Question

Simplify the following using identity ${(1025)^2} - {(975)^2}$.

Answer 1

Hint: According to the question, we have to find the value or simplify the given expression ${(1025)^2} - {(975)^2}$ with the help of using identity.
So, first of all we have to identify the identity to use for the given expression and as we can see that there are only two terms in the expressions ${(1025)^2}$ and ${(975)^2}$means both of them are given in square form and the term ${(975)^2}$ is subtracted by the term ${(1025)^2}$ so there is only one identity for this condition/formula which is given below:
${(a)^2} - {(b)^2} = (a + b) \times (a - b)............(1)$
So, on comparing the terms of the given expression with the formula we can see that the term in the formula or identity ${(a)^2}$ belongs to ${(1025)^2}$ and the term of identity ${(b)^2}$ belongs to ${(1025)^2}$ hence, easily with the help of formula above we can obtain the value or simplify the expression using identity.

Complete step-by-step answer:
Step 1: First of all we have to identify the identity to apply for the given expression and as we can see that the first term of the given expression ${(1025)^2}$ is same as ${(a)^2}$ in the form of formula/identity (1) as mentioned in the solution hint and same as ${(b)^2}$ second term of the given expression ${(975)^2}$ is same as in the form of formula/identity (1)
Step 2: On applying the identity (1) as mentioned in the solution hint in the expression given.
$ \Rightarrow {(1025)^2} - {(975)^2} = (1025 + 975) \times (1025 - 975)$…………………..(2)
Step 3: On solving the obtained expression (2) as obtained in step 2.
$
   \Rightarrow {(1025)^2} - {(975)^2} = (2000) \times (50) \\
   \Rightarrow {(1025)^2} - {(975)^2} = 100000
 $

Hence, with the help of the identity as mentioned in the solution hint we have obtained the simplification of the given expression: ${(1025)^2} - {(975)^2} = 100000$

Note: Alternative method: we can also solve the given expression without applying the identity as explained below:
Step 1: First of all we have to find the square of both of the terms ${(1025)^2}$ and ${(975)^2}$ as given in the expression.
$
   \Rightarrow {(1025)^2} = 1025 \times 1025 \\
   \Rightarrow {(1025)^2} = 1050625
 $
Now, the same as we can find the square of the term ${(975)^2}$ as given in the expression.
$
   \Rightarrow {(975)^2} = 975 \times 975 \\
   \Rightarrow {(975)^2} = 950625
 $
Step 2: Now, we have to substitute the obtained values in the expression to simplify it.
$
   = 1050625 - 950625 \\
   = 100000
 $
Hence, we have simplified the given expression ${(1025)^2} - {(975)^2} = 100000$