Question

Rút gọn biểu thức:

\(\sqrt{x+2\sqrt{x-1}}\) + \(\sqrt{x-2\sqrt{x-1}}\) với x ≥ 2

Answer 1

\(\sqrt{x+2\sqrt{x-1}}+\sqrt{x-2\sqrt{x-1}}\)

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

\(=\sqrt{\left(\sqrt{x-1}\right)^2+2\cdot\sqrt{x-1}\cdot1+1^2}+\sqrt{\left(\sqrt{x-1}\right)^2-2\sqrt{x-1}\cdot1+1^2}\)

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

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

\(=\sqrt{x-1}+1+\sqrt{x-1}-1\)

\(=2\sqrt{x-1}\)

Answer 2

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

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

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