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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 $(a + b) (a - b)$ .

Now use the FOIL method to multiply the two binomials.

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

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

So, $(a + b) (a + (- b)) = a^{2} - b^{2}$ .

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

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

Example :

Find the product.

$(3 x + 4) (3 x - 4)$

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

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

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

Simplify.

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