Question

If $\frac{(1+i{)}^2}{2-i}=x+iy,$
find $x+y.$

Answer 1

$\because \frac{(1+i{)}^2}{2-i}=x+iy$
$\Rightarrow \frac{(1+i{)}^2+2i}{2-i}=x+iy$
$\Rightarrow \frac{2i}{2-i}=x+iy$
$\Rightarrow \frac{2i}{2-i}\times \frac{2+i}{2+i}=x+iy$
$\Rightarrow \frac{4i+2i^2}{2^2-i^2}=x+iy$
$\Rightarrow \frac{4i-2}{5}=x+iy$
$\Rightarrow -\frac{2}{5}+\frac{4}{5}i=x+iy$
$\Rightarrow x=-\frac{2}{5},y=\frac{4}{5}$
{equating the real and imginary parts}
$\therefore x+y=-\frac{2}{5}+\frac{4}{5}=\frac{2}{5}$