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:

\[\begin{align}&\alpha ,\beta  = \frac{{ - 2 \pm \sqrt {4 - 25} }}{2}\\&\quad\;\;\,= \frac{{ - 2 \pm i\sqrt {21} }}{2}\end{align}\]

Download practice questions along with solutions for FREE:
Complex Numbers
grade 10 | Questions Set 1
Complex Numbers
grade 10 | Answers Set 1
Complex Numbers
grade 10 | Answers Set 2
Complex Numbers
grade 10 | Questions Set 2
Download practice questions along with solutions for FREE:
Complex Numbers
grade 10 | Questions Set 1
Complex Numbers
grade 10 | Answers Set 1
Complex Numbers
grade 10 | Answers Set 2
Complex Numbers
grade 10 | Questions Set 2
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