Question

$\frac{1}{(x-1)(x-2)}+\frac{1}{(x-2)(x-3)}=\frac{2}{3},x\ne 1,2,3$. Find sum of values of $x$

Answer 1

$\frac{(x-3)+(x-1)}{(x-1)(x-3)}=\frac{2(x-2)}{3}$

$\Rightarrow \frac{2x-4}{(x-1)(x-3)}=\frac{2x-4}{3}$

$\Rightarrow \frac{1}{x^2-4x+3}=\frac{1}{3}$

$\Rightarrow x^2-4x+3=3$

$\Rightarrow x^2-4x=0$

$\Rightarrow x^2=4x$

$x=4$ or $x=0$