[tex]2x^2+y^2+2xy-2x+2y+5=0\\\Leftrightarrow (x^2+2xy+y^2+2x+2y+1)+(x^2-4x+4)=0\\\Leftrightarrow \left [ (x+y)^2+2(x+y)+1 \right ]+(x-2)^2=0\\\Leftrightarrow (x+y+1)^2+(x-2)^2=0\\\Leftrightarrow \left\{\begin{matrix} x+y+1=0 & \\ x-2=0 & \end{matrix}\right.\\\Leftrightarrow \left\{\begin{matrix} y+3=0 & \\ x=2 & \end{matrix}\right.\\\Leftrightarrow \left\{\begin{matrix} y=-3 & \\ x=2 & \end{matrix}\right.[/tex]
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