As the exercise mentions, sometimes the product of two numbers can be written as the product of a sum and difference of two terms. In this case, the result is a difference of two squares.

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

Let's see an example applying this idea. Imagine that we want to multiply

33 \times 27 .

To rewrite this as needed, we use the number that is halfway between both factors — in this case it is

30 .

33 \times 27 \Leftrightarrow (30 + 3) (30 - 3)

Note how we used the number in between both factors —

30

— in order to rewrite the original multiplication —

33 \times 27

— as a product of the sum and difference of

30

and

3

—

(30 + 3) (30 - 3) .

Now we can use the corresponding pattern and mental math to expand the product.

(a + b) (a - b) (30 + 3) (30 - 3) (30 + 3) (30 - 3) (30 + 3) (30 - 3) = a^{2} - b^{2} = 3 0^{2} - 3^{2} = 900 - 9 = 891

However, this shortcut is not always convenient even if it is applicable. This is because for some products this method requires squaring high numbers, which is not easy to do mentally. Take a look at the following examples.

63 \cdot 17 31 \cdot 23 = (40 + 23) (40 - 23) = 4 0^{2} - 2 3^{2} = (27 + 4) (27 - 4) = 2 7^{2} - 4^{2}

As we can see, we can use the same procedure but we need to square the numbers

23

in the first case and

27

in the second. In cases like this, this method is not convenient. Nevertheless, if the factors satisfy the conditions below the use of the method is convenient.

The difference of the numbers in the tens is $1 .$ This guarantees that the term corresponding to $b,$ in the pattern $a^{2} - b^{2},$ is not greater than $10 .$
The numbers in the units add up to $10 .$ This guarantees that the number halfway between both factors is a multiple of $10 .$ Then, squaring it is not very complicated.

The diagram shown below can help us understand why these conditions work.

For example, the product

32 \times 28

satisfies both conditions since the difference in the tens is

3 - 2 = 1

and the sum of the units is

2 + 8 = 10 .

We can rewrite this multiplication as the sum and difference of two numbers as indicated in the diagram given before.

32 \times 28 \Leftrightarrow (30 + 2) (30 - 2)

We can also simplify mentally by using the pattern of the difference of two squares.

(30 + 2) (30 - 2) (30 + 2) (30 - 2) (30 + 2) (30 - 2) = 3 0^{2} - 2^{2} = 900 - 4 = 896

Exercise