Cubic Equations

Date: 05/17/2001 at 09:21:43
From: Mehek
Subject: How do you solve cubic equations?

Dear Doctor Math,

How do you solve cubic equations? Do you have a general formula for 
solving the cubic formula? I have looked at many Web sites and the 
algebra is confusing, so I was wondering if there was just a general 
formula for it without having to go through the derivation.

Thanks,
Mehek

Date: 05/17/2001 at 17:03:17
From: Doctor Rob
Subject: Re: How do you solve Cubic Equations?

Thanks for writing to Ask Dr. Math, Mehek.

The formula is very long and not easy to remember. It is simpler to 
derive the answer in each case numerically than to write down the 
formula and substitute the values of the coefficients.

Here is one version. Let the cubic be

   a*x^3 + b*x^2 + c*x + d = 0

Let

   A = (3*a*c-b^2)/(9*a^2)
   B = (9*a*b*c-2*b^2-27*a^2*d)/(54*a^3)
   C = -b/(3*a)
   w = -1/2 + (-3)^(1/2)/2

Then the three roots of the cubic are

   x = C + (B+[A^3+B^2]^[1/2])^(1/3) + (B-[A^3+B^2]^[1/2])^(1/3)
   x = C + w*(B+[A^3+B^2]^[1/2])^(1/3) + w^2*(B-[A^3+B^2]^[1/2])^(1/3)
   x = C + w^2*(B+[A^3+B^2]^[1/2])^(1/3) + w*(B-[A^3+B^2]^[1/2])^(1/3)

I warned you it was complicated!

- Doctor Rob, The Math Forum
  http://mathforum.org/dr.math/