Question

Rút gọn biểu thức P=\(\sqrt{x+2\sqrt{x-1}}+\sqrt{x-2\sqrt{x-1}}\) với \(x\ge1\)

Answer 1

\(P=\sqrt{\left(x-1\right)+2\sqrt{x-1}+1}+\sqrt{\left(x-1\right)-2\sqrt{x-1}+1}\)
\(P=\sqrt{\left(\sqrt{x-1}+1\right)^2}+\sqrt{\left(\sqrt{x-1}-1\right)^2}\)
\(P=\left|\sqrt{x-1}+1\right|+\left|\sqrt{x-1}-1\right|\)
\(\Rightarrow P=\sqrt{x-1}+1+\sqrt{x-1}-1 \left(x\ge2\right)\) hoặc \(P=\sqrt{x-1}+1-\sqrt{x-1}+1\left(1\le x\le2\right)\)
\(\Rightarrow P=2\sqrt{x-1} \left(x\ge2\right)\) hoặc \(P=2 \left(1\le x\le2\right)\)