Help with complex fractions

[LostinUtopia]

I have a test coming up, the only thing I am really messing up are complex fractions. I need someone to describe in detail the steps. I will give the example that was on my last quiz which I got wrong.

3x^2 - 2xy -y^2 / 3x^2 + 4xy + y^2
this is all over the fraction:
2x^2 + xy - y^2 / 2x^2 - 5xy + 2y^2

I am really not sure where I messed up the answer..whether it was with the LCDs or just combining/factoring.

 

[soroban]

Hello, LostinUtopia!

This is indeed a complex fraction but I wouldn't worry about LCD's, etc.

\(\displaystyle \L\:\frac{\frac{3x^2\,-\,2xy\,-\,y^2}{3x^2\,+\,4xy\,+\,y^2}}{\frac{2x^2\,+\,xy\,-\,y^2}{2x^2\,-\,5xy\,+\,2y^2}}\)

I would use "invert and multiply" and rewrite the problem:

\(\displaystyle \L\;\;\frac{3x^2\,-\,2xy\,-\,y^2}{3x^2\,+\,4xy\,+\,y^2}\,\cdot\,\frac{2x^2\,-\,5xy\,+\,2y^2}{2x^2\,+\,xy\,-\,y^2}\)

Factor: \(\displaystyle \L\:\frac{(\sout{3x\,+\,y})(x\,-\,y)}{(\sout{3x\,+\,y})(x\,+\,y)}\,\cdot\,\frac{(\sout{2x\,-\,y})(x\,-\,2y)}{(\sout{2x\,-\,y})(x\,+\,y)} \;= \;\frac{(x\,-\,y)(x\,-\,2y)}{(x\,+\,y)^2}\)

 

[LostinUtopia]

Hah, now that I'm looking at it again, I think I just made the problem a lot more difficult than I should have. Just out of curiosity though, when do you need to use LCD's to do complex fractions? I'm sure a least one will be on the test.
Thanks.

 

[stapel]

...when do you need to use LCD's to do complex fractions?

If you have a complex fraction that isn't just "(fraction)/(fraction)", but is something more complex, like "(fraction + 4)/(3 - fraction)", then you can multiply the top and bottom by the LCD of all the fractions to get a simpler expression.

Eliz.