Question

Cm 

\(\sqrt{a+4\sqrt{a-2}+2}+\sqrt{a-4\sqrt{a-2}+2}=4\) ( vs 2 ≤ a ≤ 6)

Answer 1

\(\sqrt{a+4\sqrt{a-2}+2}+\sqrt{a-4\sqrt{a-2}+2}\)

\(=\sqrt{a-2+4\sqrt{a-2}+4}+\sqrt{a-2-4\sqrt{a-2}+4}\)

\(=\sqrt{\left(\sqrt{a-2}+2\right)^2}+\sqrt{\left(\sqrt{a-2}-2\right)^2}\)

\(=\left|\sqrt{a-2}+2\right|+\left|\sqrt{a-2}-2\right|\)

Vì \(a\le6\Rightarrow\sqrt{a-2}\le2\Rightarrow\sqrt{a-2}-2\le0\Rightarrow\left|\sqrt{a-2}-2\right|=2-\sqrt{a-2}\)

Vì \(a\ge2\Rightarrow\sqrt{a-2}+2\ge2>0\)

\(\Rightarrow\text{ }\left|\sqrt{a-2}+2\right|+\left|\sqrt{a-2}-2\right|=\sqrt{a-2}+2+2-\sqrt{a-2}=4\)

 

Answer 2

Ta có: \(\sqrt{a+4\sqrt{a-2}+2}+\sqrt{a-4\sqrt{a-2}+2}\)

\(=\sqrt{a-2}+2-\sqrt{a-2}+2\)

=4