Follow this link to skip to the main content

Solving a Cubic Equation

Solving a Cubic Equation

You might remember solving quadratic equations in middle school using the quadratic formula. Cubic equations can be solved with a similar formula, though the formula is a bit more complicated.

Cubic equations are polynomial equations of the form:

x^3 + ax^2 + bx + c = 0

Cubic equations have three possible values for x, at least one of which is real. Below are the equations necessary for finding the real root.

First compute the following two quantities from the coefficients a, b, and c:

(9ab - 27c - 2a^3)/54

From these, compute the following:

S = (R + sqrt(Q3 + R^2))^(1/3)
T = (R - sqrt(Q3 + R^2))^(1/3)

Then the real solution is:

x1 = S + T - a/3


A service of the High Energy Astrophysics Science Archive Research Center (HEASARC), Dr. Alan Smale (Director), within the Astrophysics Science Division (ASD) at NASA/GSFC

NASA Logo, National Aeronautics and Space Administration