SOLUTION: combine and simplify: 3x/(x+2) + x/(x-2) - 4x/(x^2-4) the answer is 4x/(x+2), but I can't figure out how to get it.

Algebra ->  Equations -> SOLUTION: combine and simplify: 3x/(x+2) + x/(x-2) - 4x/(x^2-4) the answer is 4x/(x+2), but I can't figure out how to get it.      Log On


   

Question 553022: combine and simplify:
3x/(x+2) + x/(x-2) - 4x/(x^2-4)
the answer is 4x/(x+2), but I can't figure out how to get it.
Answer by TutorDelphia(193) About Me  (Show Source):
You can put this solution on YOUR website!
Like all fraction addition, the first thing we have to do is get the denominators the same.
Your teacher gave you a problem that works out relatively easily if you know the trick. The trick is the (x+2)*(x-2)=x^2-4. This is called the difference of two squares and you can find more info on it here: http://www.themathpage.com/alg/difference-two-squares.htm
So my common denominator will be x^2-4
To get the first fraction to equal that on the denominator I have to multiply by (x-2), for the second fraction, I have to multiply by x+2 and the third is already set

I can distribute the bottoms and get:

Since the denominator is all the same, I can now add or subtract the tops:
%283x%2A%28x-2%29%2Bx%2A%28x%2B2%29-4x%29+%2F%28x%5E2-4%29
Distributing the tops gets me
%283x%5E2-6x%2Bx%5E2%2B2x-4x%29%2F%28x%5E2-4%29
Combining like terms:
%284x%5E2-8x%29%2F%28x%5E2-4%29
Let's factor out a 4x from the top
%284x%28x-2%29%29%2F%28x%5E2-4%29
now remember x%5E2-4=%28x%2B2%29%28x-2%29
4x%28x-2%29%2F%28x%2B2%29%28x-2%29
the x-2 cancels out and we get
4x%2F%28x%2B2%29