find the number of irrational roots of $\frac{4x}{x^2 + x + 3} + \frac{5x}{x^2 - 5x + 3} = -\frac{3}{2}$

Solution 1:

Here is another way to solve this problem. It is up to you and @dxiv to decide whether this method is longer or shorter than yours.

We start by shifting the $2^{\text{nd}}$ term on LHS of the given identity to its RHS. $$\dfrac{4x}{x^2 + x + 3} = -\dfrac{3}{2}-\dfrac{5x}{x^2 - 5x + 3}$$

When RHS is simplified, we have, $$\dfrac{4x}{x^2 + x + 3} = -\dfrac{3x^2 - 5x + 9}{2x^2 - 10x + 6}.$$

Now, we add the denominators to the respective numerators to get, $$\dfrac{x^2 + 5x + 3}{x^2 + x + 3} = \dfrac{-x^2 - 5x - 3}{2x^2 - 10x + 6}.$$

This implies, $$ x^2 + 5x + 3 = 0\qquad\text{and}\qquad x^2 + x + 3= -2x^2 + 10x - 6\quad\Longrightarrow\quad x^2 -3 x + 3=0.$$

The first quadratic equation gives us two irrational roots, while the second two imaginary roots