Câu hỏi của Thiên Chỉ Hạc - Toán lớp 9 | Học trực tuyến
\(x+y+z=\sqrt{xy}+\sqrt{yz}+\sqrt{xz}\)
\(\Rightarrow2x+2y+2z=2\sqrt{xy}+2\sqrt{yz}+2\sqrt{xz}\)
\(\Rightarrow2x+2y+2z-2\sqrt{xy}+2\sqrt{yz}+2\sqrt{xz}\)= 0
\(\Rightarrow x-2\sqrt{xy}+y+y-2\sqrt{yz}+z\) \(+x-2\sqrt{xz}+z\) \(=0\)
\(\Rightarrow\left(\sqrt{x}-\sqrt{y}\right)^2+\left(\sqrt{y}-\sqrt{z}\right)^2\) \(+\left(\sqrt{x}-\sqrt{z}\right)^2=0\)
\(\Rightarrow\sqrt{x}=\sqrt{y};\sqrt{y}=\sqrt{z};\sqrt{x}=\sqrt{z}\)
\(\Rightarrow\sqrt{x}=\sqrt{y}=\sqrt{z}\)
\(\Rightarrow\)\(x=y=z\) \((x,y,z>0)\)
\(\Rightarrowđpcm\)
