College Algebra, Chapter 1, 1.4, Section 1.4, Problem 38

Evaluate the expression $\displaystyle \frac{5-i}{3+4i}$ in the form of $a + bi$.


$
\begin{equation}
\begin{aligned}
&= \frac{5-i}{3+4i}\\
\\
&= \left( \frac{5-i}{3+4i} \right) \left( \frac{3-4i}{3-4i} \right) && \text{Multiply by the complex conjugate of the denominator}
\\
&= \frac{(5-i)(3-4i)}{3^2 - (4i)^2} && \text{Use FOIL method in the denominator}\\
\\
&= \frac{15-20i-3i+4i^2}{9-(16 i^2)} && \text{Recall that } i^2 = -1\\
\\
&= \frac{15-20i-3i+4(-1)}{9-(16(-1))} && \text{Simplify}\\
\\
&= \frac{11-23i}{25} && \text{Group terms}\\
\\
&= \frac{11}{25} - \frac{23}{25}i
\end{aligned}
\end{equation}
$