Question

The number of real roots of the equation $\frac{4 x}{x^{2} + x + 3}+\frac{5 x}{x^{2} - 5 x + 3}=\frac{- 3}{2}$ is equal to

• A. $0$
• B. $1$
• C. $2$
• D. $4$

Answer 1

Answer: (C)

$x=0$ is not a root.
Dividing both the numerators and denominators by $x$ and putting $y=x+\frac{3}{x}$ , we get,
$\frac{4}{y + 1}+\frac{5}{y - 5}=\frac{- 3}{2}\Rightarrow y=-5,3$
$x+\frac{3}{x}=-5\Rightarrow x^{2}+5x+3=0 \rightarrow $ two real roots
$x+\frac{3}{x}=3\Rightarrow x^{2}-3x+3=0 \rightarrow $ two imaginary roots
$\Rightarrow $ Total two real roots