difference of two squares in Notes

# Difference of Two Squares

The term **difference of two squares** refers to the identity:

$$
a^2 - b^2 = (a - b)(a + b)
$$

## Algebraic Proof

For any numbers $a$ and $b$:

$$
\begin{aligned}
(a - b)(a + b) &= a^2 + a b - b a - b^2 \\
&= a^2 - b^2
\end{aligned}
$$

## Geometric Interpretation

For $a \gt b$, there is a geometric interpretation of this as a literal difference of two squares.  The left-hand side is viewed as the area of a shape made by taking a square of side length $a$ and removing a square of side length $b$ from one corner.  The right-hand side is viewed as the area of a rectangle of side lengths $a + b$ and $a - b$.  The equality between the two follows from the invariance of area under [[dissection]].

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[[DifferenceOfTwoSquares.png:pic]]
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