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**4DLiVing****Member**- Registered: 2011-01-08
- Posts: 22

Here is something interesting that I have been using to amaze my 8th graders~!

I have them pick a two digit number and I will square it in my head...

Here is how I do it...

(a + b)² = a² + 2ab + b²

Say they pick 24; 24 is (20 + 4); a = 20 and b = 4 so...

so 20² + 2(20)(4) + 4² = 400 + 160 + 16 = 560 + 16 = 576.

This works for all two digit numbers.

Look at 49 (this ones a little tougher to do in the head but if you practice, you will master!)

40² + 2(40)(9) + 9² = 1600 + 720 + 81 = 2320 + 81 = 2401

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 85,347

Hi;

For numbers like 49 which are under but close to a nice number I use

Now all you have to do for 49^2 is a = 50 and b = 1

50^2 - 2(50)(1) + 1

2500 - 100 +1 = 2401

**In mathematics, you don't understand things. You just get used to them.Of course that result can be rigorously obtained, but who cares?Combinatorics is Algebra and Algebra is Combinatorics.**

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**4DLiVing****Member**- Registered: 2011-01-08
- Posts: 22

aha! very cool... i like it a lot! That'll save me a few seconds! Thanks for the other side of the box!

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 85,347

Hi 4DLiVing;

You can also use that identity to multiply lots more, for instance 84 * 76.

**In mathematics, you don't understand things. You just get used to them.Of course that result can be rigorously obtained, but who cares?Combinatorics is Algebra and Algebra is Combinatorics.**

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**4DLiVing****Member**- Registered: 2011-01-08
- Posts: 22

84 * 76

would be (a + b) (a - b) = a^2 - b^2

or (80 + 4) (80 - 4) = 80^2 - 4^2 = 6400 - 16 = 6384.

Nice!

This does require a more "specialized" starting position with the numbers that are given to me, but I shall remember this one as well!

Thanks bobbym!

I have tried (a + b + c)^2, but I think I would need more of a ram upgrade in my brain in order to do these in my head. Nonetheless, I always found it interesting how these expansions can make our lives easier!

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 85,347

Not really computers make our life easier by allowing us to concentrate on the important things in life like eating.

**In mathematics, you don't understand things. You just get used to them.Of course that result can be rigorously obtained, but who cares?Combinatorics is Algebra and Algebra is Combinatorics.**

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**soroban****Member**- Registered: 2007-03-09
- Posts: 451

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