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Solve the following equation:$\frac{1}{x}-\frac{1}{x-2}=3,\ x\ is\ not\ equal\ to\ 0,\ 2$.
Given: Equation:$\frac{1}{x}-\frac{1}{x-2}=3,\ where\ x\ is\ not\ equal\ to\ 0,\ 2$.
To do: To solve the equation for $x$ is not equal to $0,\ 2$.
Solution:
$\frac{1}{x}-\frac{1}{x-2}=3$
The common denominator is $x( x-2)$
$\Rightarrow \frac{1}{x}.\frac{x-2}{x-2}-\frac{1}{x-2}.\frac{x}{x}=3$
$\Rightarrow \frac{x-2-x}{x( x-2)}=3$
$\Rightarrow -2=3x( x-2)$
$\Rightarrow -2=3x^{2}-6x$
$\Rightarrow 3x^{2}-6x+2=0$
On comparing to $ax^{2}+bx+c=0$
$a=3,\ b=-6\ and\ c=2$
Using the quadratic formula we can solve for $x$
$\Rightarrow x=\frac{-b\pm\sqrt{( b^{2}-4ac)}}{2a}$
$\Rightarrow x=\frac{6\pm\sqrt{[62-4(3)(2)]}}{2( 3)}$
$\Rightarrow x=\frac{6\pm\sqrt{36-24}}{6}$
$\Rightarrow x=\frac{6\pm\sqrt{12}}{6}$
$\Rightarrow x=\frac{6\pm2\sqrt{3}}{6}$
$\Rightarrow x=\frac{3\pm\sqrt{3}}{3}$
Updated on: 10-Oct-2022
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