Why is (n-1)(n+1)=n^2-1?
13.06.2025 05:58
The basic answer is that you cross-multiply the values in each factor:
xx
xxx
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xxx x
xxx
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The more direct answer is to look at a rectangle with sides of n + 1 and n - 1, let’s go with n = 2
If you cut off the end of that rectangle, you make it smaller, 2 * 3 and a separate nub of size 1 * 2
n^2 - n + n - 1
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n^2 - 1
Move the nub up to the top:
xxxx
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So, that’s a rectangle with sides 2 and 4.
(n - 1)(n + 1)
xxx x
And you are left with almost a square. One less than the square of side n.