Question

Giải phương trình

\(\sqrt{x-y+2010}=\sqrt{x}-\sqrt{y}+\sqrt{2010}\)

Answer 1

ĐXKĐ: ...

Bình phương 2 vế:

\(x-y+2010=x+y+2010-2\sqrt{xy}+2\sqrt{2010x}-2\sqrt{2010y}\)

\(\Leftrightarrow y-\sqrt{xy}-\left(\sqrt{2010y}-\sqrt{2010x}\right)=0\)

\(\Leftrightarrow\sqrt{y}\left(\sqrt{y}-\sqrt{x}\right)-\sqrt{2010}\left(\sqrt{y}-\sqrt{x}\right)=0\)

\(\Leftrightarrow\left(\sqrt{y}-\sqrt{x}\right)\left(\sqrt{y}-\sqrt{2010}\right)=0\)

\(\Rightarrow\left[{}\begin{matrix}y=x\\y=2010\end{matrix}\right.\)

Vậy nghiệm của pt là: \(\left[{}\begin{matrix}x=y\ge0\\\left\{{}\begin{matrix}y=2010\\x\ge0\end{matrix}\right.\end{matrix}\right.\)