After stumbling through some dense fog of algebra, I have come across the following problem - any help would be much appreciated!
I'm currently trying to solve the following equation, but my efforts to date have been inconsequential. I was wondering whether anyone had some suggestions on how to proceed with solving the following equation for $x$? Clearly, in the case where $w_3 = 0$, one can easily solve a quadratic equation, but, if possible, I'd ideally like a general method for solving the equation rather than thinking about this on a case by case basis (since the $w_i$ are themselves functions of many parameters which ideally shouldn't be restricted to special cases).
The equation I'm trying to solve for $x$ is: $$a_1x^2 + a_2x + a_3x^{1-\frac{1}{\alpha}} + a_4 = 0$$
where $a_1, a_2, a_3, a_4 > 0$, and $\alpha > 0$ is a fixed parameter.
Thank you!
For arbitrary (irrational) $\alpha$ there is an obstacle to solving the equation beyond the mere fact that it is not a polynomial equation. The coefficients $w_1,w_2,w_3,w_4$ are assumed to be positive, so that there are no changes-in-sign in the equation and Descartes' rule (generalized) tells us there are no positive roots.
But on the other hand $x^{1-\frac{1}{\alpha}}$ is only well-defined for positive $x$ when $\alpha \in (0,1)$ is irrational. So putting these two observations together would say there is not even a possibility of numerical approximation of a root (no positive root exists and it is unclear what complex root might be meaningful).