# How do you express  (x+2) / (x(x-4)) in partial fractions?

May 19, 2018

$\frac{x + 2}{x \left(x - 4\right)} = \frac{3}{2 \left(x - 4\right)} - \frac{1}{2 x}$

#### Explanation:

Here,

$\left(1\right) \frac{x + 2}{x \left(x - 4\right)} = \frac{1}{2} \left[\frac{2 x + 4}{x \left(x - 4\right)}\right]$

$\textcolor{w h i t e}{\frac{x + 2}{x \left(x - 4\right)}} = \frac{1}{2} \left[\frac{3 x - \left(x - 4\right)}{x \left(x - 4\right)}\right]$

$\textcolor{w h i t e}{\frac{x + 2}{x \left(x - 4\right)}} = \frac{1}{2} \left[\frac{3 x}{x \left(x - 4\right)} - \frac{x - 4}{x \left(x - 4\right)}\right]$

$\textcolor{w h i t e}{\frac{x + 2}{x \left(x - 4\right)}} = \frac{1}{2} \left[\frac{3}{x - 4} - \frac{1}{x}\right]$

$\textcolor{w h i t e}{\frac{x + 2}{x \left(x - 4\right)}} = \frac{3}{2 \left(x - 4\right)} - \frac{1}{2 x}$

$\left(2\right) \frac{x + 2}{x \left(x - 4\right)} = \frac{A}{x - 4} + \frac{B}{x}$

$\implies x + 2 = A x + B \left(x - 4\right)$

Take ,

$x = 4 \implies 4 + 2 = A \left(4\right) \implies 4 A = 6 \implies A = \frac{3}{2}$

$x = 0 \implies 0 + 2 = B \left(- 4\right) \implies B = - \frac{2}{4} = - \frac{1}{2}$

Subst. values of $A \mathmr{and} B$ into $\left(2\right)$

$\frac{x + 2}{x \left(x - 4\right)} = \frac{\frac{3}{2}}{x - 4} + \frac{- \frac{1}{2}}{x}$

$\frac{x + 2}{x \left(x - 4\right)} = \frac{3}{2 \left(x - 4\right)} - \frac{1}{2 x}$