SOLUTION: ADD: x/x^2 - 16 plus 4/x^2 + 8x + 16

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Question 371726: ADD: x/x^2 - 16 plus 4/x^2 + 8x + 16
Answer by jsmallt9(3758) About Me  (Show Source):
You can put this solution on YOUR website!
x%2F%28x%5E2+-+16%29+%2B+4%2F%28x%5E2+%2B+8x+%2B+16%29
In order to add fractions the denominators must be the same. So we need to start by finding the Lowest Common Denominator (LCD). And to find the LCD we must know the factors of the current denominators. So we factor the denominators first:
x%2F%28%28x%2B4%29%28x-4%29%29+%2B+4%2F%28%28x%2B4%29%28x%2B4%29%29
Looking at the above, if you have trouble figuring out the LCD I find the following can be helpful:
x^2 - 16       = (x+4) * (x-4)
x^2 + 8x + 16  = (x+4) *         (x+4)
--------------------------------------
LCD            = (x+4) * (x-4) * (x+4)

I have listed the two denominators and their factors. And I have written the factors in such a way that the common factors are lined up in the same column. Below them I have written the LCD. The LCD includes the factors from every column. (Note that the common factor, the first (x+4), shows up just once in the LCD. (The second (x+4) is in a separate column and it, too is included.))

Now that we know the LCD, we look at each fraction a figure what factor(s) of the LCD are missing from its denominator. Whatever factors are missing we will "add" by multiplying the numerator and denominator of that fraction by the missing factor(s).
The first denominator is missing the second (x+4) and the second denominator is missing the (x-4):

Multiplying out the numerators and, for reasons I'll explain later, leaving the denominators in factored form we get:

The denominators are now equal so we can add:
%28x%5E2%2B4x%2B4x-16%29%2F%28%28x%2B4%29%28x-4%29%28x%2B4%29%29
which simplifies to:
%28x%5E2%2B8x-16%29%2F%28%28x%2B4%29%28x-4%29%28x%2B4%29%29
At this point we should try to reduce this fraction. For this we try to factor the numerator and see if it has any factors in common with the denominator. (This is why we left the denominator factored.)
However the numerator will not factor. So the fraction will not reduce. So the answer is either
%28x%5E2%2B8x-16%29%2F%28%28x%2B4%29%28x-4%29%28x%2B4%29%29
or
%28x%5E2%2B8x-16%29%2F%28%28x%2B4%29%28x-4%29%28x%2B4%29%29 with the denominator multiplied out.