\(4x^2+5y^2-4xy+4y+1=0\)
\(\Leftrightarrow4x^2-4xy+y^2+4y^2+4y+1=0\)
\(\Leftrightarrow\left(2x-y\right)^2+\left(2y+1\right)^2=0\)
\(\Leftrightarrow\left[{}\begin{matrix}2x-y=0\\2y+1=0\end{matrix}\right.\Leftrightarrow\left[{}\begin{matrix}x=\frac{-1}{4}\\y=\frac{-1}{2}\end{matrix}\right.\)
4x2 +5y2 - 4xy+ 4y+1=0
(=) (4x2-4xy+ y2)+ (4y2+4y+1)=0
(=) ( 2x-y)2 + ( 2y+1)2=0
(=)\(\left\{{}\begin{matrix}\left(2x-y\right)^2=0\\\left(2y+1\right)^2=0\end{matrix}\right.\)(=) \(\left\{{}\begin{matrix}2x-y=0\\2y+1=0\end{matrix}\right.\)(=) \(\left\{{}\begin{matrix}2x-y=0\\2y=-1\end{matrix}\right.\)
(=)\(\left\{{}\begin{matrix}2x-y=0\\y=\frac{-1}{2}\end{matrix}\right.\)(=) \(\left\{{}\begin{matrix}2x-\left(\frac{-1}{2}\right)=0\\y=\frac{-1}{2}\end{matrix}\right.\)(=) \(\left\{{}\begin{matrix}2x=\frac{-1}{2}\\y=\frac{-1}{2}\end{matrix}\right.\)
(=)\(\left\{{}\begin{matrix}x=\frac{-1}{4}\\y=\frac{-1}{2}\end{matrix}\right.\)
\(4x^2+5y^2-4xy+4y+1=0\\ \Leftrightarrow\left(4x^2-4xy+y^2\right)+\left(4y^2+4y+1\right)=0\\ \Leftrightarrow\left(2x-y\right)^2+\left(2y+1\right)^2=0\\ \Leftrightarrow\left\{{}\begin{matrix}2x-y=0\\2y+1=0\end{matrix}\right.\\ \Leftrightarrow\left\{{}\begin{matrix}x=-\frac{1}{4}\\y=-\frac{1}{2}\end{matrix}\right.\)