Question

Evaluate the following by using the identity:
\[185\times 185-115\times 115\]

Answer 1

Hint: The given expression when observed carefully, contains the difference of squares of two numbers. To solve this question, we will use an identity given as \[{{a}^{2}}-{{b}^{2}}=\left( a+b \right)\left( a-b \right).\] We will assume the variable a as 185 and variable b as 115 and then by using the above stated identity, we will solve this question.

Complete step by step answer:
We are asked to evaluate the given value using an identity.
\[185\times 185-115\times 115......\left( i \right)\]
To proceed, let us assume the variables as a = 185 and b = 115.
We have an identity stated as below.
\[{{a}^{2}}-{{b}^{2}}=\left( a+b \right)\left( a-b \right)\]
Considering (i), when a = 185 and b = 115, \[185\times 185-115\times 115\] is nothing but \[{{a}^{2}}-{{b}^{2}}.\]
Using the above stated identity, we have,
\[{{a}^{2}}-{{b}^{2}}=\left( a+b \right)\left( a-b \right)\]
Substituting a = 185 and b = 115, we get,
\[\Rightarrow {{a}^{2}}-{{b}^{2}}=\left( 185+115 \right)\left( 185-115 \right)\]
Now, after adding the terms and subtracting the terms, we have the simplified equation as
\[\Rightarrow {{a}^{2}}-{{b}^{2}}=300\,\times 70\]
We will multiply the terms to simplify further and get
\[\Rightarrow {{a}^{2}}-{{b}^{2}}=21000\]
Hence, the value of \[185\times 185-115\times 115\] is 21000.

Therefore, the answer to this question is 21000.

Note: Another way to solve this question can only be done without using any identity. It is as below.
\[185\times 185-115\times 115\]
First, we will find the square of both the numbers as below,
\[\Rightarrow 185\times 185=34225\]
\[\Rightarrow 115\times 115=13225\]
Now, we can directly subtract the above obtained results and we will get
\[\Rightarrow 185\times 185-115\times 115=34225-13225\]
\[\Rightarrow 185\times 185-115\times 115=21000\]
Hence, our answer is correct.