\(a^2+b^2+c^2+6=2\left(a+2b+c\right)\)
\(\Rightarrow a^2+b^2+c^2+6-2a-4b-2c=0\)
\(\Rightarrow\left(a^2-2a+1\right)+\left(b^2-4b+4\right)+\left(c^2-2c+1\right)=0\)
\(\Rightarrow\left(a-1\right)^2+\left(b-2\right)^2+\left(c-1\right)^2=0\) (1)
Mà \(\begin{cases}\left(a-1\right)^2\ge0\\\left(b-2\right)^2\ge0\\\left(c-1\right)^2\ge0\end{cases}\)\(\Rightarrow\left(a-1\right)^2+\left(b-2\right)^2+\left(c-1\right)^2\ge0\)(2)
Từ (1) và (2) suy ra \(\begin{cases}\left(a-1\right)^2=0\\\left(b-2\right)^2=0\\\left(c-1\right)^2=0\end{cases}\)\(\Rightarrow\begin{cases}a-1=0\\b-2=0\\c-1=0\end{cases}\)\(\Rightarrow\begin{cases}a=1\\b=2\\c=1\end{cases}\)
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