Question

tìm các số thực x,y,z thỏa \(\sqrt{x-1}+\sqrt{y-2}+\sqrt{z-3}=\dfrac{1}{2}\left(x+y+z-3\right)\)

Answer 1

DKXD \(x\ge1;y\ge2;z\ge3\)

\(Pt\Leftrightarrow2\sqrt{x-1}+2\sqrt{y-2}+2\sqrt{z-3}=x+y+z-3\)

\(\Leftrightarrow\left(x-1-2\sqrt{x-1}+1\right)+\left(y-2-2\sqrt{y-2}+1\right)+\left(z-3-2\sqrt{z-3}+1\right)=0\)\(\Leftrightarrow\left(\sqrt{x-1}-1\right)^2+\left(\sqrt{y-2}-1\right)^2+\left(\sqrt{z-3}-1\right)^2=0\)

\(\Leftrightarrow\left\{{}\begin{matrix}\sqrt{x-1}=1\\\sqrt{y-2}=1\\\sqrt{z-3}=1\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}x=2\\y=3\\z=4\end{matrix}\right.\)