# 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.

$\begin{array}{l}\left(a+b\right)\left(a+\left(-b\right)\right)=a\cdot a+a\cdot \left(-b\right)+b\cdot a+b\cdot \left(-b\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}={a}^{2}-ab+ab-{b}^{2}\end{array}$

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 $b$ .

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

Example :

Find the product.

$\left(3x+4\right)\left(3x-4\right)$

By the Product of a Sum and a Difference, $\left(a+b\right)\left(a+\left(-b\right)\right)={a}^{2}-{b}^{2}$ .

Here, $a=3x$ and $b=4$ .

$\left(3x+4\right)\left(3x-4\right)={\left(3x\right)}^{2}-{\left(4\right)}^{2}$

Simplify.

$\left(3x+4\right)\left(3x-4\right)=9{x}^{2}-16$