Quadratic Equations with Complex Roots (Leaving Cert Mathematics): Revision Notes

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Quadratic Equations with Complex Roots

Introduction

Quadratic equations can sometimes have complex roots which can be solved using the quadratic formula. Given a quadratic equation in the form ax2+bx+c=0ax^2+bx+c=0, the roots can be found using :

x=−b±b2−4ac2ax=\frac{-b\pm\sqrt{b^2-4ac}}{2a}
infoNote

You may be able to find complex roots using your calculator, but you must still be able to show full working using the quadratic formula for exam marks.

chatImportant

Complex roots occur when the discriminant is negative, i.e. b2−4ac<0b^2 - 4ac < 0, leading to solutions involving ii where i2=−1i^2 = -1.

Example

infoNote

Find the roots of the following equation :

2x2+10x+13=02x^2+10x+13=0

First identify the coefficients of each of three terms :

a=2,b=10,c=13a=2,b=10,c=13

Now apply the formula :

x=−(10)±(10)2−4(2)(13)2(2)=−10±−44=−104±−44=−52±2i4=−52±12ix=−52±12i\begin{align*} x&=\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 \\\\ x&=-\frac{5}{2} \pm \frac{1}{2}i \end{align*}

Explore Leaving Cert Mathematics Model Answers by Topics

Fundamentals of Complex Numbers

The Basics

Argand Diagrams & Modulus

Solving Equations

DeMoivre's Theorem

Transformations

Explore Leaving Cert Mathematics Quizzes by Topics

Fundamentals of Complex Numbers

The Basics

Argand Diagrams & Modulus

Solving Equations

DeMoivre's Theorem

Transformations

Explore Leaving Cert Mathematics Flashcards by Topics

Fundamentals of Complex Numbers

The Basics

Argand Diagrams & Modulus

Solving Equations

DeMoivre's Theorem

Transformations

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