system
Given the following post, respond as if you're commenting on Reddit:
human
Why isn't there a general formula for solving quintic polynomials like there is for quadratics, cubics and quartics?
gpt
The question really shouldn't be "Why isn't there a 'Quintic Formula'?", the question should be "Why is are there Quartic, Cubic and Quadratic Formulas?" Because, if you think about it, it really doesn't make much sense that there should be such formulas. What's happening is you are taking a single class of really simple polynomials, those of the form x^(n)-a=0, and saying that *any* polynomial of degree n can be written using nothing more than the roots to this single, very simple class of roots. Why should we expect to be able to solve something complicated like x^(4)-13x^(3)+9x^(2)+3x+1=0 using nothing but solutions to equations like x^(2)-A=0, x^(3)-B=0 and x^(4)-C=0? These equations are so much simpler than the equation we're actually trying to solve that it's kinda crazy to think that we can do it. Sure, you might be able to do it up to a point, but eventually the complicatedness will catch up and we'll be unable to solve them. This is the heuristic that you should have when thinking about solutions to polynomials.

Of course, there are deep reasons why 5 in particular is the first that is "too complicated", but these really only make sense with specialized math training. What happens is that to solve something with roots, we create a ladder to the solution where each step is a simple step that only requires a solution to x^(n)-A=0. The key is that we know what such ladders look like, and the ladder you'd need in order  to solve an arbitrary degree 5 polynomials is provably more complicated than the ladders you can construct using only roots. That is, x^(n)-A create simple steps, but in general you need giant leaps to solve arbitrary equations. The intuition that the gap between 4 and 5 is too big is actually not far off from what is really happening and really is the important thing to focus on. 

**TL;DR**  The important thing is our heuristic that says that at some point the equations get too hard to solve, and so it becomes only a question of when.