Question

Chứng minh đẳng thức:

\(\sqrt{a+4\sqrt{a-2}+2}+\sqrt{a-4\sqrt{a-2}+2}=4\) ( với \(2\le a\le6\) ).

Answer 1

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

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

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

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

Nếu \(a=6\) thì \(VT=\sqrt{6-2}+2+\sqrt{6-2}-2=4\)

Nếu \(2\le a< 6\) thì \(VT=\sqrt{a-2}+2+2-\sqrt{a-2}=4\)