Question

How do you combine `(4-3x)/ (16-x^2) + 3/( x-4)`?

Answer 1

You need to find the lowest common denominator among these two functions. We can try to factor the first function's denominator in order to ease things for us, as shown:

`16-x^2=0`
`16=x^2`
`sqrt(16)=x`
`x=+-4`, which means `x-4=0` and `x+4=0`.

However, the factors `(x-4)(x+4)` when multiplied, give us the opposite of the first function's denominator:

`(x-4)(x+4)=x^2-16`

We just need to multiply it by `(-1)` to validate our equality.

So, we can rewrite the whole sum as

`(4-3x)/((-1)(x+4)(x-4))+3/(x-4)`

Now, our lowest common denominator is `(x+4)(x-4)`. Adopting it as l.c.d., then we have:

`((4-3x)+3(-x-4))/((-1)(x+4)(x-4))`=`(4-6x-8)/((-1)(x+4)(x-4))`

Distributing our factors in the denominator, we have the shortest answer as follows:

`color(green)((-6x-8)/(16-x^2))`