A Missing Middle

x^2 - 9 has no middle term. In the last section you expanded (x + 3)(x - 3). The cross products 3x and -3x cancelled, and the result was x^2 - 9. Factoring goes the other way. You start from x^2 - 9 and write the two factors back.

A difference of squares is x^2 minus a square number. It factors into a sum and a difference of the same two terms: x^2 - 9 = (x + 3)(x - 3). The 3 is the square root of 9. There is no middle term because the cross products cancelled.

You can also find the same answer with factor pairs. The numbers 3 and -3 have product -9 and sum 0. So x^2 - 9 = (x + 3)(x - 3). This is the sum and product method with a middle coefficient of 0.