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Question

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


Solution

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

Mathematics

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