TITLE: How to solve an nth degree polynomial equation
QUESTION [55 upvotes]: The typical approach of solving a quadratic equation is to solve for the roots
$$x=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}$$
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:
$$a_0 x^{n} + a_1 x^{n-1} + \dots + a_n = 0$$

REPLY [2 votes]: There is a very advanced way of solving polynomial equations approximately. This is the Weierstrass method also known as the Durand-Kerner method.
To understand this method requires a knowledge of several branches of advanced algebra. This method is studied under Numerical Analysis and is explained for polynomials of degree four, but generalized to polynomials of higher degree.
Another advanced method is due to Jenkins-Traub.
These two numerical solutions, the Durand-Kerner and the Jenkins-Traub method are to be found in Wikipedia.
The question was: How to solve a polynomial of degree n.
Knowing to where to find the solution is an answer to the question cited. 
In general, there are no exact solutions for solving polynomials in terms of radicals, that is in terms of square roots, cube roots , etc., for polynomials of degree five or greater, and the solutions are approximations. There are exact solutions in terms of radicals, but only for particular cases of the polynomial coefficients. 
The numerical solution to any degree of approximation of a polynomial of degree n by constructing algorithms based on algebraic procedures, for example the two methods cited above, would be studied under Computer Science and Numerical Analysis by specialists in these fields.