Why is (n-1)(n+1)=n^2-1?
Last Updated: 03.07.2025 05:13
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 x
n^2 - n + n - 1
The basic answer is that you cross-multiply the values in each factor:
xxx
n^2 - 1
xxxx
xx
xxxx
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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:
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(n - 1)(n + 1)
And you are left with almost a square. One less than the square of side n.
xxx
So, that’s a rectangle with sides 2 and 4.