\[\boxed{\text{1276.\ }\text{еуроки}\text{-}\text{ответы}\text{\ }\text{на}\text{\ }\text{пятёрку}}\]
\[x^{2} - 2x + y^{2} - 4y + 5 = 0\]
\[\left( x^{2} - 2x + 1 \right) +\]
\[+ \left( y^{2} - 4y + 4 \right) = 0\]
\[(x - 1)^{2} + (y - 2)^{2} = 0 \Longrightarrow\]
\[\Longrightarrow (x - 1)^{2} = 0\]
\[x - 1 = 0\]
\[x = 1\]
\[и\ \ \ \ \ \ \ (y - 2)^{2} = 0\]
\[\ \ \ \ \ \ \ \ \ \ y - 2 = 0\]
\[\ \ \ \ \ \ \ \ \ \ \ y = 2\]
\[Ответ:x = 1,\ \ y = 2.\]