I have a polynomial equation $f(x) = x^8 - x^6 - 2 k a x^3 - a^2 = 0$. I know that this cannot be solved analytically using radicals. However, I would like to approximate the dependence of the positive real root on the parameters $a$ and $k$. For my application, $a$ and $k$ are roughly order 1. It seems like I should be able to make an educated guess as to the form of an approximate function. However, I am unsure where to start. Here is an example of the positive real root as a function of $a$ for fixed $k$.
