**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.

x^{2} |
(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:

x^{2 }= [(x-1)(x+1)] +1

Using the FOIL method, we simplified the problem to

x^{2 }= x^{2 } + x -x -1+1

Further simplified, we see that x^{2 }= x^{2 } ;
therefore, this is true for all cases.

**Extension:**

We discovered that if we re-write the
formula as x^{2}= (x-1)(x+1), this is actually the
formula for the difference of two squares. Holy moly!