Question

rút gọn bt:\(\sqrt{x+2\sqrt{x-1}}+\sqrt{x-2\sqrt{x-1}}\)

Answer 1

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

\(A^2=x+2\sqrt{x+1}+x-2\sqrt{x+1}+2\sqrt{\left(x+2\sqrt{x+1}\right)\left(x-2\sqrt{x+1}\right)}\\ =2x+2\sqrt{\left(x-2\right)^2}\\ =2x+2\left(x-2\right)\\ =4x-2\\ \Rightarrow A=\sqrt{4x-2}=2\sqrt{x-1}\)

Answer 2

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

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

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

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