\(A=\sqrt{x-2+2\sqrt{2\left(x-2\right)}+2}+\sqrt{x-2-2\sqrt{2\left(x-2\right)}+2}\)
\(=\sqrt{\left(\sqrt{x-2}+\sqrt{2}\right)^2}+\sqrt{\left(\sqrt{x-2}-\sqrt{2}\right)^2}\)
\(=\sqrt{x-2}+\sqrt{2}+\left|\sqrt{x-2}-\sqrt{2}\right|\)
- Nếu \(2< x\le4\Rightarrow\sqrt{x-2}\le\sqrt{2}\Rightarrow\sqrt{x-2}-\sqrt{2}\le0\)
\(\Rightarrow A=\sqrt{x-2}+\sqrt{2}+\sqrt{2}-\sqrt{x-2}=2\sqrt{2}\)
- Nếu \(x>4\Rightarrow\sqrt{x-2}-\sqrt{2}>0\)
\(\Rightarrow A=\sqrt{x-2}+\sqrt{2}+\sqrt{x-2}-\sqrt{2}=2\sqrt{x-2}\)
Tóm lại \(\left\{{}\begin{matrix}x>4\Rightarrow A=2\sqrt{x-2}\\2< x\le4\Rightarrow A=2\sqrt{2}\end{matrix}\right.\)
\(\sqrt{x+2\sqrt{2x-4}}+\sqrt{x-2\sqrt{2x-4}}=\sqrt{x+2\sqrt{x-2}.\sqrt{2}}+\sqrt{x-2\sqrt{x-2}.\sqrt{2}}=\sqrt{x-2+2\sqrt{x-2}.\sqrt{2}+2}+\sqrt{x-2-2\sqrt{x-2}.\sqrt{2}+2}=\sqrt{\left(\sqrt{x-2}+\sqrt{2}\right)^2}+\sqrt{\left(\sqrt{x-2}-\sqrt{2}\right)^2}=\left|\sqrt{x-2}+\sqrt{2}\right|+\left|\sqrt{x-2}-\sqrt{2}\right|=\sqrt{x-2}+\sqrt{2}+\left|\sqrt{x-2}-\sqrt{2}\right|\)(*)
Nếu x\(\ge4\) thì (*)=\(\sqrt{x-2}+\sqrt{2}+\sqrt{x-2}-\sqrt{2}=2\sqrt{x-2}\)
Nếu 2<x<4 thì (*)=\(\sqrt{x-2}+\sqrt{2}+\sqrt{2}-\sqrt{x-2}=2\sqrt{2}\)
