An Overview of the True Quintic Formula...and Why You Should Never Use It

Many people know that the general quintic equation is unsolvable in terms of using finite elementary operations from Abel-Ruffini theorem, however, you can find a solution to it by using polynomial transformations and different non-elementary means. This video outlines one possible solution method using power series and hypergeometric functions. However, essentially all of this is a proof of concept since this solution, along with many others, is simply way too long and complicated to be used in practice, especially in back-substituted form, and it even suffers computationally with symbolic coefficients taking up so much space.

Timecodes:
0:00 - Intro and transformation
(Let principal's y² coefficient be t instead of u**)
11:40 - Bring-Jerrard principal root
20:45 - Bring-Jerrard remaining roots
(Must have u ≠ 0, could also divide out principal soln. and solve the remaining quartic**)
30:08 - Back-substitution and closer
(Elliptic soln. has actually been written back-subbed for the
original quintic, though still not with the original coefficients**)

Sources and other tidbits:
(Note that this transformation can also be rational, not just polynomial**)
(Also see Newton's Power Sum Identities for an alternate way to get coeffs**)

[joeldewitt]

I'm having unclean thoughts of writing a Fortran program to do this.

[ZXD]

I didn’t even know this was possible until I saw the video.

[captainchicky]

this is a good video, i genuinly always wondered how mathematica spits out the 4f3 hypergeometrics for quintics lol, and the scattered MSE posts never really helped much :p

[angelosterizakis]

It would be interesting to see how the Lagrange Reversion Theorem for implicit functions is derived.

[samuelogunsina]

My greatest respect for you for making this content accessible for all. I'm an EE major heading into my sophomore year. I just finished Calc 2 and about taking Calc 3 this fall.

Even though I have complete understanding of the steps to solving a cubic and quartic. I dont understand most of the techniques used in the videos.

My question is what level of Math class do I need to have completed to understand the techniques used. In a related fashion, I'm taking a math minor, starting with Bridge to Abstract Math and Linear Algebra this fall. Will they help?

[TitoTheThird]

Good work!. You got the method to transform the general quintic to Bring-Jerrard form from Math StackExchange, did you? Edit: I see from the "Sources and other tidbits" that indeed it is from MSE. :)

[isaakyhsialf]

nice, i mean the idea of solving a quintic is solving a quintic