Answer:
A. (C+D)^2 is the largest expression
Step-by-step explanation:
Squaring Properties
The square of a number N is shown as N^2 and is the product of N by itself, i.e.
[tex]N^2=N*N[/tex]
If N is positive and less than one, its square is less than N, i.e.
[tex]N^2<N, \ for\ 0<N<1[/tex]
If N is greater than one, its square is greater than N
[tex]N^2>N, \ for\ N>1[/tex]
We have the following information: C and D represent two different school populations, C > D, and C and D must be positive. We can safely assume C and D are also greater or equal than 1. Let's evaluate the following expressions to find out which is the largest
A. (C+D)^2
Expanding
[tex](C+D)^2=C^2+2CD+D^2[/tex]
Is the sum of three positive quantities. This is the largest of all as we'll prove later
B. 2(C+D)
The extreme case is when C=2 and D=1 (recall C>D). It results:
2(C+D)=2(3)=6
The first expression will be
(3)^2=9
Any other combination of C and D will result smaller than the first option
C. [tex]C^2 + D^2[/tex]
By comparing this with the first option, we see there are two equal terms, but A. has one additional term 2CD that makes it greater than C.
D. [tex]C^2 - D^2[/tex]
The expression can be written as
(C+D)(C-D)
Comparing with A.
[tex](C+D)^2=(C+D)(C+D)[/tex]
The subtracting factor (C-D) makes this product smaller than A which has two adding factors.
Thus A. is the largest expression
