Complex Numbers and Quadratic Equations
Consider an arbitrary quadratic equation where the coefficients are all real:
\[a{x^2} + bx + c = 0\]
We have seen that the nature of the roots is decided by the sign of the discriminant \(D = {b^2}  4ac\).

If \(D > 0\), then the roots are real and distinct.

If \(D = 0\), then the roots are real and identical

If \(D < 0\), then the roots are nonreal complex numbers.
Let us take an example. Consider the quadratic equation
\[{x^2} + x + 1 = 0\]
Using the quadratic formula, the roots of this equation are given by
\[\begin{align}&\alpha ,\beta = \frac{{  1 \pm \sqrt {1  4} }}{2} = \frac{{  1 \pm \sqrt {  3} }}{2}\\&\;\;\,\quad = \frac{{  1 \pm i\sqrt 3 }}{2}\end{align}\]
Thus, the roots are nonreal complex numbers.
We see that with complex numbers, we now have the tool to handle and solve quadratic equations with nonreal roots.
Example 1: Determine the roots of the following quadratic equation:
\[{x^2} + 2x + 5 = 0\]
Solution: We have:
\[\begin{align}&\alpha ,\beta = \frac{{  2 \pm \sqrt {4  25} }}{2}\\&\quad\;\;\,= \frac{{  2 \pm i\sqrt {21} }}{2}\end{align}\]