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.

( a+b )( a+( b ) )=aa+a( b )+ba+b( b ) = a 2 ab+ab b 2

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 ab is the square of a minus the square of b .

( a+b )( a+( b ) )= a 2 b 2

Example :

Find the product.

( 3x+4 )( 3x4 )

By the Product of a Sum and a Difference, ( a+b )( a+( b ) )= a 2 b 2 .

Here, a=3x and b=4 .

( 3x+4 )( 3x4 )= ( 3x ) 2 ( 4 ) 2

Simplify.

( 3x+4 )( 3x4 )=9 x 2 16