How to solve an nth degree polynomial equation

The typical approach of solving a quadratic equation is to solve for the roots

Here, the degree of x is given to be 2

However, I was wondering on how to solve an equation if the degree of x is given to be n.

For example, consider this equation:

For higher degrees, no general formula exists (or more precisely, no formula in terms of addition, subtraction, multiplication, division, arbitrary constants and $n$-th roots). This result is proved in Galois theory and is known as the Abel-Ruffini theorem. Edit: Note that for some special cases (e.g., $x^n - a$), solution formulas exist, but they do not generalize to all polynomials. In fact, it is known that only a very small part of polynomials of degree $\ge 5$ admit a solution formula using the operations listed above.