How to solve $y=2x^3+x$ for $x$

Question

I'm doing something that has throw out: $$y=nx^{2n-1}+x$$ and do make any progress with this problem I needed to make x the subject. Correct me if I'm wrong but this seems impossible to work with for any values other than $n=0,1,2$.

$n=2$ gives $y=2x^3+x$, and putting that into wolfram alpha gives a very messy:

But I have no idea how I'd have gotten with this without wolfram alpha, and can't seem to find anything helpful online. Any kind of help on this would be appreciated.

*edit: forget to mention I only care about real values of x and y

Answer 1

The general cubic equation has a "closed form" solution and this is probably what it looks like for your particular equation. Wikipedia gives the general solution in the article "cubic function".

Starting with the fifth degree ($n=3$) even that is no longer guaranteed and you are looking at numerical solution methods.

Answer 2

The equation $2x^3+x-y=0$ shows only one real root. If you look here, you will notice that, in such a case, the is also an hyperbolic method; applied to your case $(p=\frac 12$, $q=-\frac y2)$, the solution write $$x=\sqrt{\frac{2}{3}} \sinh \left(\frac{1}{3} \sinh ^{-1}\left(3 \sqrt{\frac{3}{2}} y\right)\right)$$ which I personally find much nicer than the one Wolfram Alpha gave (it corresponds to the real root obtained using Cardano method).