Transcript Using the Quadratic Formula to Find Complex Roots (Including

Using the Quadratic Formula to
Find Complex Roots (Including
Complex Conjugates)
Complex Number
A number consisting of a real and imaginary part.
Usually written in the following form (where a
and b are real numbers):
a  bi
Example: Solve 0 = 2x2 – 2x + 10
x
2
 2 
2
 4110 
21

a = 1 b = -2 c = 10
2 36
2
1 3i and 1 3i

2 6i
2
 1  3i
Classifying the Roots of a Quadratic
Describe the amount of roots and what number set
they belong to for each graph:
1 Repeated
22Real
Complex
Roots
Real Root
Roots
because
because
it has
because it has
it has
twono
one x-intercept
x-intercepts
(bounces off)
A Quadratic
ALWAYS has
two roots
Determining whether the Roots are Real or
Complex
What part of the Quadratic Formula
determines whether there will be real or
complex solutions?
Discriminant < 0
b  b  4ac
x
2a
2
Complex Conjugates
a  bi
The Complex Conjugate is: a  bi
For any complex number:
The sum and product of complex conjugates are
always real numbers
Example: Find the sum and product of 2 – 3i and its
complex conjugate.
Complex Conugate: 2  3i
Sum: 2  3i  2  3i  4
2
Product:  2  3i  2  3i   4  6i  6i  9i  4  9  13
Complex Roots Are Complex Conjugates
A quadratic equation y = ax2 + bx + c in which
b2 – 4ac < 0 has two roots that are complex
conjugates.
Example: Find the zeros of y = 2x2 + 6x + 10
 6 2  4 2 10 
2 4 
x
6 
x
x
6 44
4
6 2 11i
4
x
3 11i
2
x
3
2

11
2
i
11
2
i
and
x
3
2

Complex
Conjugates!