
Solution to a Cubic Equation
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. 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 = (3ba^{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
x_{1} = S + T  a/3.
