but I am wondering if anyone has worked on this idea:
Notice that among all monic quadratics, the non-trivial equation $x^2+x-2$ has it's coefficient equal to it's roots, ie. $x=1,-2$. Let's consider the concept that this is the 'ideal' monic quadratic, and all other monic quadratics have transformed from this 'ideal' equation.
Now is there a systematic method where we can compare the transformation of other monic quadratics $x^2 + bx +c$ with respect to our ideal quadratic $x^2+x-2$ - such that depending on how b and c have transformed from 1 and -2 respectively, we can calculate exactly how the roots of $x^2 + bx +c = 0$ changed from 1 and -2?
I hope this makes sense. Obviously one can compare the vertex of a monic quadratic and see how far it moved from the vertex of $x^2+x-2$. So is there a clever way we can compute the roots based on how the coefficients changed with respect to our ideal parabola?
Note that I'm sure it isn't necessary to compare to $x^2+x-2$, however I am wondering that if it's unique characteristics allows a method similar to completing the square, where quadratic equations are 'compared' to $x^2$ which is another monic quadratic where the roots are same as it's coefficient.