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Simplify the expression $\frac{4-7i}{5+i}\text{.}$

Solution

To divide, we rationalize the denominator using the conjugate $5-i\text{:}$

$\begin{aligned} \frac{4-7i}{5+i}\amp=\frac{(4-7i)}{(5+i)}\multiplyright{\frac{(5-i)}{(5-i)}}\\ \amp=\frac{20-4i-35i+7i^2}{25-5i+5i-i^2}\\ \amp=\frac{20-39i+7(-1)}{25-1(-1)}\\ \amp=\frac{20-39i-7}{25+1}\\ \amp=\frac{13-39i}{26}\\ \amp=\frac{13}{26}-\frac{39i}{26}\\ \amp=\frac{1}{2}-\frac{3}{2}i \end{aligned}$