In Ramanujan's Notebooks Volume IV pg. 31 by Bruce C. Berndt, he describes an easy way to solve the general quartic by starting with the system$$x^2+ay=b\\y^2+cx=d\tag1$$ And solving for $x$; which gives you the depressed form of the quartic.

Now, I'm wondering whether Ramanujan had his own methods for solving the general cubic, quintic, etc.

And according to this MSE post, Ramanujan had his own method for solving a solvable quintics.

Question: What was Ramanujan's method for solving the solvable quintic and quartic?

Note: I mean the sort of sneaky system (like $(1)$) employed by Ramanujan, and some substitution.

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    $\begingroup$ Could you please clarify the meaning of the word "solving"? I am asking bacuse it is impossible to solve the general quintic in radicals. $\endgroup$
    – SashaP
    Oct 30, 2016 at 19:36
  • $\begingroup$ According to your own reference in Does anyone know about Ramanujan's method of solving the quartic?, "Ramanujan devised his own method of solving the Quartic Equation after he learnt to solve the Cubic Equation", so the method for solving the cubic was not his own. As for the quintic and higher, Abel proved that there is no method for solving those by algebraic manipulations at all. $\endgroup$
    – Conifold
    Oct 30, 2016 at 21:08
  • $\begingroup$ @SashaP Hm.. true. Perhaps I should add the most important term "solvable" in there... $\endgroup$
    – Frank
    Oct 30, 2016 at 22:44
  • $\begingroup$ @Conifold Think of this question as an extension for the linked one... :) $\endgroup$
    – Frank
    Oct 31, 2016 at 5:54


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