## A Handy SAT Math Shortcut – N^{2} – 1

Many of us have a pretty good sense of what the squares of the first 15 integers are. Sure, you might be a bit shaky on ‘13’ and ‘14’ but you should be comfortable with the rest. Just to be sure, I’ll reproduce those below:

1^2 = 1

2^2 = 4

3^2 = 9

4^2 = 16

5^2 = 25

6^2 = 36

7^2 = 49

8^2 = 64

9^2 = 81

10^2 = 100

11^2 = 121

12^2 = 144

13^2 = 169

14^2 = 196

15^2 = 225

Here are more squares students tend to know:

16^2 = 256

20^2 = 400

25^2 = 625

30^2 = 900

(If you know all these, that’s pretty solid! You don’t have to memorize anymore.)

**Why did I even bring this up in the first place? Well, I have a cool mental math shortcut.** Assuming you know the above, you also know the following:

11 x 13, 14 x 16, 15×17 and even the crazy 29×31.

How is that possible?

Well, what if I told you that n^2 – 1 = (n – 1)(n + 1)

Big deal, you say. You already know basic algebra? And what does this have to do with squares?

Well, let’s say n = 20.

See, by knowing that 20^2 = 400, then the product of one integer less than 20—the number 19—and one integer greater than 20—the number 21—will be 400-1, which equals 399.

Try it with any of the numbers above. For instance, we know that 12^2 = 144. Therefore, 11 x 13 = 143.

29 x 31?

Well, what’s 30^2 – 1.

Just like that, voila! You’ve doubled your knowledge of squares above.

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