MHB Why x2 +1 and x2 -1 are Not/Are Difference of Squares

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The expression x² + 1 is not a difference of squares because it cannot be factored into the form a² - b², which requires a real number b. Instead, it can be expressed as x² - (-1), leading to complex factors (x - i)(x + i). In contrast, x² - 1 is a difference of squares, as it can be factored into (x - 1)(x + 1) using real numbers. The distinction lies in the nature of the terms involved; x² + 1 involves imaginary numbers, while x² - 1 remains within the realm of real numbers. Understanding these differences is crucial for proper factorization in algebra.
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Explain why x2 +1 is not a difference of squares and x2 -1 is
 
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Oh, but it is ...

$x^2+1 = x^2 - (-1) = (x - i)(x + i)$
 

Thread 'Area of Overlapping Squares'

Here is a little puzzle from the book 100 Geometric Games by Pierre Berloquin. The side of a small square is one meter long and the side of a larger square one and a half meters long. One vertex of the large square is at the center of the small square. The side of the large square cuts two sides of the small square into one- third parts and two-thirds parts. What is the area where the squares overlap?
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