Many people know that the general quintic equation is unsolvable in radicals 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:
The quintic equation: https://mathworld.wolfram.com/QuinticEquation.html#:~:text=Unlike%20quadratic%2C%20cubic%2C%20and%20quartic,Abel's%20impossibility%20theorem)%20and%20Galois
Check if a quintic is solvable in radicals: https://math.stackexchange.com/a/1904919/833563
Tschirnhaus transformation: https://en.wikipedia.org/wiki/Tschirnhaus_transformation
(Note that this transformation can also be rational, not just polynomial**)
Resultant of two polynomials: https://en.wikipedia.org/wiki/Resultant
(Also see Newton's Power Sum Identities for an alternate way to get coeffs**)
Transforming a quintic guide: https://math.stackexchange.com/a/542228
Lagrange inversion/reversion theorem: https://mathworld.wolfram.com/LagrangeInversionTheorem.html
Bring radical: https://en.wikipedia.org/wiki/Bring_radical
Generalized hypergeometric function: https://en.wikipedia.org/wiki/Generalized_hypergeometric_function
Alternate differential resolvent solution: https://math.stackexchange.com/a/3789323
Alternate elliptic function solution: https://math.stackexchange.com/a/541027
Overall quintic solution summary: https://arxiv.org/pdf/math/0005026.pdf
Wolfram Mathematica guide to solving the quintic: https://web.archive.org/web/20140701060849/http://library.wolfram.com/examples/quintic/
Passare-Tsikh solution to the principal quintic (series only, not analytic continuation): https://arxiv.org/abs/0910.2957
(If you're insane, modular functions can solve any polynomial equation: https://en.wikipedia.org/wiki/Thomae%27s_formula)
Twitter - https://twitter.com/michaelmaths_
Instagram - https://www.instagram.com/michaelmaths_