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Subtraction of Algebraic Expressions

What should b...

Question

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

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Solution

Let $p$ should be added.
Then according to question,
$x^{2} + x y + y^{2} + p = 2 x^{2} + 3 x y \Rightarrow p = 2 x^{2} + 3 x y - (x^{2} + x y + y^{2})$
$\Rightarrow p = 2 x^{2} + 3 x y - x^{2} - x y - y^{2}$
$\Rightarrow p = 2 x^{2} - x^{2} - y^{2} + 3 x y - x y$
$\Rightarrow p = x^{2} - y^{2} + 2 x y$
Hence, $x^{2} - y^{2} + 2 x y$ should be added.

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