Others have pointed out that there is no general solution for polynomials degree 5 and above. Of course, there are certain higher degree polynomials that are easily solvable, like $x^5-32=0$.
And there are methods that yield an inexact solution, such as numerical methods.
Here is an approach to the quadratic that is a bit off the beaten path:
re-arrange $ax^2+bx+c=0$ to get $x = \frac{-b}{a} - \frac{c}{ax}$. Now all you need to know to calculate $x$ is... $x$ itself. The cool thing is that when you assume a value for $x$, you plug that into the formula, and you get an improved estimate. Then use that improved estimate with the formula to get an even better one. What you end up with is called a CONTINUED FRACTION.
You can do something similar by using $x = \sqrt{\frac{-bx}{a} -{c}{a}}$. This gives you a continued radical approximation.