\(2x^2-2xy+2y^2-3y+\dfrac{3}{2}=0\)
\(\Leftrightarrow4x^2-4xy+4y^2-6y+3=0\)
\(\Leftrightarrow\left(4x^2-4xy+y^2\right)+\left(3y^2-6y+3\right)=0\)
\(\Leftrightarrow\left(2x-y\right)^2+3\left(y-1\right)^2=0\)
\(\Leftrightarrow\left\{{}\begin{matrix}2x-y=0\\y-1=0\end{matrix}\right.\)
\(\Leftrightarrow\left\{{}\begin{matrix}y=1\\x=\dfrac{1}{2}\end{matrix}\right.\)