Question

Find the square of 95 using the identity.

Answer 1

Hint: We can find the square of a number using many methods, but we are going to use the mathematical identity to find its square. The identity we will be using is
$ \to {\left( {a - b} \right)^2} = {a^2} - 2ab + {b^2}$
Here, we need to express 95 as the difference of two numbers and then use the above identity.

Complete step-by-step answer:
In this question, we have to find the square of 95 using the identity.
Now, we can find the square using many different methods like log method, but in this question, we are going to use the mathematical identity to find the square of a number.
The identity is
$ \to {\left( {a - b} \right)^2} = {a^2} - 2ab + {b^2}$
So, we are going to use this identity to find the square of 95.
For that, we need to express 95 as a difference of two numbers. Now, we could do this in many ways like $\left( {97 - 2} \right)$, $\left( {98 - 3} \right)$ but we need to express it in such a way that we do not need to use the calculator. So, we will be expressing 95 as $\left( {100 - 5} \right)$.
Hence, $a = 100$ and $b = 5$. So, using the identity we will get
$ \to {\left( {95} \right)^2} = {\left( {100 - 95} \right)^2}$
                 $
   = {\left( {100} \right)^2} - 2\left( {100} \right)\left( 5 \right) + {\left( 5 \right)^2} \\
   = 10000 - 1000 + 25 \\
   = 9000 + 25 \\
   = 9025 \\
 $
Hence, the square of 95 is 9025.

Note: We can also find the square of 95 using another property, that is ${\left( {a + b} \right)^2} = {a^2} + 2ab + {b^2}$.
Here, we need to express 95 as a sum of two numbers instead of difference of two numbers. So, we can express 95 as $\left( {90 + 5} \right)$. Hence, using the identity we will get,
$ \to {\left( {95} \right)^2} = {\left( {90 + 5} \right)^2}$
                 $
   = {\left( {90} \right)^2} + 2\left( {90} \right)\left( 5 \right) + {\left( 5 \right)^2} \\
   = 8100 + 900 + 25 \\
   = 9000 + 25 \\
   = 9025 \\
 $
Hence, the square of 95 is 9025.