I don't need to know solutions. I figured that the logic would be similar to computing the discriminant and testing whether it is positive. For example in quadratic systems with real coefficients,
$$ ax^2 + bx + c = 0$$
if $b^2 - 4ac < 0$ then both solutions are complex, otherwise they are real. Then, if c > 0 they are the same sign, and then the sign of $b$ determines what their sign is if so.
Thus if $b^2 - 4ac > 0$ and $c < 0$ I know the quadratic equation has exactly one real, positive solution.
Are there general versions of this criteria?