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### 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/
```
Associated Topics:
High School Basic Algebra
High School Polynomials

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