What should be subtracted from $x^{2} - x y + y^{2} - x + y + 3$ to obtain $- x^{2} + 3 y^{2} - 4 x y + 1$ ?

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Solution

Let $^{'} A^{'}$ be the required expression.
Then, we have
$(x^{2} - x y + y^{2} - x + y + 3) - A = - x^{2} + 3 y^{2} - 4 x y + 1$
$\Rightarrow A = (x^{2} - x y + y^{2} - x + y + 3) - (- x^{2} + 3 y^{2} - 4 x y + 1)$
$\Rightarrow A = x^{2} - x y + y^{2} - x + y + 3 + x^{2} - 3 y^{2} + 4 x y - 1$
$\Rightarrow A = (x^{2} + x^{2}) + (- x y + 4 x y) + (y^{2} - 3 y^{2}) - x + y + 2$
$\Rightarrow A = 2 x^{2} + 3 x y - 2 y^{2} - x + y + 2$

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x^{2} - x y + y^{2} - x + y + 3

to obtain

- x^{2} + 3 y^{2} - 4 x y + 1 ?

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3 x^{2} - 4 y^{2} + 5 x y + 20

to obtain

- x^{2} - y^{2} + 6 x y + 20

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What should be subtracted from x2−xy+y2−x+y+3 to obtain −x2+3y2−4xy+1?

Let ′A′ be the required expression.Then, we have(x2−xy+y2−x+y+3)−A=−x2+3y2−4xy+1⇒A=(x2−xy+y2−x+y+3)−(−x2+3y2−4xy+1)⇒A=x2−xy+y2−x+y+3+x2−3y2+4xy−1⇒A=(x2+x2)+(−xy+4xy)+(y2−3y2)−x+y+2⇒A=2x2+3xy−2y2−x+y+2

What should be subtracted from $x^{2} - x y + y^{2} - x + y + 3$ to obtain $- x^{2} + 3 y^{2} - 4 x y + 1$ ?