Why is (n-1)(n+1)=n^2-1?

12.06.2025 00:26

Why is (n-1)(n+1)=n^2-1?

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n^2 - n + n - 1

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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

n^2 - 1

The basic answer is that you cross-multiply the values in each factor:

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So, that’s a rectangle with sides 2 and 4.

If you cut off the end of that rectangle, you make it smaller, 2 * 3 and a separate nub of size 1 * 2

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Move the nub up to the top:

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I’m wondering about attachment and transference with the therapist and the idea of escape and fantasy? How much do you think your strong feelings, constant thoughts, desires to be with your therapist are a way to escape from your present life? I wonder if the transference serves another purpose than to show us our wounds and/or past experiences, but is a present coping strategy for managing what we don’t want to face (even if unconsciously) in the present—-current relationships, life circumstances, etc. Can anyone relate to this concept of escape in relation to their therapy relationship? How does this play out for you?

And you are left with almost a square. One less than the square of side n.

(n - 1)(n + 1)

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