\(A=\left(x^2-2xy+y^2\right)-\left(x-y\right)+\dfrac{1}{4}+\left(y^2-y+\dfrac{1}{4}\right)-\dfrac{1}{2}\\ A=\left[\left(x-y\right)^2-\left(x-y\right)+\dfrac{1}{4}\right]+\left(y-\dfrac{1}{2}\right)^2-\dfrac{1}{2}\\ A=\left(x-y-\dfrac{1}{2}\right)^2+\left(y-\dfrac{1}{2}\right)^2-\dfrac{1}{2}\ge-\dfrac{1}{2}\\ A_{min}=-\dfrac{1}{2}\Leftrightarrow\left\{{}\begin{matrix}x=y+\dfrac{1}{2}=1\\y=\dfrac{1}{2}\end{matrix}\right.\)
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