Page 60 - Start Up Mathematics_8 (Non CCE)

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2
Using the identity, (x + y) = x + 2xy + y 2
2
2
2
2
(108) = (100 + 8) = (100) + 2(100)(8) + (8) = 10,000 + 1,600 + 64 = 11,664
2
\ (108) = 11,664
2
2
2
Sometimes using the identity (x – y) = x – 2xy + y makes it easier to find the square of a number. For
example, if we wish to find the square of 98, first we write 98 as 98 = 100 – 2.
3. Diagonal Method
Diagonal method is applicable to two or more digit numbers.

Step 1: Let there be a number x say, 23. It has 2 digits. Draw twice the number
of squares, i.e., 2 × 2 = 4 squares. Divide each square by a diagonal 2 3
into two parts.

Step 2: Starting from top left square, write the number along the side of the 2
squares vertically and horizontally. Remember each digit of the number
occupies one square. 3
Step 3: Multiply each digit on the left of the square with the digit on the top 2 3
and place the product in the corresponding square.
2 0
The square has been divided into two parts by the diagonal. The tens 4
digit of the product should be placed above the diagonal and the units 3
digit below the diagonal in that square.

If the product is a single digit number, place 0 in the space above the diagonal.

Step 4: Repeat the multiplication 2 3
i.e., (a) Row 1 × Column 1, Row 1 × Column 2, ... 2 0 0
(b) Row 2 × Column 1, Row 2 × Column 2, ... 4 6
 3 0 6 0 9
and so on.
Step 5: Starting from right bottom, add the digits diagonally. If the sum 2 3
is a two-digit number, encircle the units digit and carry over the 2 0 0
tens digit to the sum of the next diagonal. 4 6
Step 6: Arrange the encircled digits starting from extreme left. This 0 0 3 0 0
gives the square of the given number. 4 6 9
2
\ (23) = 529 0 6
+ 1 0 9
Easy Methods of Finding Squares 5 + 6
1 2
1. For numbers ending in 5
The rule says, ‘‘One more than previous’’.
25
( xx +
1)
If a number is in the form x5, its square will be   .
2
For example, in (35) , x = 3. 1st part 2nd part
To get the first part of the answer, multiply 3 with (3 + 1), i.e., 3 × (3 + 1) = 3 × 4 = 12.
12 25
The answer is   .
1st part 2nd part
2
So, (35) = 1,225.
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