I’m trying to solve an equation of the kind

f1(x)*m^2 + f2(x)mn + f3(x)*n = 0

for given x real defined up to a certain precision, f1, f2, f3 linear functions and m and n positive integers.
To clarify what I mean, consider that for an x like i. e. 1.34 every rational approximation like 67/50 or 267/200 that gives x to the desired precision can be tried till a solution for the equation is found.

My idea was to use brute force recursion over all the possible values 1.34 -> (1.336, …, 1.344) -> ( … ) until the solution is found through a Diophantine solver, but it is of course very inefficient and often the recursion limit is reached before a solution is found.

Any idea for a more efficient algorithms?