Question

Find the roots of the equation $\frac{1}{x}-\frac{1}{x-2}=3,x\ne 0,2$.

Answer 1

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

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

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

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

$x=\frac{6\pm \sqrt{6^2-4\times 3\times 2}}{2\times 3}$

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

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

$=\frac{6\pm 2\sqrt{3}}{6}$

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

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