 0$ . From there the sign of the discriminant $\dfrac{q^2}{4} + \dfrac{p^3}{27}$ decides how to carry on.

The three cases

A positive discriminant means one real root, found by adding two cube roots in Cardano's formula. Zero means there is a repeated root, leaving one or two real roots. A negative discriminant means three distinct real roots, but Cardano's formula then asks for the cube root of a negative number and cannot be written with real numbers, so trigonometry is used instead.

Example

For $x^3 - 6x^2 + 11x - 6 = 0$ , the shift gives $p = -1$ and $q = 0$ , so the discriminant is $-\dfrac{1}{27}$ , which is negative. There are three distinct real roots: $x = 1, 2, 3$ . Expanding $(x-1)(x-2)(x-3)$ returns the original equation.

Notes

Only distinct real roots are listed, smallest first. A repeated root counts once, so $x^3 = 0$ has the single root 0. Any unused slots are filled with a hyphen.

Complex roots are not given. With $a$ equal to 0 this is not a cubic, so use the quadratic calculator instead.