\(ĐKXĐ:\left\{{}\begin{matrix}x\ge-1\\y\ge-2\\z\ge-3\end{matrix}\right.\)
\(PT\Leftrightarrow\left(\sqrt{x+1}-2\right)^2+\left(\sqrt{y+2}-3\right)^2+\left(\sqrt{z+3}-4\right)^2=0\)
\(\Leftrightarrow\left\{{}\begin{matrix}\sqrt{x+1}=2\\\sqrt{y+2}=3\\\sqrt{z+3}=4\end{matrix}\right.\)
\(\Leftrightarrow\left\{{}\begin{matrix}x=3\\y=7\\z=13\end{matrix}\right.\)
ĐKXĐ : \(\left\{{}\begin{matrix}x+1\ge0\\y+2\ge0\\z+3\ge0\end{matrix}\right.\) => \(\left\{{}\begin{matrix}x\ge-1\\y\ge-2\\z\ge-3\end{matrix}\right.\)
Ta có : \(x+y+z+35=2\left(2\sqrt{x+1}+3\sqrt{y+2}+4\sqrt{z+3}\right)\)
=> \(x+y+z+35=4\sqrt{x+1}+6\sqrt{y+2}+8\sqrt{z+3}\)
=> \(x-4\sqrt{x+1}+y-6\sqrt{y+2}+z-8\sqrt{z+3}+35=0\)
=> \(x+1-2.2\sqrt{x+1}+4+y+2-2.3\sqrt{y+2}+9+z+3-4.2\sqrt{z+3}+16=0\)
=> \(\left(\sqrt{x+1}-2\right)^2+\left(\sqrt{y+2}-3\right)^2+\left(\sqrt{z+3}-4\right)^2=0\)
Ta thấy : \(\left\{{}\begin{matrix}\left(\sqrt{x+1}-2\right)^2\ge0\\\left(\sqrt{y+2}-3\right)^2\ge0\\\left(\sqrt{z+3}-4\right)^2\ge0\end{matrix}\right.\)
=> \(\left(\sqrt{x+1}-2\right)^2+\left(\sqrt{y+2}-3\right)^2+\left(\sqrt{z+3}-4\right)^2\ge0\)
- Dấu "=" xảy ra
<=> \(\left\{{}\begin{matrix}\sqrt{x+1}-2=0\\\sqrt{y+2}-3=0\\\sqrt{z+3}-4=0\end{matrix}\right.\)
<=> \(\left\{{}\begin{matrix}\sqrt{x+1}=2\\\sqrt{y+2}=3\\\sqrt{z+3}=4\end{matrix}\right.\)
<=> \(\left\{{}\begin{matrix}x+1=4\\y+2=9\\z+3=16\end{matrix}\right.\)
<=> \(\left\{{}\begin{matrix}x=3\\y=7\\z=13\end{matrix}\right.\) ( TM )
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