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)(ab).

Now use the FOIL method to multiply the two binomials.

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

So, (a + b)(a + (– b)) = a2 b2 .

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) = a2b2

Example :

Find the product.

(3x + 4)(3x – 4)

By the Product of a Sum and a Difference, (a + b)(a b) = a2b2 .

Here, a = 3x and b = 4.

(3x + 4)(3x – 4) = (3x)2 – (4)2

Simplify.

(3x + 4)(3x – 4) = 9x2 – 16