Question

gpt:

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

Answer 1

ĐKXĐ: \(x\ge1\)

\(\Leftrightarrow\sqrt{x-1+2\sqrt{x-1}+1}-2\left(x-1\right)=0\)

\(\Leftrightarrow\sqrt{\left(\sqrt{x-1}-1\right)^2}-2\left(x-1\right)=0\)

\(\Leftrightarrow\left|\sqrt{x-1}-1\right|-2\left(x-1\right)=0\)

TH1: \(x\ge2\)

\(\Leftrightarrow\sqrt{x-1}-1-2\left(x-1\right)=0\)

Phương trình vô nghiệm (đặt \(\sqrt{x-1}=a\))

TH2: \(1\le x< 2\)

\(\Leftrightarrow-2\left(x-1\right)-\sqrt{x-1}+1=0\)

\(\Rightarrow\left[{}\begin{matrix}\sqrt{x-1}=\frac{1}{2}\\\sqrt{x-1}=-1\left(l\right)\end{matrix}\right.\) \(\Rightarrow x-1=\frac{1}{4}\Rightarrow x=\frac{5}{4}\)