Question

If $\frac{{\left(1+i\right)}^2}{2-i}=x-iy,$ then find the value of $x+y$.

Answer 1

$\frac{(1+i{)}^2}{2-i}=x-iy$

$=\frac{1+i^2+2i}{2-i}$

$=\left(\frac{2i(2+i)}{2-i(2+i)}\right)$

$=\frac{4i-2}{4+1}$

$=\frac{4i}{5}-\frac{2}{5}$

$x=\frac{-2}{5},y=\frac{-4}{5}$

$x+y=\frac{-6}{5}$