It would help if you kept your notation consistent. You called your quadratic polynomials f(x) and g(x) in one post, then p(x) and q(x) in the next.
I'll stick with f(x) and g(x), and say p = f(x1), q = g(x1).
I note that (assuming the coefficients are integers), for two quadratic polynomials to have equal and opposite discriminants, the xcoefficients b and B have to be even. For if
b^2  4*a*c + B^2  4*A*C = 0, we have
b^2 + B^2 = 4*(a*c + A*C).
The only way for the sums of the squares of two integers to be divisible by 4 is, for both squares to be even, so their roots are also even.
This does allow for some reformulation, e.g.
a*f(x) = (a*x + b/2)^2 + a*c  b^2/4 and
A*g(x) = (A*x + B/2)^2 + A*C  B^2/4
with a*c  b^2/4 and A*C  B^2/4 are equal and opposite.
I do not know of any way to combine the tests of p = f(x1) and q = g(x1).
