If (1+i)^2 2-i =x+iy, find x, y. - Vidyadip

Question

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

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Answer

$\frac{(1+\text{i})^2}{2-\text{i}}=\text{x}+\text{iy}$
$\Rightarrow\frac{(1+2\text{i}-1)}{2-\text{i}}=\text{x}+\text{iy}$
$\Rightarrow\frac{2\text{i}}{2-\text{i}}=\text{x}+\text{iy}$
$\Rightarrow\frac{2\text{i}(2+\text{i})}{(2-\text{i})(2+\text{i})}=\text{x}+\text{iy}$ [Rationalizing the denominator]
$\Rightarrow\frac{2(2\text{i}-1)}{4+1}=\text{x}+\text{iy}$
$\Rightarrow\frac{4\text{i}-2}{5}=\text{x}+\text{iy}$
$\Rightarrow-\frac{2}{5}+\text{i}\frac{4}{5}=\text{x}+\text{iy}$
Comparing the real and imaginary parts, we get
$\text{x}=-\frac{2}{5},\text{y}=\frac{4}{5}$
$\text{x}+\text{y}=\frac{2}{5}$

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