# How do you simplify (2)/(x) + (2)/(x-1) - (2)/(x-2) ?

Mar 19, 2017

$\frac{6 {x}^{2} - 12 x + 4}{{x}^{3} - 3 {x}^{2} + 2 x}$

#### Explanation:

The idea is to get everything into one big fraction by multiplying everything by the common denominator. In this case, the common denominator is $x \left(x - 1\right) \left(x - 2\right)$. So, just like with regular fractions, we can multiply the numerator and denominator of each fraction to get them to have a common denominator.

$\frac{2}{x} + \frac{2}{x - 1} + \frac{2}{x - 2}$

$\frac{2 \left(x - 1\right) \left(x - 2\right)}{x \left(x - 1\right) \left(x - 2\right)} + \frac{2 \left(x\right) \left(x - 2\right)}{x \left(x - 1\right) \left(x - 2\right)} + \frac{2 \left(x\right) \left(x - 1\right)}{x \left(x - 1\right) \left(x - 2\right)}$

$\frac{2 {x}^{2} - 6 x + 4}{{x}^{3} - 3 {x}^{2} + 2 x} + \frac{2 {x}^{2} - 4 x}{{x}^{3} - 3 {x}^{2} + 2 x} + \frac{2 {x}^{2} - 2 x}{{x}^{3} - 3 {x}^{2} + 2 x}$

$\frac{6 {x}^{2} - 12 x + 4}{{x}^{3} - 3 {x}^{2} + 2 x}$

Mar 21, 2017

$\frac{2 \left({x}^{2} - 4 x + 2\right)}{x \left({x}^{2} - 3 x + 2\right)}$

#### Explanation:

$\frac{2}{x} + \frac{2}{x - 1} - \frac{2}{x - 2}$

$\therefore = \frac{2 \left(x - 1\right) \left(x - 2\right) + 2 x \left(x - 2\right) - 2 x \left(x - 1\right)}{x \left(x - 1\right) \left(x - 2\right)}$

$\therefore = \frac{2 \left({x}^{2} - 3 x + 2\right) + 2 {x}^{2} - 4 x - 2 {x}^{2} + 2 x}{x \left({x}^{2} - 3 x + 2\right)}$

$\therefore = \frac{2 {x}^{2} - 6 x + 4 + 2 {x}^{2} - 4 x - 2 {x}^{2} + 2 x}{x \left({x}^{2} - 3 x + 2\right)}$

$\therefore = \frac{2 {x}^{2} - 8 x + 4}{x \left({x}^{2} - 3 x + 2\right)}$

$\therefore = \frac{2 \left({x}^{2} - 4 x + 2\right)}{x \left({x}^{2} - 3 x + 2\right)}$