What should be added to $x^{2} + x y + y^{2}$ to obtain $2 x^{2} + 3 x y$ ?

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Solution

Finding the term to be added to obtain the equation
Let $S$ be the term which when added to $x^{2} + x y + y^{2}$ gives $2 x^{2} + 3 x y$
$\Rightarrow$ $S + x^{2} + x y + y^{2} = 2 x^{2} + 3 x y$
$\Rightarrow$ $S = (2 x^{2} + 3 x y) - (x^{2} + x y + y^{2})$
$\Rightarrow$ $S = 2 x^{2} + 3 x y - x^{2} - x y - y^{2}$
$\Rightarrow$ $S = 2 x^{2} - x^{2} + 3 x y - x y - y^{2}$
$\Rightarrow$ $S = x^{2} + 2 x y - y^{2}$
Hence, $x^{2} + 2 x y - y^{2}$ is the term that needs to be added to $x^{2} + x y + y^{2}$ to obtain $2 x^{2} + 3 x y$ .

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