5.6 – Quadratic Equations and Complex Numbers

Objectives:Classify and find all roots of a quadratic equation.

Graph and perform operations on complex numbers.Standard:

2.5.11.C. Present mathematical procedures and results clearly, systematically, succinctly and correctly.

The Solutions to a Quadratic Equation can referred to as ANY of the following:

x – interceptsSolutionsRootsZeroes

Discriminant• The expression b2– 4ac is called the discriminant of a

quadratic equation.• If b2– 4ac > 0 (positive), the formula will give two real

number solutions.• If b2– 4ac = 0, there will be one real number solution,

called a double root.• If b2– 4ac < 0 (negative), the formula gives no real

solutions.

Ex 1. Find the discriminant for each equation. Then determine the number of real solutions for each equation

by using the discriminant.

Cardano

Imaginary NumbersIf r > 0, then the imaginary number is defined as follows: = =

r

r r 1 ri

Ex 1: Use the quadratic formula to solve each equation.

a)

2b) 4 5 3 0x x

5 23

8

ix

Complex Numbers

Example 1a and b*

Operations with Complex Numbers

Conjugate of a Complex Number

• The conjugate of a complex number a + bi is a – bi.

• To simplify a quotient with an imaginary number in the denominator, multiply by a fraction equal to 1, using the conjugate of the denominator.

• This process is called rationalizing the denominator.

(2 3 )

(2 3 )

i

i

2

2

4 10 6 15

4 6 6 9

i i i

i i i

11 16

13

i

11 16

13 13i

(2 )

(2 )

i

i

2

2

6 3 8 4

4 2 2

i i i

i i i

2 11

5 5i