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What is the intuitive explanation for the Sum and Difference of two squares?
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We represent geometrically an algebraic concept namely the sum and difference of two squares. The difference of two squares states you can factor $a^2 – b^2$ into $(a+b)(a-b)$. You may use a piece of paper to follow the procedure below.

Geometric Representation

1. Take a square with length $a$ (any convenient side length)

2. Cut out a small square with side length $b$ from one of the corners of the large square. The area of the remaining figure is $a^2 – b^2$. Can you see why?

3. Fold and cut the remaining figure so that when the two pieces are put together, they form a rectangle. The rectangle has length $a + b$ and width $a – b$.

The second figure has area $a^2 – b^2$ and the rectangle has area $(a+b)(a-b)$. Since we did not take away any portion, the two areas are equal. Therefore,

$a^2 – b^2= (a + b)(a-b)$.

The geometric representation of algebraic proofs such as the third figure is called proof without words.

by Diamond (55,292 points)