Question

Suppose A and B represent two different school populations where A > B and A and B must be greater than 0. Which of the following expressions is the largest? Explain why. Show all work necessary.
2(A + B)
(A + B)2
A2 + B2
A2 − B2

Answer 1

The first one is 
left at 2(A + B)

The second one is
A^2 + 2AB + B^2 The expanded version is greater than A^2 + B^2 See below for why.

(A + B)^2 / [2(A + B) ] = (A + B)/2 which is what is  left over when (A + B)^2 is divided by the first one.
 
(A + B)^2 / (A^2 - B^2) = (A + B) * (A + B) / [(A + B) (A - B)] = (A + B) / (A - B) which is less than (A + B)/2 for (A - B)>2

The third one is just A^2 + B^2 which is smaller than the second one by an amount equal to 2AB. Since A and B are both > 0, 2 AB must be > 0



A^2 - B^2