Which polynomial equations of higher degree will have a solution formula?

Question

A polynomial equation of degree greater than four will in general have no solution formula. But what are some typical cases one should be aware of as a practical person in which there are solutions?

Answer 1

Another example that come up fairly often is $a x^{2n} + b x^n + c = 0$

Answer 2

The example of greatest practical importance is $x^n=1$. The solutions are the $n$-th roots of unity. They come up moderately often in applied work, and are omnipresent in pure mathematics.