Question

Find the roots of each of the following equations, if they exist, by applying the quadratic formula:
$\frac{1}{x}-\frac{1}{x-2}=3,x\ne 0,2$

Answer 1

We have

$\frac{1}{x}-\frac{1}{x-2}=3,-\frac{\left(1\right)\left(x\right)}{(x-2)\left(x\right)}=3$


The common denominator is x(x-2)

$\to$ $\frac{1.(x-2)}{x.(x-2)}$ - $\frac{\left(1\right)\left(x\right)}{(x-2)\left(x\right)}=3$

$\to$ $\frac{(x-2)}{x.(x-2)}$ - $\frac{x}{(x-2)\left(x\right)}=3$

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

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



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

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

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

Using the quadratic formula and solving for x we have
$\frac{-b\pm \sqrt{b^2-4ac}}{2a}$

$\to$ $\frac{6\pm \sqrt{6^2-4\left(3\right)\left(2\right)}}{2\left(3\right)}$

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

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

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


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