How can I solve a cubic equation in terms of a variable?

[ollie456]

Can anyone help solve S³+101S²+(10K+100)S+100K to find the roots of S in terms of K

 

[Boorglar]

If you want an exact solution (i.e. not a numerical approximation) then you will have to use Cardano's method. The cubic formula is a bit messy in its generality though.
http://en.wikipedia.org/wiki/Cubic_function#Roots_of_a_cubic_function

You could also try guessing some obvious solutions and factor them, which will simplify the cubic to a quadratic. But I didn't see any just by glancing at the equation.

 

[lurflurf]

The trigonometric form is more simple.
Any cubic can be put into the form.
$$t^3+pt+q=0$$
which has solution
$$t_k=2\sqrt{-\frac{p}{3}}\cos\left(\frac{1}{3}\arccos\left(\frac{3q}{2p}\sqrt{\frac{-3}{p}}\right)-k\frac{2\pi}{3}\right) \quad \text{for} \quad k=0,1,2$$
see here
http://en.wikipedia.org/wiki/Cubic_function#Trigonometric_.28and_hyperbolic.29_method