Question

The value of squared \[19\] lies between which two consecutive integers ?

Answer 1

Hint: In the above problem, we have to find the two consecutive integers between which the square of the number \[19\] lies. Since the square of any real number is positive or non-negative therefore we can say that those two consecutive integers are positive integers because \[19 > 0 \Rightarrow {19^2} > 0\] i.e. the two numbers are natural numbers.
Now before getting to know those two consecutive natural numbers between which the square of \[19\] lies, first we have to find the value of the square of \[19\] .
To find the square of \[19\] we do not have to do the calculation directly. Instead of that we can find the square of \[19\] by using the well known mathematical formula of \[{\left( {a - b} \right)^2} = {a^2} + {b^2} - 2ab\] .


Complete step by step solution:
First, we have to find the square of \[19\]
Since we can write \[19\] as
 \[ \Rightarrow 19 = 20 - 1\]
Therefore, squaring both sides we get
\[ \Rightarrow {\left( {19} \right)^2} = {\left( {20 - 1} \right)^2}\]
Now, using the mathematical identity \[{\left( {a - b} \right)^2} = {a^2} + {b^2} - 2ab\]
We can write,
\[ \Rightarrow {\left( {20 - 1} \right)^2} = 2{0^2} + {1^2} - 2 \cdot 20 \cdot 1\]
That gives,
\[ \Rightarrow {\left( {20 - 1} \right)^2} = 20 \times 20 + 1 - 40\]
So we get,
\[ \Rightarrow {\left( {20 - 1} \right)^2} = 400 + 1 - 40\]
Again,
\[ \Rightarrow {\left( {20 - 1} \right)^2} = 360 + 1\]
\[ \Rightarrow {\left( {20 - 1} \right)^2} = 361\]
Therefore,
\[ \Rightarrow {19^2} = {\left( {20 - 1} \right)^2} = 361\]
Here we have found the square of \[19\] to be \[361\].
Now we can easily know the two consecutive numbers.
Since in the number system, \[360\] , \[361\] and \[362\] are three consecutive natural numbers where \[361\] is the square of \[19\] .
As we can see that the number \[361\] lies between two consecutive natural numbers \[360\] and \[362\].
Therefore the value of squared \[19\] lies between the two positive integers \[360\] and \[362\] .

Note:
We can use the various mathematical identities such as \[{\left( {a - b} \right)^2} = {a^2} + {b^2} - 2ab\] and \[{\left( {a + b} \right)^2} = {a^2} + {b^2} + 2ab\] to find the square of those numbers which are close to any number whose square we already know or they are easy and quick to calculate.
For example,
\[{\left( {198} \right)^2}\] can be written as \[{\left( {200 - 2} \right)^2}\] .
\[{\left( {103} \right)^2}\] can be written as \[{\left( {100 + 3} \right)^2}\] .
Also multiplication of two numbers can be done by an easy method after using the mathematical identity \[\left( {a - b} \right)\left( {a + b} \right) = {a^2} - {b^2}\] and \[\left( {x + a} \right)\left( {x + b} \right) = {x^2} + \left( {a + b} \right)x + ab\] for those numbers which are close to any number whose square we already know or we can easily calculate.
For example
\[98 \times 102\] can be written as \[\left( {100 - 2} \right)\left( {100 + 2} \right)\] .
\[101 \times 103\] can be written as \[\left( {100 + 1} \right)\left( {100 + 3} \right)\] .
\[97 \times 101\] can be written as \[\left( {100 - 3} \right)\left( {100 + 1} \right)\] .