Question

The roots of the equation $\frac{1}{x+1}+\frac{2}{x-4}=2$ are

• A. real and equal.
• B. rational and distinct.
• C. imaginary.
• D. irrational and distinct.

Answer 1

The correct option is D irrational and distinct.

$\frac{1}{x+1}+\frac{2}{x-4}=2\Rightarrow \frac{x-4+2x+2}{(x+1)(x-4)}=2\Rightarrow \frac{3x-2}{(x+1)(x-4)}=2$
Assuming $x\ne -1,4$
$\Rightarrow 3x-2=2(x+1)(x-4)\Rightarrow 3x-2=2x^2-6x-8\Rightarrow 2x^2-9x-6=0$
$D=81+48=129$
$D$ is positive and not a perfect square.
Hence, roots are irrational and distinct.