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Product of a Sum and a Difference

What happens when you multiply the sum of two quantities by their difference? This calculation occurs so commonly in mathematics that it's worth memorizing a formula. Write the product as $\left(a+b\right)\left(a-b\right)$ .

Now use the FOIL method to multiply the two binomials.

$\begin{array}{l}\left(a+b\right)\left(a+\left(-b\right)\right)=a\cdot a+a\cdot \left(-b\right)+b\cdot a+b\cdot \left(-b\right)\\ =a^2-ab+ab-b^2\end{array}$

Notice that the middle terms are opposite and add to a zero pair.

So, $\left(a+b\right)\left(a+\left(-b\right)\right)=a^2-b^2$ .

In other words, the product of $a+b$ and $a-b$ is the square of $a$ minus the square of $b$ .

$\left(a+b\right)\left(a+\left(-b\right)\right)=a^2-b^2$