How do you simplify (8x)/ (x^2-4) div (4x^2)/ (x^2+4x+4)?

1 Answer
Jun 11, 2016

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

Explanation:

The first step here is to factorise the denominators of both fractions.
$\text{-------------------------------------------}$

${x}^{2} - 4 \text{ is a "color(blue)"difference of squares}$

which in general factorises as follows.

$\textcolor{red}{| \overline{\underline{\textcolor{w h i t e}{\frac{a}{a}} \textcolor{b l a c k}{{a}^{2} - {b}^{2} = \left(a - b\right) \left(a + b\right)} \textcolor{w h i t e}{\frac{a}{a}} |}}}$

here ${x}^{2} = {\left(x\right)}^{2} \Rightarrow a = x$

and $4 = {\left(2\right)}^{2} \Rightarrow b = 2$

$\Rightarrow {x}^{2} - 4 = \left(x - 2\right) \left(x + 2\right)$
$\text{-----------------------------------------------}$
To factorise ${x}^{2} + 4 x + 4$ we have to consider the factors of +4 (constant term) which at the same time sum to give +4 , the coefficient of the middle term (+4x)
The only pair which satisfy these conditions are 2 , 2

$\Rightarrow {x}^{2} + 4 x + 4 = \left(x + 2\right) \left(x + 2\right) = {\left(x + 2\right)}^{2}$
$\text{----------------------------------------------------}$

The original division now becomes

(8x)/((x-2)(x+2))÷(4x^2)/((x+2)(x+2))

We can now convert the operation from division to multiplication as follows.

color(red)(|bar(ul(color(white)(a/a)color(black)(a/b÷c/d=a/bxxd/c)color(white)(a/a)|)))

Changing division to multiplication gives

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

Now we can cancel any common factors on the numerators with any on the denominators.

(color(red)cancel(8)^2color(black)cancel(x))/((x-2)color(blue)cancel((x+2)))xx(color(blue)cancel((x+2))(x+2))/(color(red)cancel(4)^1color(black)cancel(x^2)^1

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