A Difference of Squares

One more special product is a sum times a difference of the same two terms, such as $\left(x+3\right)\left(x-3\right)$. The four products are $x^{2}$, $-3x$, $3x$ and $-9$. The two cross products are opposites, so they cancel.

What remains is $x^{2}-9$, a difference of squares. In letters, $\left(a+b\right)\left(a-b\right) = a^{2}-b^{2}$. Because the cross products cancel, the product has no middle term.

You can use the difference of squares for mental arithmetic. To work out 21 times 19, write it as $21\cdot 19 = \left(20+1\right)\left(20-1\right) = 400-1 = 399$. The two numbers are 20 plus 1 and 20 minus 1, so the product is 20 squared minus 1 squared.