How to find $r$ in an equation like this: $r^3= xr+y$

Question

Can anyone give me an an idea how to solve this and find $r$, where $r^3= xr+y$ and $x$ and $y$ are known numbers?

Answer 1

I'm sure there are more elegant methods, but one simple approach is to turn it into a polynomial in $r$ as such:

$$r^3 -xr - y = 0$$

and then apply the cubic formula (it's not so bad since there is no $r^2$ term) to obtain

$$r = \sqrt[3]{\frac{y}{2} + \sqrt{\left(\frac{y}{2}\right)^2 + \left(-\frac{x}{3}\right)^3}} + \sqrt[3]{\frac{y}{2} - \sqrt{\left(\frac{y}{2}\right)^2 + \left(-\frac{x}{3}\right)^3}}$$