Why is (n-1)(n+1)=n^2-1?
10.06.2025 23:59

xx
So, that’s a rectangle with sides 2 and 4.
(n - 1)(n + 1)
Panthers' winning formula goes missing in Stanley Cup Final Game 1 loss to Oilers - NHL.com
xxx x
If you cut off the end of that rectangle, you make it smaller, 2 * 3 and a separate nub of size 1 * 2
xxxx
Evan Engram embraces the “Joker” role in the Broncos’ offense - NBC Sports
The more direct answer is to look at a rectangle with sides of n + 1 and n - 1, let’s go with n = 2
xxx
xxx x
The basic answer is that you cross-multiply the values in each factor:
xxxx
n^2 - n + n - 1
Move the nub up to the top:
And you are left with almost a square. One less than the square of side n.
xxx
NASA’s Webb Rounds Out Picture of Sombrero Galaxy’s Disk - NASA Science (.gov)
n^2 - 1