What is the minimum value of z if z=x^2+2y^2+6x-4y+22?

z = x^2+2y^2+6x-4y+22   complete the square on x and y

[x^2 + 6x + 9]  +  2 [ y^2 - 2y + 1]  +  22   =  z + 11

(x +3)^2   + 2(y -1)^2   =   z - 11

This will be minimized when  the right side = 0    ⇒   z = 11