# How do you express (5x - 4 ) / (x^2 -4x)  in partial fractions?

Oct 22, 2016

$\frac{5 x - 4}{{x}^{2} - 4 x} = \frac{4}{x - 4} + \frac{1}{x}$

#### Explanation:

The denominator can be written as $x \left(x - 4\right)$.

We can rewrite this as follows:

$\frac{A}{x - 4} + \frac{B}{x} = \frac{5 x - 4}{{x}^{2} - 4 x}$

Put on a common denominator.

$A \left(x\right) + B \left(x - 4\right) = 5 x - 4$

$A x + B x - 4 B = 5 x - 4$

$\left(A + B\right) x - 4 B = 5 x - 4$

We can hence write the following system of equations.

$A + B = 5$
$- 4 B = - 4$

Solving, we have that $B = 1$ and that $A = 4$

We can now reinsert into the original equation to get our partial fraction decomposition.

$\frac{4}{x - 4} + \frac{1}{x} = \frac{5 x - 4}{{x}^{2} - 4 x}$

Let's quickly check the validity of our answer.

What is the sum of $\frac{4}{x - 4} + \frac{1}{x}$?

Put on a common denominator.

$\frac{4 \left(x\right)}{\left(x - 4\right) \left(x\right)} + \frac{1 \left(x - 4\right)}{x \left(x - 4\right)} = \frac{4 x + x - 4}{x \left(x - 4\right)} = \frac{5 x - 4}{{x}^{2} - 4 x}$

This verifies the initial expression, so we have done our decomposition properly.

Hopefully this helps!