Simplify Math Problem: How to Get Final Answer?

[rockytriton]

I have a problem where I got the final answer:

       1              1
-  ---------   -  --------
   (u - 1)^2       (u + 1)^2

which is correct, but the book further simplifies it to:

     2(1 + u^2)
-  -------------
    (u^2 - 1)^2

I tried and tried, but couldn't figure out how to simplify it to that result. Can someone please explain the process to me?

Thanks!

 

[HallsofIvy]

Do you remember adding fractions from fourth grade arithmetic?

Get a common denominator and add.

The denominator of one fraction is (u-1)²= (u-1)(u-1) and the denominator of the other is (u+1)^{2[/sup= (u+1)(u+1). The "least common denominator" is (u-1)(u-1)(u+1)(u+1)= (u-1)(u+1)(u-1)(u+1)= (u2-1)(u2-1)= (u2-1)2.
$$-\frac{1}{(u-1)^2}- \frac{1}{(u+1)^2}= -\frac{(u+1)^2}{(u-1)^2(u+1)^2}-\frac{(u-1)^2}{(u-1)^2(u+1)^2}$$
$$= -\frac{u^2+ 2u+ 1}{(u^2-1)^2}-\frac{u^2-2u+1}{(u^2-1)^2}$$
$$= -\frac{2u^2+ 2}{(u^2-1)^2}= -\frac{2(u^2+1)}{(u^2-1)^2}$$}

 

[tacman]

To simplify this math problem, we can first combine the fractions on the right side of the equation by finding a common denominator. In this case, the common denominator is (u-1)^2(u+1)^2. This means we need to multiply the first fraction by (u+1)^2 and the second fraction by (u-1)^2. This gives us:

1(u+1)^2 - 1(u-1)^2
---------------------
(u-1)^2(u+1)^2

Next, we can expand the parentheses and simplify the numerator by combining like terms:

(u^2+2u+1) - (u^2-2u+1)
-----------------------
(u^2-1)^2

This simplifies to:

(2u+2)
-------
(u^2-1)^2

Finally, we can factor out a 2 from the numerator and factor the denominator as the difference of squares:

2(u+1)
-------
(u^2-1)(u^2-1)

This can be further simplified to:

2(u+1)
-------
(u^2-1)^2

Which is the same result as the book's simplified answer. So, the process for simplifying this math problem involves finding a common denominator, expanding and simplifying the numerator, and then factoring the resulting expression.