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Solution to a Cubic Equation

Solution to a Cubic Equation

Cubic equations are polynomial equations of the form x3 + ax2 + bx + c = 0.

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

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

  • Q = (3b-a2)/9
  • R = (9ab - 27c - 2a3)/54

From these, compute

  • S = (R + sqrt(Q3 + R2))1/3
  • T = (R - sqrt(Q3 + R2))1/3

Then the real solution is

x1 = S + T - a/3.
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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

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