Tukukkk

up vote 2 down vote favorite Let $ax^2 + bx + c = 0$ and $dy^2 + ey + f = 0$ such that variables $a$ through $f$ are fixed integer parameters. I'm trying to find integers $g$ , $h$ , and $j$ such that $g(xy)^2 + h(xy) + j = 0$ . I imagine the equation will be different depending on which roots of the original polynomials we consider. I'm looking for a proof either they exist or don't exist and if they do exist what they are in terms of the original integer values. The motivation is that I'm trying to represent certain numbers using the form $x = [S, A, B, C]$ where $S$ denotes which root $x$ is to a quadratic polynomial $Ax^2 + Bx + C$ . I'm trying to figure out how to multiply such numbers. algebra-precalculus