\(\left(x-1\right)^2=\left(1-x\right)^3\)
\(\Leftrightarrow x^2-2x+1=1-3x+3x^2-x^3\)
\(\Leftrightarrow x^3-2x^2+x=0\)
\(\Leftrightarrow x\left(x^2-2x+1\right)=0\)
\(\Leftrightarrow x\left(x-1\right)^2=0\)
\(\Leftrightarrow\left[{}\begin{matrix}x=0\\\left(x-1\right)^2=0\end{matrix}\right.\)
\(\Leftrightarrow\left[{}\begin{matrix}x=0\\x-1=0\end{matrix}\right.\)
\(\Leftrightarrow\left[{}\begin{matrix}x=0\\x=1\end{matrix}\right.\)
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