   SEARCH HOME Math Central Quandaries & Queries  Question from Sarah, a student: Using the "difference of squares" formula how do I compute 27 * 33? Hi Sarah,

Look at the difference of squares expression

x2 - y2 = (x - y)(x + y)

The two numbers on the right hand side are (x - y) and (x + y) and their difference is

(x + y) - (x - y) = 2y

There are two facts to notice here, one is that the difference is even and the second is that the two numbers on the right of the difference of squares expression are x minus half this difference and x plus half this difference.

Here is an example

find 32 48

I am looking to see if this can be the right hand side of a difference of squares expression.

48 - 32 = 16 which is even and hence 2y = 16 and y = 8.

Hence

48 = 40 + 8 and 32 = 40 - 8 and I have

32 48 = (40 - 8)(40 + 8) = 402 - 82 = 1600 - 64 = 1536

This difference of squares technique worked because 48 - 32 is even and 40 and 8 are numbers I can easily square.

Here is another example

find 13 17

17 - 13 = 4 so y = 2 and thus

13 17 = (15 - 2)(15 + 2) = 152 - 22 = 225 - 4 = 221

easy if you know that 152 = 225.     