Question

Find the roots for the given equation ${{x}^{2}}+3=0$.
(a) $\pm 3i$,
(b) $\pm 9i$,
(c) $\pm \sqrt{3}i$,
(d) $\pm \sqrt{4}i$.

Answer 1

Hint: We start solving the problem by finding the range of the values of the function $\left( {{x}^{2}}+3 \right)$. We then recall the definition of complex number and the value of the complex number $i=\sqrt{-1}$. We then use the value of complex number $i$ while solving the equation ${{x}^{2}}+3=0$ to get the required solution for the roots.

Complete step-by-step answer:
According to the problem, we are given an equation ${{x}^{2}}+3=0$ and we need to find the roots for this equation.
We know that the value of the function $\left( {{x}^{2}}+3 \right)$ is always positive as the function ${{x}^{2}}$ is always greater or equal to 0 and a positive number 3 is added to it which makes the function $\left( {{x}^{2}}+3 \right)$ always greater than or equal to 0. We know that the roots will be complex numbers for such types of functions.
Before solving for the roots, we recall the concepts about the complex number.
A complex number is a number that can be written in the form of $a+ib$, here a, b are real numbers and i is a solution of the equation ${{x}^{2}}=-1$. This is because no real value satisfies the equation ${{x}^{2}}+1=0$ or ${{x}^{2}}=-1$, hence ‘i’ is called an imaginary number. So, this tells us that $i=\sqrt{-1}$. For the complex number $a+ib$, a is considered as real part and b as imaginary part, Despite the historical nomenclature “imaginary”, complex numbers are regarded in the mathematical sciences as just as “real” as real numbers and are fundamental in any aspect of scientific description of the natural world.
Now, let us solve for the roots of ${{x}^{2}}+3=0$.
So, we have ${{x}^{2}}=-3$.
$\Rightarrow {{x}^{2}}=-1\times 3$.
$\Rightarrow x=\pm \sqrt{\left( -1\times 3 \right)}$.
We know that $\sqrt{ab}=\sqrt{a}\times \sqrt{b}$.
\[\Rightarrow x=\pm \left( \sqrt{-1}\times \sqrt{3} \right)\].
\[\Rightarrow x=\pm \left( \sqrt{3}i \right)\].
So, we have found the roots of the equation ${{x}^{2}}+3=0$ as $\pm \left( \sqrt{3}i \right)$.

So, the correct answer is “Option (c)”.

Note: We can also solve for the roots of the given equation ${{x}^{2}}+3=0$ by using the fact that the roots of the quadratic equation $a{{x}^{2}}+bx+c=0$ are $\dfrac{-b\pm \sqrt{{{b}^{2}}-4ac}}{2a}$, which also gives us the same answer. Before solving the problem, we need to check whether the roots we need to find are real or complex. If the roots are needed to be strictly real, we should not proceed as we solve this problem. We need to keep in mind that any equation is solvable using the complex numbers.