The goal of the linked question is to transform the general quintic
into Bring-Jerrard normal form.
Tito Piezas III begins his answer with the quadratic Tschirnhausen transformation,
and by using resultants which may be calculated by WolframAlpha, one can write the result as
where we proceed to make $c_1=c_2=0$.
However, it is not immediately obvious to me how one performs this step, particularly the process of eliminating $x$ and replacing it with $y$.
How can I perform this step without referring to resultants and anything outside of simple algebra?
Or, if it makes any difference, how can I go from