What should be added to $6 x^{2} - 3 x y + 4 y^{2}$ to get $2 y^{2} + x y - 4 x^{2}$ ?

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Solution

$Let p (x) be added to 6 x^{2} - 3 x y + 4 y^{2} to get 2 y^{2} + x y - 4 x^{2} . i . e ., 6 x^{2} - 3 x y + 4 y^{2} + p (x) = 2 y^{2} + x y - 4 x^{2} \Rightarrow p (x) = (2 y^{2} + x y - 4 x^{2}) - (6 x^{2} - 3 x y + 4 y^{2}) \Rightarrow p (x) = 2 y^{2} + x y - 4 x^{2} - 6 x^{2} + 3 x y - 4 y^{2} \Rightarrow p (x) = 2 y^{2} - 4 y^{2} + x y + 3 x y - 4 x^{2} - 6 x^{2} \Rightarrow p (x) = - 2 y^{2} + 4 x y - 10 x^{2} \Rightarrow p (x) = - 10 x^{2} - 2 y^{2} + 4 x y ∴ - 10 x^{2} - 2 y^{2} + 4 x y can be added to 6 x^{2} - 3 x y + 4 y^{2} to get 2 y^{2} + x y - 4 x^{2} .$

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