This is a writing problem from AOPS, so we cannot say if you got it correct or not. Have you tried putting numbers into the boxes to figure out if the maximum and minimum values make sense? Also, try using details from the maximum instead of using just logic. I am taking the course right now. Don't be afraid to use the message board ;).
+84 How can I find the minimum with completing the square? Thats the part I dont understand.
+118687 I'd answer both of these just through logic.
I am interested in seeing an algebric way too.
Show me the algebraic way to do the max please VWoid.
+84 For the maximum, I started off by adding up all the integers. 1+2+3...+10=55.
So with that I gave a variable to each group, x and y.
Thus, x+y=55, then I switched that to a slope intercept form, so y= -x+55
So that gave me P= y * x, which I then simplified to P= x* (-x+55), which became -(x^2-55x)
Then we can complete the square to get -(x^2-55x+765.25), and that simplified is -(x-27.5), so 27.5* 27.5 will give us the highest number, but since we only have intergers we can go +1 and -1 to get 27*28 and that will equal 756.
+118687 Thanks VVSmiley,
Can you explain why you have completed the square?
I know you have been taught to do this but can you explain to me the logic behind it?
+84 Sorry I am not 100% how to explain this but basically by completing the square, we can eventually put the equation in vertex form, which in this case will give us the maximum. In this the equation became -1(x+27.5)^2+765.25. So we know that 765.25 is the maximum.
Essentially, we just complete the square so we can easily fine the vertex through vertex form.
+118687 If one cannot explain why they do something then it almost always means they do not fully understand.
That is ok, we all do mathematical steps that we have just memorized but it is good to understand that that is what one has done.
I will give it further thought myself.
Thanks for showing me :)
+84 I am new to quadratic optimzation so I just tried out a trick and it worked lol. I learned the way of expanding and such, but I wanted to try completing the square, sorry I could not help you.
+118687 Actually I thought this was a method that AoPS had instructed you in. Was i wrong?
You did show me, that is what I asked of you 
I will look at it some more. Are you sure it is correct?
+84 AOPS directed us in the using vieta (-b/2a) or the average of the roots which is close to the idea of vietas. I am pretty sure it is correct, I ca not see how anything else is the maximum, but I am not the smartest so I could wrong as well.
