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

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


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