Question

Using a shortcut method, find the squares of the following numbers.
A) 35
B) 65
C) 145
D) 95

Answer 1

Hint:
One of the easiest methods to calculate the square of a number is by representing it into a sum of two numbers whose squares can be easily calculated. Then use the identity \[{(a + b)^2} = {a^2} + {b^2} + 2ab\] to find the square of the number. There are many other shortcut methods, but they vary with the number of digits used. The identity method works with all numbers as long as you can represent it into a sum of two numbers whose squares can easily be calculated.

Complete step by step solution:
A) 35
35 can be expressed as the sum of 35 and 5.
So, \[{35^2} = {(30 + 5)^2}\]
This is of the form \[{(a + b)^2}\].
So, let’s expand it using the identity \[{(a + b)^2} = {a^2} + {b^2} + 2ab\]
\[
   = {30^2} + {5^2} + 2 \times 30 \times 5 \\
   = 900 + 25 + 300 \\
   = 1225 \\
\]

B) 65
65 can be expressed as the sum of 60 and 5.
So, \[{65^2} = {(60 + 5)^2}\]
This is of the form \[{(a + b)^2}\].
So, let’s expand it using the identity \[{(a + b)^2} = {a^2} + {b^2} + 2ab\]
\[
   = {60^2} + {5^2} + 2 \times 60 \times 5 \\
   = 3600 + 25 + 600 \\
   = 4225 \\
\]

C) 145
145 can be expressed as the sum of 150 and -5.
So, \[{145^2} = {(150 - 5)^2}\]
This is of the form \[{(a + b)^2}\].
So, let’s expand it using the identity \[{(a + b)^2} = {a^2} + {b^2} + 2ab\]
\[
   = {150^2} + {( - 5)^2} + 2 \times 150 \times ( - 5) \\
   = 22500 + 25 - 1500 \\
   = 21025 \\
\]

D) 95
95 can be expressed as the sum of 100 and -5.
So, \[{95^2} = {(100 - 5)^2}\]
This is of the form \[{(a + b)^2}\].
So, let’s expand it using the identity \[{(a + b)^2} = {a^2} + {b^2} + 2ab\]
\[
   = {100^2} + {( - 5)^2} + 2 \times 100 \times ( - 5) \\
   = 10000 + 25 - 1000 \\
   = 9025 \\
 \]

Note:
Formula for calculating the square ending with 5 is easy. Suppose we have to calculate the square of 85. Then follow these steps:
• Multiply 5 by 5 and put composite digit 25 on the right-hand side.
• Add 1 to the upper left-hand side digit i.e. 8 i.e. \[8 + 1 = 9\]
• Multiply 9 to the lower hand digit 8, i.e. \[9 \times 8 = 72\]
• Our answer is 7225