Question

[Nguyen Anh]

Tìm tất cả các giá trị x , y, z thỏa mãn đẳng thức :

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

Answer 1

[Son Goku]

Bình phương 2 vế, ta đc:\(x-y+z=x+y+x-2\sqrt{xy}-2\sqrt{yz}+2\sqrt{zx}\Rightarrow y-\sqrt{xy}-\sqrt{yz}+\sqrt{zx}=0\Rightarrow\left(\sqrt{y}-\sqrt{x}\right)\left(\sqrt{y}-\sqrt{z}\right)=0\)Tự lm nốt nha.

Answer 2

[Nguyễn Tấn An]

\(\sqrt{x-y+z}=\sqrt{x}-\sqrt{y}+\sqrt{z}\Leftrightarrow\left(\sqrt{x-y+z}\right)^2=\left(\sqrt{x}-\sqrt{y}+\sqrt{z}\right)^2\Leftrightarrow x-y+z=x+y+z-2\sqrt{xy}-2\sqrt{yz}+2\sqrt{xz}\Leftrightarrow2y-2\sqrt{xy}-2\sqrt{yz}+2\sqrt{xz}=0\Leftrightarrow y-\sqrt{xy}-\sqrt{yz}+\sqrt{xz}=0\Leftrightarrow\sqrt{y}\left(\sqrt{y}-\sqrt{x}\right)-\sqrt{z}\left(\sqrt{y}-\sqrt{x}\right)=0\Leftrightarrow\left(\sqrt{y}-\sqrt{x}\right)\left(\sqrt{y}-\sqrt{z}\right)=0\Leftrightarrow\left[{}\begin{matrix}\sqrt{y}-\sqrt{x}=0\\\sqrt{y}-\sqrt{x}=0\end{matrix}\right.\Leftrightarrow}\left[{}\begin{matrix}\sqrt{y}=\sqrt{x}\\\sqrt{y}=\sqrt{z}\end{matrix}\right.\Leftrightarrow}\left[{}\begin{matrix}y=x\\y=z\end{matrix}\right.\)