Suppose a function
$$ g(x) = a\cdot q(x) + b\cdot f(x) $$ where $q(x)$ is an arbitrary quadratic function, $f(x)$ is an arbitrary function that's decreasing and $f(x) \leq 0$ for all $x$.
Is it possible to show that $g(x)$ only has 0,1,or 2 roots? Here a and b are arbitrary positive scalars.