I'm working on an implementation of a numeric solver for quintic equations and have a question regarding usage of the principal form.
Say you have some quintic equation in general form. For example, the following:
$$x^5 + 2x^4 + 3x^3 + 4x^2 + 5x + 6 = 0\tag{1}$$
Converting to the principal form via a quadratic Tschirnhausen transformation, we end up with
$$y^5 - 14y^2 - 105y - 126 = 0\tag{2}$$
which has an approximate real root of $\alpha = 3.71725922555816$.
To turn this root into an approximate root of the original quintic we must solve $$ y=x^2+mx+n\tag{3}$$ for $x$ given $y=\alpha, m = 1, n = 0$, yielding $x = -1.49179798813990$ and $x = 2.49179798813990$, of which the former is indeed an approximate root of the general quintic $(1)$.
My question is how can I determine which root of $(3)$ will be a root of $(1)$ without simply plugging into $(1)$ and testing?