# Complex Numbers and Quadratic Equations

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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 non-real 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 non-real complex numbers.

We see that with complex numbers, we now have the tool to handle and solve quadratic equations with non-real 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}

Complex Numbers
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