# Thread: Simplifying Complex Rational Function

1. ## Simplifying Complex Rational Function

Hey guys, I just signed up for forum looking forward to contributing.

I am stuck on this problem, I have attached a picture in this posting.

I have tried solving it several times, and I cannot seem to get it correct.

Can someone guide me through ? Thanks!

2. ## Re: Simplifying Complex Rational Function

Originally Posted by ruusky
Hey guys, I just signed up for forum looking forward to contributing.

I am stuck on this problem, I have attached a picture in this posting.

I have tried solving it several times, and I cannot seem to get it correct.

Can someone guide me through ? Thanks!
\displaystyle \begin{align*} \frac{10}{2x - 8} + \frac{14}{x^2 - 4x} &= \frac{10}{2(x - 4)} + \frac{14}{x(x - 4)} \\ &= \frac{5}{x - 4} + \frac{14}{x(x - 4)} \\ &= \frac{5x}{x(x - 4)} + \frac{14}{x(x - 4)} \\ &= \frac{5x + 14}{x(x - 4)} \end{align*}

\displaystyle \begin{align*} \frac{1}{x^2 + x - 20} - \frac{2}{x^2 - 16} &= \frac{1}{(x - 4)(x + 5)} - \frac{2}{(x - 4)(x + 4)} \\ &= \frac{x + 4}{(x - 4)(x + 4)(x + 5)} - \frac{2(x + 5)}{(x - 4)(x + 4)(x + 5)} \\ &= \frac{x + 4 - 2(x + 5)}{(x - 4)(x + 4)(x + 5)} \\ &= \frac{x + 4 - 2x - 10}{(x - 4)(x + 4)(x + 5)} \\ &= \frac{-x - 6}{(x - 4)(x + 4)(x + 5)} \end{align*}

Therefore

\displaystyle \begin{align*} \frac{\frac{10}{2x - 8} + \frac{14}{x^2 - 4x}}{\frac{1}{x^2 + x - 20} - \frac{2}{x^2- 16}} &= \frac{\frac{5x+14}{x(x-4)}}{\frac{-x-6}{(x-4)(x+4)(x+5)}} \\ &= \frac{(5x+14)(x - 4)(x + 4)(x + 5)}{x(x - 4)(-x-6)} \\ &= -\frac{(5x+14)(x + 4)(x + 5)}{x(x + 6)} \end{align*}

3. ## Re: Simplifying Complex Rational Function

Thank you so much, I got it now