Notes
difference of two squares

Difference of Two Squares

The term difference of two squares refers to the identity:

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

Algebraic Proof

For any numbers aa and bb:

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

Geometric Interpretation

For a>ba \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 aa and removing a square of side length bb from one corner. The right-hand side is viewed as the area of a rectangle of side lengths a+ba + b and a−ba - b. The equality between the two follows from the invariance of area under dissection.

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Created on June 24, 2021 10:51:01 by Andrew Stacey?