Why is (n-1)(n+1)=n^2-1?
12.06.2025 01:12

The basic answer is that you cross-multiply the values in each factor:
xxx
The more direct answer is to look at a rectangle with sides of n + 1 and n - 1, let’s go with n = 2
Ford Stock Rises After Strong May Sales. The Pain Is Coming. - Barron's
n^2 - 1
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
Move the nub up to the top:
xx
xxxx
xxxx
xxx
n^2 - n + n - 1
Natural Tick Repellent Found—on Donkey Skin - Newser
xxx x
So, that’s a rectangle with sides 2 and 4.
(n - 1)(n + 1)
The Largest Black Hole Jet Ever Found Is 3 Times the Size of the Milky Way - The Daily Galaxy
And you are left with almost a square. One less than the square of side n.