The Shape in Letters

The difference of squares works for any two terms, not only $x$ and a number. Written in letters, the rule is $a^{2}-b^{2} = \left(a+b\right)\left(a-b\right)$. The difference of two squares equals the sum of their square roots times the difference of their square roots.

You can check the rule by expanding the right side. The cross terms $ab$ and $-ab$ cancel, and what is left is $a^{2}-b^{2}$.

The rule also works with plain numbers. To work out $41\cdot 39$, write it as $\left(40+1\right)\left(40-1\right) = 1600-1 = 1599$. The same rule is used for multiplying and for factoring. When you multiply, you go from the right side of the rule to the left. When you factor, you go from the left side to the right.