differenceofsquares

Difference of squares refers to the algebraic expression a^2 − b^2. It is a special product that factors as (a − b)(a + b). This identity holds for any real or complex numbers a and b, and in general over any commutative ring.

If a and b are integers, the factorization provides a convenient way to factor expressions or solve equations of the form a^2 − b^2 = c, since c = (a − b)(a + b). For example, 3^2 − 2^2 = (3 − 2)(3 + 2) = 1 · 5 = 5, and 7^2 − 4^2 = (7 − 4)(7 + 4) = 3 · 11 = 33. Note that a^2 − b^2 can be zero when a = b, and it is negative when a < b.

A key property is that the two factors a − b and a + b have the same parity: both are even or both are odd. This follows from their sum, (a + b) + (a − b) = 2a, which is even. The expression is nonnegative if a ≥ b and can be used to analyze sign patterns in problems.

Applications of the difference of squares include factoring polynomials, simplifying expressions, and solving quadratic-type equations. Geometrically, it represents the difference between the areas of two squares with side lengths a and b. The concept also underpins many algebraic techniques and appears in number theory and problem solving as a basic factoring tool.