ta có : \(\sqrt{x^2-2x+1}+\sqrt{x^2-6x+9}=2\)
\(\Leftrightarrow\sqrt{\left(x-1\right)^2}+\sqrt{\left(x-3\right)^2}=2\Leftrightarrow\left|x-1\right|+\left|x-3\right|=2\) (1)
th1: \(x< 1\)
\(\left(1\right)\Leftrightarrow1-x+3-x=2\Leftrightarrow x=1\left(loại\right)\)
th2: \(1\le x< 3\)
\(\left(1\right)\Leftrightarrow x-1+3-x\Leftrightarrow2=2\Rightarrow1\le x< 3\)
th3: \(x\ge3\)
\(\left(1\right)\Leftrightarrow x-1+x-3=2\Leftrightarrow x=3\left(tmđk\right)\)
lậy hợp các nghiệm \(\Rightarrow1\le x\le3\)
vậy \(1\le x\le3\)
\(\sqrt{x^2-2x+1}+\sqrt{x^2-6x+9}=2\)
\(\Leftrightarrow\sqrt{\left(x-2\right)^2}+\sqrt{\left(x-3\right)^2}=2\)
\(\Leftrightarrow\left|x-2\right|+\left|x-3\right|=2\)
TH1: \(x\ge0\)
Ta có: \(\left|x-1\right|+\left|x-3\right|=2\)
\(\Leftrightarrow x-1+x-3=2\)
\(\Leftrightarrow2x-4=2\)
\(\Leftrightarrow2x=6\)
\(\Leftrightarrow x=3\left(tm\right)\)
TH2: \(x< 0\)
Ta có: \(\left|x-1\right|+\left|x-3\right|=2\)
\(\Leftrightarrow1-x+3-x=2\)
\(\Leftrightarrow4-2x=2\)
\(\Leftrightarrow2x=2\)
\(\Leftrightarrow x=1\left(ktm\right)\)
Vậy \(x=3\)