ĐKXĐ: \(x\ge\frac{2}{3}\)
\(\Leftrightarrow\left(x-1\right)\left(x-2\right)+\left(x-1\right)\sqrt{3x-2}=0\)
\(\Leftrightarrow\left(x-1\right)\left(x-2+\sqrt{3x-2}\right)=0\)
\(\Leftrightarrow\left[{}\begin{matrix}x=1\\x-2+\sqrt{3x-2}=0\left(1\right)\end{matrix}\right.\)
\(\left(1\right)\Leftrightarrow\sqrt{3x-2}=2-x\) (\(x\le2\))
\(\Leftrightarrow3x-2=\left(2-x\right)^2\)
\(\Leftrightarrow x^2-7x+6=0\Rightarrow\left[{}\begin{matrix}x=1\\x=6\left(l\right)\end{matrix}\right.\)