Intuitive reasoning why are quintics unsolvable I know that quintics in general are unsolvable, whereas lower-degree equations are solvable and the formal explanation is very hard. I would like to have an intuitive reasoning of why it is so, accessible to a bright high school student, or even why it should be so. I have also read somewhere that any $n$-degree equation can be depressed to the form $ax^n + bx + c$. I would also like to know why or how this happens, at least for lower degree equations. 
I know that this question might be too broad and difficult, but this is a thing that has troubled me a lot. To give some background, I recently figured out how to solve the cubic and started calculus, but quartics and above elude me. 
EDIT: It was mentioned in the comments, that not every $n$-degree equation can be depressed to the form $ax^n + bx + c$, although I recall something like this I have read, anyways, I wanted to find out the same for quintics.

@Sawarnik, what are you referring to exactly when you write that "the formal explanation is very hard"? The basic idea is actually quite simple. I looked through the comments above and none seem to mention the smallest nonabelian simple group which happens to be the group $A_5$ of order $60$.  This group is not contained in any $S_n$ for $n\leq 4$, which implies that all those are solvable, or equivalently any equation of order $\leq 4$ is solvable. Solvability of the group corresponds to the solvability in radicals of the polynomial. The lowest degree of a polynomial which makes it possible to have a simple Galois group is therefore $5$.
