Question

Answer 1

\(B=x^2+2y^2+2xy+8y+7=\left(x^2+2xy+y^2\right)+\left(y^2+8y+16\right)-9=\left(x+y\right)^2+\left(y+4\right)^2-9\ge-9\)\(B_{min}=-9\Leftrightarrow x=4;y=-4\)

Answer 2

\(E=x^2+y^2+xy-3x-3y=\dfrac{2x^2+2y^2+2xy-6x-6y}{2}=\dfrac{\left(x^2+y^2+4+2xy-4x-4y\right)+\left(x^2-2x+1\right)+\left(y^2-2y+1\right)-6}{2}=\dfrac{\left(x+y-2\right)^2+\left(x-1\right)^2+\left(y-1\right)^2-6}{2}\ge\dfrac{-6}{2}=-3\)

\(E_{min}=-6\Leftrightarrow x=y=1\)