Binomial formulas

Some brackets show up again and again, for example $(a + b)^2$. Instead of multiplying them out laboriously every time, there are three fixed patterns, the binomial formulas. Whoever recognises them saves time, makes fewer mistakes, and can even multiply large numbers in their head.

The first binomial formula

When you multiply $(a + b)$ by itself, every term meets every other. You get $a^2$, twice the mixed product $ab$, and $b^2$. The middle term is often forgotten, yet it makes all the difference.

The second binomial formula

If the bracket holds a minus, only the sign of the middle term changes. The square of a difference is $a^2 - 2ab + b^2$. The last term keeps its plus sign, because $(-b) \cdot (-b)$ gives $+b^2$ again.

The third binomial formula

When you multiply the sum $(a + b)$ by the difference $(a - b)$, the mixed terms cancel. What remains is $a^2 - b^2$. This pattern is a real calculation trick: $32 \cdot 28$ is $(30 + 2)(30 - 2) = 900 - 4 = 896$, with no written multiplication at all.