Short-cut for squaring of a Number

Short-cut for squaring of a Number

Squaring:
Squaring is multiplying the numbers by itself.

Examples:
9² = 9 × 9 = 81.
145² = 145 × 145 = 21025.

Table of square of some numbers:

Short-cut Method for Squaring a Number:
Let a and b denote the two numbers.
We can write,
a² = a²
or, a² = a² - b² + b²

[As addition and subtraction of the same quantity do not change the number.]
or, a² = (a² - b²) + b²
or, a² = [(a + b) × (a - b)] + b²

Example 1:
Find the value of 97².

Solution:
We can write, 97² = (93 + 3) × (97 - 3) + 32.
Here, a = 97, b = 3, we can choose value of b as per our convenient.
Now, 97² = 100 × 94 + 9;
or, 97² = 9400 + 9 = 9409.

Example 2:
Find the value of 1213².

Solution:
1213² = (1213 + 13)(1213 - 13) + 132
1213² = (1200 × 1226) + 169
= 1471200 + 169 = 1471369.

Squaring of a number ending in 5:
The square of a number ending in 5, the last two digits will always be 25. The digits before that in the answer will be got by multiplying the leading up to digit 5 in the number 1 more than itself.

Illustration:
85² = _ _25.
The missing digits in above answer will be got by 8 × (8 + 1) = 8 × 9 = 72. Hence, the square of 85 is given by 7225. i.e.
85² = 7225.

Square of number consisting of 9s only:
Let the number consists of n number of 9s.
Write down (n - 1) number of 9s, followed by one 8, then (n-1) number of zeros and finally annex 1 at the end.

Example:
(999999)² = ?

Solution:
(999999)² = 999998000001[square is consisting five 9s then one 8 then five zeros then 1.]

Square of number consisting of 1s only:
11² = 121.
111² = 12321.
1111² = 1234321

We increase number from 1 to number of digits contains in given number then decrease it to one.

Click Here for Solved Examples on Number System