Question

Evaluate the following
$(97)^2$
A) 9409
B) 9049
C) 9949
D) 4949

Answer 1

Hint:
This question can be easily solved by using the difference of squares formula, which has a lot of applications in the field of mathematics, statistics, physics, et cetera. This formula helps us evaluate the squares of large numbers by using the most basic squares. It helps in saving time, as we do not need to multiply the large quantities by making a multiplication table and things like that.

Formula Used:
We are going to use one and just one formula for the calculation of the result, which is the difference of the squares formula, and it is written as -
\[{a^2} - {b^2} = \left( {a + b} \right)\left( {a - b} \right)\]

Complete step by step solution:
Now, in solving questions like these it is always advised to take the nearest number divisible by 10 (say t) to our original number (say n) and note it down. Then, what we need to do is find the difference (say d) between t and n. Then, subtract d from n and note it down (s), add d to n and note it down (a). Then, multiply the two. And finally, add \[{{\rm{d}}^2}\] to \[a \times s\]. So, the final equation becomes:
\[\left( {\left( {a \times s} \right) + {d^2}} \right)\]
or, \[\left( {n - d} \right) \times \left( {n + d} \right) + {d^2}\]
According to the question, the given number \[\left( n \right)\] is \[97\].
So, for this number, here the nearest number divisible by \[10\] is \[100{\rm{ }}\left( t \right)\].
The difference between the two numbers (\[97\]and\[100\]) is\[3{\rm{ }}\left( d \right)\].
Applying the above expression to evaluate the result, we have:
\[{\left( {97} \right)^2} = \left( {97 - 3} \right)\left( {97 + 3} \right) + {3^2} = 94 \times 100 + 9 = 9400 + 9 = 9409\]

Hence, the correct option is A.

Note:
In solving these kinds of questions, it is best to first identify the nearest number divisible by 10. Then, just apply the above written square difference method to get to the answer without having to do a whole lot of writing by traditionally multiplying the number.