Question

\(\sqrt{x-1+2\sqrt{2x-6}}\) + \(\sqrt{x-1-2\sqrt{2x-6}}\) với \(3\le x\le5\)

Answer 1

\(=\sqrt{x-1+2\sqrt{2\left(x-3\right)}}+\sqrt{x-1-2\sqrt{2\left(x-3\right)}}\)

\(=\sqrt{x-1+2\sqrt{2}.\sqrt{\left(x-3\right)}-2+2}+\sqrt{x-1-2\sqrt{2}.\sqrt{\left(x-3\right)}-2+2}\)

\(=\sqrt{x-3+2\sqrt{2}.\sqrt{\left(x-3\right)}+2}+\sqrt{x-3-2\sqrt{2}.\sqrt{\left(x-3\right)}+2}\)

\(=\sqrt{\left(\sqrt{x-3}+\sqrt{2}\right)^2}+\sqrt{\left(\sqrt{x-3}-\sqrt{2}\right)^2}\)

\(=\left|\sqrt{x-3}+\sqrt{2}\right|+\left|\sqrt{x-3}-\sqrt{2}\right|\)

\(=\sqrt{x-3}+\sqrt{2}+\sqrt{2}-\sqrt{x-3}\left(3\le x\le5\right)\)

\(=2\sqrt{2}\)