## Solution to a Cubic Equation

# Solution to a Cubic Equation

^{3}+ ax

^{2}+ 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-a
^{2})/9 - R = (9ab - 27c - 2a
^{3})/54

From these, compute

- S = (R + sqrt(Q
^{3}+ R^{2}))^{1/3} - T = (R - sqrt(Q
^{3}+ R^{2}))^{1/3}

Then the real solution is

_{1}= S + T - a/3.

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