Finding $t(r,\mu)$ for $t\left (\frac{2\mu}{3}-\mu r \right )+t^2 \left (\frac{\mu}{4}-\frac{\mu r}{3} \right )+...=r-\mu$

Question

For example, if I have:

$$t\left (\frac{2\mu}{3}-\mu r \right )=r-\mu$$

I find very easily what is $t(r,\mu)$:

$$t(r,\mu)=\frac{r-\mu}{ \left (\frac{2\mu}{3}-\mu r \right ) }$$

Now, I need to find the whole expression what is $t(r,\mu)$ in this case below: $$t\left (\frac{2\mu}{3}-\mu r \right )+t^2 \left (\frac{\mu}{4}-\frac{\mu r}{3} \right )+t^3 \left (\frac{\mu}{15}-\frac{\mu r}{12} \right )=r-\mu$$

Any help about this would mean a lot for me.

Answer 1

Just a follow up to the comment by @Delta-u...Using Mathematica one gets the following solutions to the cubic equation:

FullSimplify[Solve[t (2 m/3 - m r) + t^2 (m/4 - m r/3) + t^3 (m/15 - m r/12) == r - m, t]]