Multiplying Numbers Using the Square of Binomial Formula: Lesson 1

Now let’s try to find the answer for the squares of these numbers:

18, 27, and 38

Example 1:

1-    18 = (20-2) and 18²= (20 – 2)²

The reason we want to change it to 20 – 2 is so that it is easier to find the square of 20 than the square of 18. The square of 20 = 400.

Let’s apply the following formula (a - b)² = a² - 2ab + b².  In this example (a = 20) and (b = 2)

2-      a² = 20² = 400

3-      2ab = 2(20 × 2) = 2(40) = 80. Which is double the product of (a × b)

4-      b² = 2² = 2 × 2 = 4

Now let’s add all together using the Square of Binomial formula.

 a² - 2ab + b² = 400 – 80 + 4 the numbers we got from steps 2,3 and 4.

(400 – 80) + 4 = 320 + 4 = 324

So 18 × 18 = 324

Let’s do another one.

Example 2:

1-    27 = (30-3) and 27²= (30 – 3)²

The reason we want to change it to 30 – 3 is so that it is easier to find the square of 30 than the square of 27. The square of 30 = 900

Let’s apply the following formula (a - b)² = a² - 2ab + b².  In this example (a = 30) and (b = 3)

2-      a² = 30² = 900

3-      2ab = 2(30 × 3) = 2(90) =180. Which is double the product of (a × b)

4-      b² = 2² = 3 × 3 = 9

Now let’s add all together using the Square of Binomial formula.

 a² - 2ab + b² = 900 – 180 + 9 the numbers we got from steps 2,3 and 4.

(900 – 180) + 9 = 720 + 9 = 729

So 27 × 27 = 729

For your own practice, try repeating the same process with the number 38. If it would help, try copying and pasting one of the examples and just replace the numbers with the new numbers that correspond to the number 38. You would use (40 - 2) to represent the number 38 in the formula. Since 4 × 4 = 16, 40 × 40 will equal to 1600.

Once you do several examples, you will start finding that it is much easier than you think. The more you practice, the better you will get at it. After you’ve done a few examples, try doing it mentally to see how fast you can come up with the answer.

In the next lesson we will do examples using (a + b)² = a² + 2ab + b². This one should be a little easier since it is all addition, no subtraction involved. Lesson 2

Multiplication Utilizing The Square of Binomial

Now let’s take a look at the Square of Binomial, and how we can utilize this formula to ease multiplication of two digit numbers.

Here is the formula again (a ± b)² = a² ± 2ab + b².  

We want to first break the formula down so that it is easier to read.  Rather than the ± sign, we will break it down using the “+” sign once, and utilizing the “-“sign the other time.

(a + b)² = a² + 2ab + b².  This is what the formula looks like with the plus sign—the sum of two numbers raised to the power of 2. We will replace the ‘a’ and ‘b’ with digits and perform the multiplication as prescribed by the formula.

Let (a=30), for instance and we will let (b=1). So (a + b)² will look like this (30 + 1)².

We know that 30 + 1 = 31. So, what we are doing now is finding the square root of the number 31, or in other words 31 × 31.

The formula says that (a + b)² = a² + 2ab + b². or (30 + 1)² = 30² +( 2 ×30×1) + 1².  

This might look as if it makes the multiplication a bit more complicated, but when you get the hang of it, you will be able to do it mentally without using a pen and paper.

We will do it one segment at a time:

1- 30² or 30×30 = 900

2- (2 ×30×1) = 2 ×30 = 60. And 60 × 1 = 60. So the result of the middle part = 60.

3- The last part b², which is 1².  Or 1 × 1 = 1

Now let’s add 1,2 and 3 together as depicted in the formula:

900 + 60 + 1 = 961. This means that 31 × 31 = 961.

Now we will do the square of different between two numbers. (a - b)² = a² - 2ab + b².  

In this example we will be finding the square of the 29.

We will let a=30 and b=1. We want to stick with the number 30 to keep simple and relevant.

Now (30 – 1) is actually 29, which is the number we want to multiply. Let’s apply the formula now. Notice this time that the signs have changed a bit from the previous one. There is a ‘-‘ sign before the middle (2ab).

This is what it should look like with numbers instead of the letters ‘a’ and ‘b’.

(30 - 1)² = 30² -( 2 ×30×1) + 1² . Now let’s look at the sum of each segment.

1- 30² or 30×30 = 900. Same as above

2- (2 ×30×1) = 2 ×30 = 60. And 60 × 1 = 60. So the result of the middle part = 60.

3- The last part b², which is 1².  Or 1 × 1 = 1

The only difference here is, instead of adding the middle part; we will be subtracting it from the first sum, as shown here (a - b)² = a² - 2ab + b².  So here is the breakdown:

900 - 60 + 1 = 841. This means that 29 × 29 = 841.

If you’ve gotten the hang of it, good for you. If not, we will continue breaking it down in the upcoming post. Stay tuned, and feel free to post any questions that you might have.

Fun Learning Way to Square a Number

There are several ways to perform multiplications. You can use a pen and paper, use a calculator, or memorize the results of particular numbers. Off course, trying to memorize the results of multiplications is not practical. You can only memorize so many. However, memorizing the multiplication table up to 10 or 12 might be a good idea to facilitate learning advanced multiplications. If you are able to memorize even up to 99, more power to you. But beyond that, I think is a waste of time and energy and takes the fun learning concept out of the equation. What a person needs to do instead is memorize and practice a method that will allow him/her to instantly process numbers in their brain to come up with the answer. I’m sure there are several methods that can practice with numbers. In this post, I will discuss only one fun and easy learning method that can be easily applied to a variety of two digit numbers.

If you know the basic multiplication table, you will be able to instantaneously answer the results of 10 × 10. 20 × 20. 30 × 30. and so on. I’m sure this should not be a problem for you. If so, here are the answers:

10 × 10 = 100. Basically this is the result of 1 × 1, and just adding the number of zeroes to the result 2 in this case. I hope I’m not insulting your intelligence or confusing you here.

20 × 20 = 400

30 × 30 = 900

If you can instantaneously know the answer for 8 × 8, for instance, which is 64, you would know

that 80 × 80 is equal to 6400.

How many seconds does it take you to do that? I hope it’s no more than 3 seconds.

Now we will only deal with multiplying like numbers—the square of a 2 digit number—so that the concept is easily understood, and then we can move on to other numbers. Let’s start with a different numbers than the, fun and easy one’s, that end with zero. How about (19²), (29²) and (39²), how can we multiply these numbers in our mind without using a pen and paper or calculator?

To do so, what we’re going to do is round them to the nearest number ending with a zero and take it from there. 

Let’s try the first number 19. The nearest zero number to it is 20. Why? Because 19 is actually (20 – 1). In other words, 19² is the same (20 – 1 )².  Realizing this much, we will first know that the answer will be close to 400, but not quite.

So the product of 19 × 19  is actually (400  - 19 – 20) = 361. 

And 29² is actually 30 × 30 = 900 – (30 – 29) = 900 – 59 = 841

The last one is 39² = (40)² - (40 – 39) = 1600 – 79 = 1521

If you didn't quite understand the concept here, don't worry, you'll get it later. In the next post, we will do similar multiplications utilizing a different approach, the Square of a Binomial. This approach should make things much easier than the previous method.

(a ± b)² = a² ± 2ab + b².