Find the number of irrational roots of the equation $$\dfrac{4x}{x^2 + x + 3} + \dfrac{5x}{x^2 - 5x + 3} = -\dfrac{3}{2}.$$
I got a solution : divide both denominator and numerator by $x$. Let $x+\dfrac{3}{x} = y$. Then the equation becomes $$\dfrac{4}{y + 1} + \dfrac{5}{y - 5} = \dfrac{3}{2}.$$ Now simplifying this we get $y = -5, 3$.
Finally, $x+\dfrac{3}{x} = -5$ has 2 irrational roots, and $x+\dfrac{3}{x} = 3$ has 2 imaginary roots.
But my question is, since this question is for a competitive exam, is there any other quick approach to solve this question?