Quintic Formula [duplicate]

$\begingroup$

I've recently been watching some 3blue1brown videos and I watched one about the Monster set in group theory. Now, I have my questions about that but most I'm curious about something else he mentioned, namely quintic equations. I probably spelled that wrong, but what I mean is, an equation one-step above a quadratic equation. For example, something that looks like this: $$ 6x^{5}+3x^{4}+3x^{3}+5x+6=0 $$ So, my question is, what is the universal formula for solving an equation like this. And, where exactly does it come from?

$\endgroup$ 6

2 Answers

$\begingroup$

There is no formula to find the roots of a general quintic polynomial. The Abel-Ruffini Theorem is the formal statement of this:

$\endgroup$ $\begingroup$

There is no general formula to solve a quintic, but the proof that no such formula exists requires some machinery. It is one of the famous results of a branch of abstract algebra called Galois theory. See this answer for more details.

$\endgroup$ 1