[tex] x^{2} + y^{2}-4x+2y [/tex]
can be written as follows:
[tex] x^{2} -4x+ y^{2}+2y[/tex]
[tex](x^{2} -4x+4)-4+ (y^{2}+2y+1)-1[/tex]
[tex] (x-2)^{2}+(y+1)^{2} -5[/tex]
since [tex](x-2)^{2}[/tex] and [tex](y+1)^{2}[/tex] are always positive, except for x=2 and y=-1, when they become 0,
the minimum value of this expression is when x=2, and y=-1, that is -5
Answer: -5
