I'm interesting in finding analytical solutions for the equation 
$$\alpha K x^\alpha + x -N = 0,$$
where $\alpha$ is a positive integer and both $K$ and $N$ are positive real constants. 


Based on the meaning of the equation (derived from a problem in chemistry), there must be one (and only one) real solution between zero and $N$. Actually, I'm interested in only such solution, not any other outside the $(0, \, N)$ range. 


I know that I can use numerical methods to solve for specific $\alpha$, $N$ and $K$. I'm seeking a more elegant method. 


I also know that no general solution for a general polynomial of order 5 or more can exist, but maybe this one falls in some special case I'm not aware of.


I'm a chemist, so I apology if the question is malformed. Let me know if you need more details.



Are there analytical solutions for the polynomial $\alpha K x^\alpha + x -N = 0$?
Dividing by $\alpha K$ we obtain a polynomial equation of the form $$x^{\alpha} + d\_1x + d\_0=0.$$


Already for $\alpha=5$ it is known that such an equation (called in *Bring-Jerrard normal form*) is *not* solvable by radicals for general $d\_0$, $d\_1$. A solution of the Bring-Jerrard quintic in terms of hypergeometric functions can be found in the corresponding [Wikipedia article](https://en.wikipedia.org/wiki/Bring_radical#Bring.E2.80.93Jerrard_normal_form). 


