- #1

Frogeyedpeas

- 80

- 0

So I was working on a little project the other day and it was in regards to the cubic formula...

Which can be found here:

http://en.wikipedia.org/wiki/Cubic_formula#General_formula_of_roots

basically given an equation of the form:

ax

^{3}+ bx

^{2}+ cx + d = 0 the formulas on the attached link will give you the three values of x that satisfy this equation.

Here is my dilemma,

I was just kind of playing around with the quadratic equation which is:

x = [itex]\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}[/itex]

It can be conveniently "re arranged" to look like this:

(2ax + b)

^{2}= b

^{2}- 4ac

So in dealing with the cubic equation I tried the same thing which was fairly simple:

(3ax + b) = Ω (where Ω is all of our other business that remains)...

Can somebody tell me the simplified value of Ω

^{3}?

Its terribly difficult to solve by hand and I do not own a symbolic calculator can manipulate it so if somebody could do the solution that would be convenient...

Additionally I would like to know another value:

Ω can be seen as the sum of (σ + μ) (see the two giant cube roots in the wiki article, those are sig and mu for this expression).

What is (σ

^{2}- σμ + μ

^{2})?

and What is (σ

^{3}- μ

^{3})

Hopefully between those three questions I can resolve to find a pattern between the quadratic and cubic formulas.

Thanks!