minimum value

uptowngirl92 wrote:What is the minimum value of the expression 2(x^2) + 3(y^2) - 4x - 12y + 18?
a) 18
b) 10
c) 6
d) 4
e) 0

Detailed explanations please!

Try to rework the expression:
2(x^2) + 3 (y^2) - 4x - 12y + 18
=> [2(x^2)- 4x] + [3(y^2) -12y] + 18
=> 2[(x^2)-2x] + 3 [(y^2) - 4y] + 18
=> 2[ (x-1)^2 -1] + 3[(y-2)^2 - 4] + 18
=> 2(x-1)^2 + 3 (y-2)^2 -2 - 12 + 18
=> 2 (x-1)^2 + 3(y-2)^2 + 4
Since min value of (x-1)^2 and (y-2)^2 is 0
the min value of the expression is 4.