Solution 3.1:2a

A quotient of two complex numbers is calculated by multiplying the top and bottom of the fraction by the complex conjugate of the denominator:

\displaystyle \frac{3-2i}{1+i} = \frac{3-2i}{1+i}\frac{1-i}{1-i}.

Then, the conjugate rule gives that the new denominator is a real number

\displaystyle \begin{align}\frac{3-2i}{1+i}\frac{1-i}{1-i}&=\frac{(3-2i)(1-i)}{(1+i)(1-i)}\\ &=\frac{(3-2i)(1-i)}{1^2-i^2}\\ &=\frac{(3-2i)(1-i)}{1+1}\\ &=\frac{(3-2i)(1-i)}{2}\end{align}

All that remains is to multiply together what is in the numerator:

\displaystyle \begin{align}\frac{(3-2i)(1-i)}{2}&=\frac{3\cdot 1 -3\cdot i - 1\cdot 2i -2i\cdot(-i)}{2}\\ &=\frac{3-3i-2i+2i^2}{2}\\ &=\frac{3-(3+2)i+2(-1)}{2}\\ &=\frac{1-5i}{2}\\ &=\frac{1}{2}-\frac{5}{2}i.\end{align}