Welcome to Day 7

We have already covered a lot and today seems to be one of my favourite topics and I am sure you are going to enjoy all the neat tricks I am going to teach. Have fun!

Let’s learn square using the add and subtract method.
                       
In this method we again choose a base. This method is nothing but the above method in a more simplified manner for calculating squares.
           
7 x 7 = (7-3)/3 x 3 = 4/9
By subtract and add method. Taking the base 10
            7 – 3
          7 – 3
          (7-3)/ (-3 x -3) = 49

Look at the following examples

12 x 12 = (12 + 2)/2 x 2 = 14/ 4

14 x 14 = (14 + 4)/4 x 4 = 18/16 = 19/6 (remember the base is 10)

91 x 91 = (91 – 9)/9 x 9 = 82/81

95 x 95 = (95 – 5)/5 x 5 = 90/25

108 x 108 = (108 + 8)/8 x 8 =116/64

988 x 988 = (988 – 12)/12 x 12 = 976/144 (The base is 1000. Therefore we do not need to carry 1 of 144 on the left hand side.

45 x 45 = (45 – 5)/5 x 5= 40/25 = 20/25 (The base is taken as 50 (100/2). Therefore divide by 40 by 2)

45 x 45 = (45 – 5)/5 x 5 = 40/25 = 200/25 = 202/5 (The base is taken as 50 (10 x 5). Therefore we multiply 40 x 5 = 200 and the carry forward 2 from left hand side to right hand side as the base is 10)

Squares using the formula:
(a + b) (a + b) = a2 + 2ab + b2
(a - b) (a - b) = a2 - 2ab + b2

This is simple enough to understand.

97 x 97 = (100 -3)(100-3) = 10000 – 600 + 9

108 x 108 = (100 + 8) (100 + 8) = 10000 + 1600 + 64

29 x 29 = (30 – 1) (30 – 1) = 900 -60 + 1 = 841

49 x 49 = (50 -1) (50 – 1) = 2500 – 100 + 1 = 2401

78 x 78 = (80 – 2) (80 – 2) = 6400 – 320 + 4 = 6084

Let’s try a different method to find squares now:

Method:

  1. Square the units digit.
  2. Tens digit x units digit x 2
  3. Square of tens digit

Examples

42 x 42 = 16/16/4 = 1764 (we carry forward 1 from the centre calculation to the left hand side)

Method:

  1. Square the units digit. We get 4
  2. Tens digit x units digit x 2. We get 16
  3. Square of tens digit is 16.

23 x 23 = 4/12/9 = 5/2/9 (we carry forward 1 from the centre calculation to the left hand side)

Method:

  1. Square the units digit - 9
  2. Tens digit x units digit x 2 - 12
  3. Square of tens digit - 4

 

62 x 62 = 36/24/4=38/4/4

Method:

  1. Square the units digit - 4
  2. Tens digit x units digit x 2 - 24
  3. Square of tens digit – 36

73 x 73 = 49/42/9 =53/2/9

26 x 26 = 4/24/36= 6/7/6

(We carry forward 3 to 24. That makes it 27. We put down 7 and carry forward to the left 2. Thus we get 6 on the extreme left. And our answer is 676)

34 x 34 = 9 / 24 / 16 = 1156

42 x 42 = 16/16/4 = 1764

63 x 63 = 36/36/9 = 3969


We learned 3 different ways to find out square. Choose the one you think was simpler and stick to it. Do not try to learn all the method, it will just confuse more and more.

I hope this lesson was fun and you enjoyed it as much as I enjoyed writing it down for you!

See you tomorrow.

Rahul Makhija

If you have missed any of the previous days, then you can visit them now:

Day 1
Day 2
Day 3
Day 4
Day 5
Day 6



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