\(A=\sqrt{x+4\sqrt{x-2}+2}+\sqrt{x-4\sqrt{x-2}+2}\)
\(=\sqrt{x-2+4\sqrt{x-2}+4}+\sqrt{x-2-4\sqrt{x-2}+4}\)
\(=\sqrt{\left(\sqrt{x-2}+2\right)^2}+\sqrt{\left(\sqrt{x-2}-2\right)^2}\)
\(=\sqrt{x+2}+2+\left|\sqrt{x-2}-2\right|\)
Với \(x\ge6\) \(\Rightarrow\) \(\sqrt{x-2}-2\ge0\)\(\Rightarrow A=2\sqrt{x-2}\)
Với \(2\le x< 6\) \(\Rightarrow\sqrt{x-2}-2< 0\) \(\Rightarrow A=\sqrt{x-2}+2+2-\sqrt{x+2}\)
\(\Rightarrow A=4\)
Đặt \(u=\sqrt{x-2},u\ge0\Rightarrow u^2=x-2\Rightarrow x=u^2+2\)
Ta có:
\(A=\sqrt{u^2+2+4u+2}+\sqrt{u^2+2-4u+2}\)
\(A=\sqrt{\left(u+2\right)^2}+\sqrt{\left(u-2\right)^2}\\ A=\left|u+2\right|+\left|u-2\right|\)
Thay vào lại:
\(A=\left|\sqrt{x-2}+2\right|+\left|\sqrt{x-2}-2\right|\)
\(TH_1:\sqrt{x-2}+2+\sqrt{x-2}-2=2\sqrt{x-2}\\ TH_2:\sqrt{x-2}+2-\left(\sqrt{x-2}-2\right)=\sqrt{x-2}+2-\sqrt{x-2}+2=4\)
em cảm mơn nhìu nhìu 