Question

How to find two square roots of a number?

Answer 1

Hint: For this question we are asked how we can find two square roots a certain number. So, for solving this question we will explain the concept of the possible square roots for a given number or question in a certain interval or the conditions where the root exists and also we will use an example to make the explanation much easier.

Complete step by step solution:
Generally, for any given question or a number \[a\] (in \[R\] or \[C\]) that is in real numbers or complex numbers, if \[r\] is a square root of \[a\] then \[-r\] is also square root of the number \[a\].
If \[a\in R\] and \[a>0\] then \[a\] has two real square roots positive and negative ones. The positive square root we represent as \[\sqrt{a}\] and the negative root we represent as \[-\sqrt{a}\].
If the number \[a\in R\] and \[a<0\] then the number has two pure imaginary square roots, \[\sqrt{-a}\times i\] and \[-\sqrt{-a}\times i\].
So, we will take an example with the number as \[2\].
The number \[2\] is greater than zero and belongs to real numbers. So, it has two real square roots which can be represented as follows.
\[\Rightarrow \sqrt{2}\text{ and -}\sqrt{2}\]
Now, we will take an example which is \[-2\].
The number \[-2\] is less than zero and belongs to real numbers then the number has two pure imaginary square roots which can be represented as follows.
\[\Rightarrow \sqrt{2}i\text{ and -}\sqrt{2}i\]
Therefore, in this way we find the two square roots of a number.

Note: Students must have good knowledge in the concept of square roots of a number and also good knowledge in complex numbers. For example when the given number is zero then there is one (repeated) square root which is zero itself. Students should not do mistakes in calculation like for example in the second case if we write the solution as \[\Rightarrow \sqrt{-2}i\text{ and -}\sqrt{-2}i\] instead of \[\Rightarrow \sqrt{2}i\text{ and -}\sqrt{2}i\] then our solution will be wrong.