Squares and Differences

Pick any number. Subtract one from the number to create a second number, and add one to the first number to create a third number. For example, if 6 is my first number, then 5 is my second number and 7 is my third number.

How is the square of the first number related to the product of the second and third numbers? Is this relationship true for all cases? Explain your reasoning.
 


To solve this problem, we simply followed the instructions given and looked for a pattern. We did this using an Excel spreadsheet.

1st number   2nd number   3rd number
x   x-1   x+1
       
4   3   5
5   4   6
6   5   7
7   6   8
8   7   9
9   8   10
10   9   11
11   10   12

We called the first number x, the second number x-1 and the third number x+1. When we plugged the numbers into the spreadsheet, we noticed that the product of the second and third numbers was one less than the square of the first number.

x2 (x-1)(x+1)
16 15
25 24
36 35
49 48

Our next quest was to find out if this was true for all numbers. We came up with the following algebraic formula for our problem:

                    x2 = [(x-1)(x+1)] +1

Using the FOIL method, we simplified the problem to

                    x2 = x2  + x -x -1+1

Further simplified, we see that x2 = x2  ; therefore, this is true for all cases.

Extension:

We discovered that if we re-write the formula as x2 -1 = (x-1)(x+1), this is actually the formula for the difference of two squares. Holy moly!