{%hackmd @cimeesia/SJv5Zgj1yl %}

# chefo's HW 2

## Photo


## Q15

\begin{align} \\
f(x) = & \frac{2}{4p-20} + \frac{p-6}{p^2-8p+15} \\
f(x) = & \frac{2}{4(p-5)} + \frac{p-6}{(p-5)(p-3)}
\end{align}

Multiply both the numerator and denominator of ==the first fraction== by ==$(p - 3)$==, and multiply both the numerator and denominator of ==the second fraction== by ==$4$==.

\begin{gather}
\\
f(x) = & \frac{2(p-3)}{4(p-5)(p-3)} + \frac{4(p-6)}{4(p-5)(p-3)}
\end{gather}

Now the denominator is the same, so we can combine the numerator now.

\begin{gather}
f(x) = & \frac{2(p-3) + 4(p-6)}{4(p-5)(p-3)}
\end{gather}

~~I don't think I need to continue.~~ OK, I was wrong.

\begin{gather}
f(x) = & \frac{2p-6+4p-24}{4(p-5)(p-3)} \\
f(x) = & \frac{6p-30}{4(p-5)(p-3)} \\
f(x) = & \frac{6(p-5)}{4(p-5)(p-3)} \end{gather}

At here, we can cancel that ==$(p-5)$==.

\begin{gather}
f(x) = & \frac{6}{4(p-3)} \\
f(x) = & \frac{3}{2(p-3)} \\
f(x) = & \frac{3}{2p-6} \\
\end{gather}