# What is (x^2-4)/(12x) -: (2-x)/(4xy)?

Feb 15, 2016

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

#### Explanation:

$\frac{{x}^{2} - 4}{12 x} \div \frac{2 - x}{4 x y}$

Whenever we have a complex division, may it is simpler to turn it into a mutiplication $a \div \left(\frac{b}{c}\right) = a \times \left(\frac{c}{b}\right)$:

$\frac{{x}^{2} - 4}{12 x} \times \frac{4 x y}{2 - x}$

We can now exchange the denominators, because multiplication is permutable:

$\frac{{x}^{2} - 4}{2 - x} \times \frac{4 x y}{12 x}$

Let's turn $2 - x$ in a expression that begins by $x$. Doesn't have any effect, but I need it to develope the reasoning:

$\frac{{x}^{2} - 4}{- x + 2} \times \frac{4 x y}{12 x}$

Now, let's take the minus sign of x to outside of the expression:

$- \frac{{x}^{2} - 4}{x - 2} \times \frac{4 x y}{12 x}$

${x}^{2} - 4$ is on the form ${a}^{2} - {b}^{2}$, which is (a+b)(a-b):

$- \frac{\left(x - 2\right) \left(x + 2\right)}{x - 2} \times \frac{4 x y}{12 x}$

Now we can cut the factors in common between numerators and denominators:

$- \frac{\cancel{x - 2} \left(x + 2\right)}{\cancel{x - 2}} \times \frac{4 \cancel{x} y}{12 \cancel{x}}$

$- \left(x + 2\right) \times \frac{4 y}{12}$

Now, you only need to divide 12 by 4:

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