\(Q=x^2+2y^2-2x-6y+2021\)
\(=\left(x^2-2x+1\right)+2\left(y^2-3y+\dfrac{9}{4}\right)+\dfrac{4031}{2}\)
\(=\left(x-1\right)^2+2\left(y-\dfrac{3}{2}\right)^2+\dfrac{4031}{2}\ge\dfrac{4031}{2}\)
\(minQ=\dfrac{4031}{2}\Leftrightarrow\) \(\left\{{}\begin{matrix}x=1\\y=\dfrac{3}{2}\end{matrix}\right.\)
\(Q=\left(x^2-2x+1\right)+\left(2y^2-6y+\dfrac{9}{2}\right)+\dfrac{4031}{2}\\ Q=\left(x-1\right)^2+2\left(y^2-2\cdot\dfrac{3}{2}y+\dfrac{9}{4}\right)+\dfrac{4031}{2}\\ Q=\left(x-1\right)^2+2\left(y-\dfrac{3}{2}\right)^2+\dfrac{4031}{2}\ge\dfrac{4031}{2}\\ Q_{min}=\dfrac{4031}{2}\Leftrightarrow\left\{{}\begin{matrix}x=1\\y=\dfrac{3}{2}\end{matrix}\right.\)
