Đặt \(N=\sqrt{x-1+2\sqrt{x-2}}+\sqrt{x-1-2\sqrt{x-2}}\)
\(\Rightarrow N^2=x-1+2\sqrt{x-2}+x-1-2\sqrt{x-2}+2\sqrt{\left(x-1+2\sqrt{x-2}\right)\left(x-1-2\sqrt{x-2}\right)}\)
\(\Leftrightarrow N^2=2x-2+2\sqrt{\left(x-1\right)^2-\left(2\sqrt{x-2}\right)^2}\)
\(\Leftrightarrow N^2=2x-2+2\sqrt{x^2-2x+1-4\left(x-2\right)}\)
\(\Leftrightarrow N^2=2x-2+2\sqrt{x^2-2x+1-4x+8}\)
\(\Leftrightarrow N^2=2x-2+2\sqrt{x^2-6x+9}\)
\(\Leftrightarrow N^2=2x-2+2\sqrt{\left(x-3\right)^2}\)
\(\Leftrightarrow N^2=2x-2+2\left|x-3\right|\)
* Với \(x\ge3\)thì \(N^2=2x-2+2\left(x-3\right)=4x-8=4\left(x-2\right)\Rightarrow N=2\sqrt{x-2}\)
* Với \(2\le x\le4\)thì \(N^2=2x-2-2\left(x-3\right)=4\Rightarrow N=\sqrt{4}=2\)
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