Question

Simplify. Show and explain your work.

Answer 1

Hey there!!

First of all , let us learn the basic formula -

a² - b² = ( a - b ) ( a + b )

First , let's solve for the numerator

( x / x - 2 ) + x

The LCM would be ( x - 2 )

So, this would become

x + x² - 2x / x - 2

x² - x / x - 2

..................................

Now let's solve for the denominator

1 / x² - 4

We can write 4 as 2²

1 / x² - 2²

As we have discussed the formula a² - b² and now is the right time to implement it .

a = x , b = 2

1 / ( x - 2 ) ( x + 2 )

The total equation :

( x² - x / x - 2  ) / ( 1 / ( x - 2 ) ( x + 2 )  

Remember :

When we have a / b

We can also write it as ( a × 1 / b )

Over here ,

a = ( x² - x / x - 2  ) b =  ( 1 / ( x - 2 ) ( x + 2 )

( x² - x / x - 2  )  /  ( 1 / ( x - 2 ) ( x + 2 )  

( x² - x / x - 2  )  × ( x - 2 ) ( x + 2 )  

The two x-2 would get cancelled

(x² - x )(  x + 2) is the final answer

(x² - x )( x + 2)

Hope my answer helps!

Answer 2

$\dfrac{\frac{x}{x-2}+x}{\frac{1}{x^2-4}}=\left(\dfrac{x}{x-2}+\dfrac{x(x-2)}{x-2}\right):\left(\dfrac{1}{x^2-2^2}\right)\\\\\text{use distributive property}\\\\=\left(\dfrac{x}{x-2}+\dfrac{x^2-2x}{x-2}\right)\cdot\left(\dfrac{x^2-2^2}{1}\right)\\\\\text{use}\ a^2-b^2=(a-b)(a+b)\\\\=\dfrac{x+x^2-2x}{x-2}\cdot\dfrac{(x-2)(x+2)}{1}\\\\(x-2)\ \text{is canceled}\\\\=(x^2-x)(x+2)\\\\\text{use distributive property}\\\\=(x^2)(x)+(x^2)(2)-(x)(x)-(x)(2)=x^3+2x^2-x^2-2x\\\\=\boxed{x^3+x^2-2x}$