Squaring Numbers in Range 26-50
Let A be such a number. Subtract 25 from A to get x. Subtract x from 25 to get, say, a. Then
A² = a² + 100x. For example, if A = 26, then x = 1 and a = 25 - 1 = 24. Hence
Similarly, if A = 37, then x = 37 - 25 = 12, and a = 25 - 12 = 13. Therefore,
| | 37² = 13² + 100·12 = 1200 + 169 = 1369. |
Why does this work?
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| (25 + x)² - (25 - x)² | = [(25 + x) + (25 - x)]·[(25 + x) - (25 - x)] |
| | = 50·2x |
| | = 100x. |
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Another way: recollect that (a ± b)² = a² ± 2ab + b². This leads to the following derivation:
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| (25 + x)² - (25 - x)² | = [25² + 50x + x²] - [25² - 50x + x²] |
| | = [625 -625] + [x² - x²] + [50x + 50x] |
| | = 100x. |
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Either way, it follows that
| |
(25 + x)² | = (25 - x)² + 100x.
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Copyright © 1996-2008 Alexander Bogomolny
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