Question

What is the value of \[{{248}^{2}}-{{249}^{2}}\]?

Answer 1

Hint: In this type of question we have to use the concept of algebraic identities. We know that, we can simplify the expression of the type \[{{a}^{2}}-{{b}^{2}}\] by using the algebraic identity \[{{a}^{2}}-{{b}^{2}}=\left( a-b \right)\left( a+b \right)\]. By using this identity we can easily find the value of the required expression.

Complete step by step answer:
Now we have to find the value of the expression \[{{248}^{2}}-{{249}^{2}}\]
Let us consider the expression
\[\Rightarrow {{248}^{2}}-{{249}^{2}}\]
We can easily observe that the given expression is of the form \[{{a}^{2}}-{{b}^{2}}\]
As we know that one of the algebraic identity which is given by \[{{a}^{2}}-{{b}^{2}}=\left( a-b \right)\left( a+b \right)\].
Hence, by using this identity in the above expression we can write
\[\Rightarrow {{248}^{2}}-{{249}^{2}}=\left( 248-249 \right)\left( 248+249 \right)\]
Now, by performing addition and subtraction we get,
\[\Rightarrow {{248}^{2}}-{{249}^{2}}=\left( -1 \right)\left( 497 \right)\]
Finally by performing multiplication we get the value of the expression as
\[\Rightarrow {{248}^{2}}-{{249}^{2}}=-497\]
Hence, the value of the expression \[{{248}^{2}}-{{249}^{2}}\] is \[-497\].

Note: In this type of question students have to remember all the algebraic identities. Also students have to take care in the calculations. One of the students may solve the same example by using another identity i.e. \[{{\left( a-b \right)}^{2}}={{a}^{2}}-2ab+{{b}^{2}}\] as follows:
Let us consider the expression
\[\Rightarrow {{248}^{2}}-{{249}^{2}}\]
Now, we know that we can express \[248\] and \[249\]as \[\left( 250-2 \right)\] and \[\left( 250-1 \right)\] respectively
Hence, we can rewrite the above expression as
\[\Rightarrow {{248}^{2}}-{{249}^{2}}={{\left( 250-2 \right)}^{2}}-{{\left( 250-1 \right)}^{2}}\]
By using the identity \[{{\left( a-b \right)}^{2}}={{a}^{2}}-2ab+{{b}^{2}}\] we can write the above expression as
\[\Rightarrow {{248}^{2}}-{{249}^{2}}=\left( 62500-1000+4 \right)-\left( 62500-500+1 \right)\]
On simplification we get,
\[\begin{align}
  & \Rightarrow {{248}^{2}}-{{249}^{2}}=61504-62001 \\
 & \Rightarrow {{248}^{2}}-{{249}^{2}}=-497 \\
\end{align}\]
Here, students have to remember that as we are subtracting larger numbers from the smaller one and the answer must be a negative number.
Hence, the value of the expression \[{{248}^{2}}-{{249}^{2}}\] is \[-497\].