Suppose c and d represent two different school populations where c > d and c and d must be greater than 0. whitch of the following expressions is the largest? explain why. a. (c+d)^2 b. 2(c+d) c. c^2 + d^2 d. c^2 - d^2

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.  

If N is positive and less than one, its square is less than N, i.e.

If N is greater than one, its square is greater than N

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  

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.

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.

The expression can be written as

(C+D)(C-D)

Comparing with A.

The subtracting factor (C-D) makes this product smaller than A which has two adding factors.

Thus A. is the largest expression

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