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The imaginary number i is defined such that ${{i}^{2}}=-1$. What is the value of $\left( 1-i\sqrt{5} \right)\left( 1+i\sqrt{5} \right)$?
(a) $\sqrt{5}$
(b) $5$
(c) $6$
(d) $\sqrt{6}$

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Last updated date: 19th Jul 2024
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Answer
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Hint: First of all, we are going to simplify the given expression $\left( 1-i\sqrt{5} \right)\left( 1+i\sqrt{5} \right)$. As you can see that this expression is written in the form of the following algebraic identity: $\left( a-b \right)\left( a+b \right)$ and we know the expansion of this expression which is equal to: $\left( a-b \right)\left( a+b \right)={{a}^{2}}-{{b}^{2}}$. After that, we need to use the square of the imaginary number “i” which is given above: ${{i}^{2}}=-1$.

Complete step-by-step solution:
The expression given in the above problem is as follows:
$\left( 1-i\sqrt{5} \right)\left( 1+i\sqrt{5} \right)$
Now, the above expression is written in the form of following algebraic identity:
$\left( a-b \right)\left( a+b \right)={{a}^{2}}-{{b}^{2}}$
Substituting $a=1$ and $b=i\sqrt{5}$ in the above equation and we get,
$\left( 1-i\sqrt{5} \right)\left( 1+i\sqrt{5} \right)={{\left( 1 \right)}^{2}}-{{\left( i\sqrt{5} \right)}^{2}}$
We know that when we do the square of a square root then the square root sign will get removed so the square of $\left( \sqrt{5} \right)$ is equal to 5. Also, squaring one will give us 1. Now, using this transformation in the R.H.S of the above equation and we get,
$\Rightarrow \left( 1-i\sqrt{5} \right)\left( 1+i\sqrt{5} \right)=1-{{i}^{2}}\left( 5 \right)$
It is given in the above problem that ${{i}^{2}}=-1$ so using this relation in the above equation we get,
$\Rightarrow \left( 1-i\sqrt{5} \right)\left( 1+i\sqrt{5} \right)=1-\left( -1 \right)\left( 5 \right)$
We know the multiplication of negative sign with negative sign will give us positive sign so multiplying -1 with negative sign in the R.H.S of the above equation will give us +1.
$\begin{align}
  & \Rightarrow \left( 1-i\sqrt{5} \right)\left( 1+i\sqrt{5} \right)=1+5 \\
 & \Rightarrow \left( 1-i\sqrt{5} \right)\left( 1+i\sqrt{5} \right)=6 \\
\end{align}$
From the above solution, we have simplified the given expression to 6.
Hence, the correct option is (c).

Note: The alternate approach to the above solution is that instead of using the algebraic identity $\left( a-b \right)\left( a+b \right)={{a}^{2}}-{{b}^{2}}$, we can just multiply the two brackets. The catch here is that if you use the algebraic identity then your time got saved in the exam and when you multiply the two brackets then you will get the exact same expression that you got while using the algebraic identity.