# 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

26² = 24² + 100 = 676.

Similarly, if

37² = 13² + 100·12 = 1200 + 169 = 1369.

Why does this work?

(25 + x)² - (25 - x)² | = [(25 + x) + (25 - x)]·[(25 + x) - (25 - x)] |

= 50·2x | |

= 100x. |

Another way: recollect that (a ± b)² = a² ± 2ab + b². This leads to the following derivation:

(25 + x)² - (25 - x)² | = [25² + 50x + x²] - [25² - 50x + x²] |

= [625 -625] + [x² - x²] + [50x + 50x] | |

= 100x. |

Either way, it follows that

(25 + x)²= (25 - x)² + 100x.

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