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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?

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By "solution formula" do you mean "solution in terms of radicals"? "Galois Theory" is the branch of algebra that answers your question. – GEdgar Mar 23 '12 at 21:47
It depends on what you mean by "formula." I want to caution you that the answers below which use the word "solvable" are doing so in a rather precise technical sense of the word (see…) and it's not clear to me whether you actually care about this notion of solvability. – Qiaochu Yuan Mar 23 '12 at 22:01
up vote 3 down vote accepted

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

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Ah, that actually makes me think that maybe every polynomial equation, which can be written down as $a·X^4+...+d·X+e=0$, where $X$ made up from something solvable like $X=x^n$ might lead to a solution. (?) – NikolajK Mar 23 '12 at 22:53
Yes, of course. If $f$ is solvable and $g - c$ is solvable for every constant $c$, then $f \circ g$ is solvable. – Robert Israel Mar 24 '12 at 1:35

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.

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More generally, every abelian extension is solvable ;). – Martin Brandenburg Mar 23 '12 at 20:39

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