Definition: A difference of squares is an expression of the form a^2 – b^2, which always factors as (a + b)(a – b).
What is the pattern?
Two perfect squares subtracted. When you multiply (a + b)(a – b), the middle terms cancel, leaving a2 – b2. So the factoring is just that multiplication in reverse.
How do you spot it?
Check for exactly two terms, a minus sign between them, and both terms perfect squares. Numbers like 49, 121, and 400 qualify, as do variable powers with even exponents.
Example. Factor 81x2 – 100y2. Here a = 9x and b = 10y, so the result is (9x + 10y)(9x – 10y). A harder case is x4 – 16, which becomes (x2 + 4)(x2 – 4), and then (x2 + 4)(x + 2)(x – 2).
What goes wrong?
A sum of squares such as x2 + 25 does not factor over the real numbers. Students also write (a – b)2 instead of (a + b)(a – b); those are different. And a common factor should come out first: 5x2 – 45 is 5(x2 – 9) = 5(x + 3)(x – 3).