What is the minimum value of the expression x^2 + y^2 + 2x - 4y + 8 - 10y + 3y for real x and y?
Completing the square on $x$ and $y$ independently to write the given expression as $(x+a)^2+(y+b)^2+c$. The answer is $c$.
Put into standard circle form :
(x +1)^2 + (y-5.5)^2 = -8 +1+ 30.25 r = sqrt 23.25 center = -1, 5.5
Minumum 'x' will be - 1 - sqrt 23.25
Minumum 'y' will be 5.5 - sqrt 23.5 (check my math)