Fractions Add/Subtract Factoring

[Samara]

I have the correct answer to this question, however, I am unsure of the process of this type of problem...If someone could please assist me, I would really appreciate the assistance!

 

[soroban]

Hello, Samara!

Perform the indicated operations: .\(\displaystyle \dfrac{2}{4r^2 - 7rs - 2s^2} - \dfrac{6}{6r^2 - 13rs + 2rs} + \dfrac{9}{24r^2 + 2rs - s^2} \)

Factor the denominators:

. . \(\displaystyle \dfrac{2}{(4r+s)(r-2s)} - \dfrac{6}{(6r-s)(r-2s)} + \dfrac{9}{(6r-s)(4r+s)}\)


The LCD is: \(\displaystyle (4r+s)(r-2s)(6r-s)\)


Convert the fractions to the LCD:

. . \(\displaystyle \dfrac{2}{(4r+s)(r-2s)}\cdot\dfrac{6r-s}{6r-s} \;-\; \dfrac{6}{(6r-s)(r-2s)}\cdot\dfrac{4r+s}{4r+s} \;+\; \) .\(\displaystyle \dfrac{9}{(6r-s)(4r+s)}\cdot\dfrac{r-2s}{r-2s} \)


. . \(\displaystyle =\;\dfrac{2(6r-s) - 6(4r+s) + 9(r-2s)}{(4r+s)(r-2s)(6r-s)}\)


. . \(\displaystyle =\;\dfrac{12r - 2s - 24r - 6s + 9r - 18s}{(4r+s)(r-2s)(6r-s)}\)


. . \(\displaystyle =\;\dfrac{-3r - 26s}{(4r+s)(r-2s)(6r-s)}\)

 

[Samara]

Thank you kindly!