Question

Find the square of the following numbers:
95.

Answer 1

Hint: First we will write the given number in terms of the number for which we know the square. Now we will assume those 2 numbers as variable and derive a formula to use here to find the square of the number. So first find 2 numbers whose mathematical operation gives this number.

Complete step-by-step solution -
Given number for which we need to find square is given as:
95
We must find 2 numbers whose square is known and by applying mathematical operations we must get 95.
By basic trial and error we get the number as:
100, 5
So, the operation we choose here to get 95 is:
Subtraction
Our aim in the question is to find value of:
${95^2}$
By above 2 numbers and given operation, we say:
$95 = 100 - 5$
By substituting this into the square given above, we get:
${\left( {100 - 5} \right)^2}$.
Now we need two variables to denote these numbers:
a, b
Let the value of a is assumed to be as 100:
$100 = a$
Let the value of b is assumed to be as 5:
$5 = b$
By substituting this both into the square, we get it as:
 ${\left( {a - b} \right)^2}$
As we know square can be written as product of it twice:
$\left( {a - b} \right)\left( {a - b} \right)$
By distributive law: $a\left( {b + c} \right) = ab + ac,$we can write it:
$a\left( {a - b} \right) - b\left( {a - b} \right)$
By applying distributive law again, we get the terms as:
${a^2} - ab - ba + {b^2}$
As a, b are numbers we know that ab=ba, is true always.
By substituting above equation, we get the terms of equation as:
${a^2} - ab - ab + {b^2}$
By simplifying the above terms, we can write it in form:
${a^2} - 2ab + {b^2}$
Equating it to its original term, we get the equation as:
${\left( {a - b} \right)^2} = {a^2} - 2ab + {b^2}$
By substituting \[{\rm{a}} = {\rm{1}}00,{\rm{ b}} = {\rm{5}}\] in above equation, we get it as:
${\left( {100 - 5} \right)^2} = {\left( {100} \right)^2} - 2\left( {100} \right)\left( 5 \right) + {\left( 5 \right)^2}$
By simplifying al the squares in the equation, we get:
${95^2} = 10000 - 1000 + 25$
Bu simplifying the addition terms, we get the equation as:
${95^2} = 10025 - 1000$
By simplifying the above terms, we get it as:
${95^2} = 9025$
Therefore, the required value is 9025.

Note: Alternative method-1: Write 95 as \[\left( {{\rm{9}}0 + {\rm{5}}} \right)\] and use ${\left( {a + b} \right)^2}$formula.
Alternative method-2: Write ${95^2} {\rm{ as }} {{\rm{5}}^{\rm{2}}} \times {\rm{1}}{{\rm{9}}^{\rm{2}}}$ which is ${\left( {12 - 7} \right)^2} \times {\left( {12 + 7} \right)^2}$ and then solve it as you know squares of 12, 7 you can solve this equation easily. Just multiply after calculating the values of squares.