Question

Factorise : $\frac{1}{x}-\frac{1}{x-2}=3,x\ne 0,2$

Answer 1

Given,

$\frac{1}{x}-\frac{1}{x-2}=3$

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

$\Rightarrow$ $-2=3x(x-2)$

$\Rightarrow$ $3x^2-6x+2=0$

We have a formula for solving quadratic equation $a{x}^2+bx+c=0$ is

$x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}$

Here, $x=\frac{6\pm \sqrt{36-4\left(3\right)\left(2\right)}}{6}$

$=\frac{1\pm \sqrt{36-24}}{6}$

$=\frac{1\pm \sqrt{12}}{6}$

$=1\pm \frac{\sqrt{3}}{3}$

$\therefore x=1+\frac{1}{\sqrt{3}}or1-\frac{1}{\sqrt{3}}$