How to solve for p in Akra-Bazzi method for analyzing time complexity?
You don't solve for $p$ by hand. You use a computer to get a numerical solution. In general, there is no reason to expect that $p$ have some nice form.

Since the left-hand side of the equation is monotone decreasing, there is at most one solution. Furthermore, when $p \to -\infty$ the left-hand side tends to $\infty$, whereas when $p \to \infty$ it tends to $0$. This shows that a solution does exist. The computer can find it using standard root-finding methods.

This kind of situation is encountered in many other places in computer sciences. Here are two examples:


The approximation ratio of the Goemans–Williamson algorithm is
$$ \min_{0 \leq \theta \leq \pi} \frac{2}{\pi} \frac{\theta}{1-\cos\theta} \approx 0.878, $$
which doesn't have a closed form. Surprisingly, assuming the unique games conjecture, this approximation ratio is the best that polynomial time algorithms can achieve in the worst case.
Running times of fast matrix multiplication algorithms are of the form $O(n^\rho)$ where $\rho$ is in simple cases the solution of an equation similar to the one you describe, and in other cases the solution of a much more complicated optimization problem. Again, there is no closed form for $\rho$ in most cases (other than Strassen's algorithm, perhaps).