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Find the value of $x$ and $y$ for which $(2 + 3 i) x^{2} - (3 - 2 i) y = 2 x - 3 y + 5 i$ where $x, y ϵ R .$

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Solution

$(2 + 3 i) x^{2} - (3 - 2 i) y = 2 x - 3 y + 5 i$
$\Rightarrow$ $2 x^{2} - 3 y = 2 x - 3 y$
$\Rightarrow$ $x^{2} - x = 0$
$\Rightarrow$ $x = 0, 1$ and $3 x^{2} + 2 y = 5$
$\Rightarrow$ if $x = 0, y = \frac{5}{2}$ and if $x = 1, y = 1$
$∴$ $x = 0, y = \frac{5}{2}$ and $x = 1, y = 1$
are two solutions of the given equation which can also be represented as $(0, \frac{5}{2})$ & $(1, 1), (0, \frac{5}{2}) (1, 1)$

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