What must be added to $2 a^{2} - 7 a b + 5 b^{2}$ to get $a^{2} + 4 a b - 3 b^{2}$ ?

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Solution

Finding the required expression
Let, $A$ must be added to $2 a^{2} - 7 a b + 5 b^{2}$ to get $a^{2} + 4 a b - 3 b^{2}$
According to the question,
$A + 2 a^{2} - 7 a b + 5 b^{2} = a^{2} + 4 a b - 3 b^{2} A = a^{2} + 4 a b - 3 b^{2} - (2 a^{2} - 7 a b + 5 b^{2}) = a^{2} + 4 a b - 3 b^{2} - 2 a^{2} + 7 a b - 5 b^{2} = a^{2} - 2 a^{2} - 3 b^{2} - 5 b^{2} + 4 a b + 7 a b = - a^{2} - 8 b^{2} + 11 a b$
Hence, $- a^{2} - 8 b^{2} + 11 a b$ must be added to $2 a^{2} - 7 a b + 5 b^{2}$ to get $a^{2} + 4 a b - 3 b^{2}$ .

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