Quadratic Equations with Complex Roots Simplified Revision Notes for Leaving Cert Mathematics

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Quadratic Equations with Complex Roots

Introduction

Quadratic equations can sometime have complex roots which can be solved using the quadratic formula. Given a quadratic equation in the form $az^2+bz+c=0$ , the roots can be found using :

z=\frac{-b\pm\sqrt{b^2-4ac}}{2a}

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You won't be able to derive the roots directly in your calculator as you will get an error because of the negative number under the square root. For these types of questions you will have perform the calculations manually.

Example

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Find the roots of the following equation :

2z^2+10z+13=0

First identify the coefficients of each of three terms :

a=2,b=10,c=13

Now apply the formula :

\begin{align*} z&=\frac{-(10)\pm\sqrt{(10)^2-4(2)(13)}}{2(2)} \\\\ &=\frac{-10\pm\sqrt{-4}}{4} \\\\ &= -\frac{10}{4} \pm \frac{\sqrt{-4}}{4} \\\\ &= -\frac{5}{2} \pm \frac{-2i}{4} \\\\ &= -\frac{5}{2} \pm \frac{-1}{2}i \\\\ z&=-\frac{5}{2} + \frac{1}{2}i, z=-\frac{5}{2} - \frac{1}{2}i \end{align*}

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