Simplifying a rational expression

[J.Ole]

The instructions just say to simplify. The problem is:

4 / 2a-2 + 3a / a-a^2

Factoring the denominators, I get:

2(a-1) and a(1-a) so the problem becomes:

4a-4a^2 + 6a^2 -6a
_________ + __________

(a-1)(1-a)(2+a)

Adding the numerators, I get 2a^2 - 2a, which I factor again and get 2a(a-1) over (a-1)(1-a)(2+a). And the (a-1) cancels out, so I get:

2a
____
(1-a)(2+a)

But I've gone off the rails somewhere, because the answer given is: - 1/a-1

If I leave the denominator 2(a-1) a(a-1), I can see how the 2a cancels, but I don't then understand why it doesn't turn into (2+a), or how it becomes a negative/opposite.

Help?

 

[tkhunny]

4 / 2a-2 + 3a / a-a^2

\(\displaystyle \frac{4}{2a-2} + \frac{3a}{a-a^{2}}\)

You got this

\(\displaystyle \frac{4}{2(a-1)} + \frac{3a}{a(1-a)}\)

I didn't much understand what you were doing, after that. Ponder this:

\(\displaystyle \frac{2}{2}\frac{2}{(a-1)} + \frac{a}{a}\frac{3}{(1-a)}\)

Then:

\(\displaystyle \frac{2}{(a-1)} + \frac{3}{(1-a)}\)

One last thing, then you take it...

\(\displaystyle \frac{2}{(a-1)} - \frac{3}{(a-1)}\)

Ah, life is so much simpler when the denominators are the same.

I skipped a few steps. Please fill in the details.

 

[HallsofIvy]

The instructions just say to simplify. The problem is:

4 / 2a-2 + 3a / a-a^2

Please, please please use parentheses. I am sure you mean 4/(2a-2) and 3a/(a- a^2) which is not at all what you wrote!

Factoring the denominators, I get:

2(a-1) and a(1-a) so the problem becomes:

But a-1= -(1- a) so you don't need both, just one or the other. Further, 4/(2a- 2)= 4/(2(a- 1))= 2/(a- 1) so you don't need the "2" in the denominator and 3a/(a- a^2)= 3a/(a(1- a))= 3/(1- a) (as long as a is not 0). What you really have is 2/(a-1)- 3/(a- 1)= (2- 3)/(a- 1)= -1/(a- 1).

4a-4a^2 + 6a^2 -6a
_________ + __________

(a-1)(1-a)(2+a)


Adding the numerators, I get 2a^2 - 2a, which I factor again and get 2a(a-1) over (a-1)(1-a)(2+a). And the (a-1) cancels out, so I get:

2a
____
(1-a)(2+a)

But I've gone off the rails somewhere, because the answer given is: - 1/a-1

If I leave the denominator 2(a-1) a(a-1), I can see how the 2a cancels, but I don't then understand why it doesn't turn into (2+a), or how it becomes a negative/opposite.

Help?