A.6 Complex
Numbers & Solving
Equations
It is our choices…who show who we really are, far more
than our abilities.
-Albus Dumbledore
Solving Equations
Let’s recall solving the following equation.
x2  4  0
If we extend our number system to include complex numbers,
we can solvethe previous equation. How cool is that!
If N is a positive real number, we define the principal square
root of –N, denoted by N , as
N  Ni

x   1  i 
x   4  2i
x   8  2 2i
N In Complex Number System
Evaluate the square root of the following negative numbers
in the complex number system.
1)
3
2)
 18
Equations in Complex Number System
Solve the following equations in the complex number system.
x2  9  0
4 x 2  8 x  16  0
Equations in Complex Number System
Solve the following equations in the complex number system.
2x 3  3x 2  6x  9  0
x  6x  12x  0
3
2
x 3  27  0
10x 2  6x  1  0
Equations in Complex Number System
2x 3  3x 2  6x  9  0
x  3 ,  3i
2
x 3  6x 2  12x  0
x  0,  3 3i
3
x  3,  3i
2
3  i
x
10
x  27  0
3
10x 2  6x  1  0


Now that we have extended our number system to include
as many solutions as the
 complex numbers, we should get
trying to solve!
degree of the polynomial we are
Character of Solutions
For the equation ax 2  bx  c  0
b 2  4ac  0 the equation has two unequal real solutions
2 
b  4ac  0 the equation has a repeated real solution
the equation has two complex solutions that
b  4ac  0 are not real and are conjugates of each other
2
Now look at your worksheet for determining the character of
the equations!
Character of Solutions
Ex. x  6 x  13  0
2
b 2  4ac  0 the equation has two unequal real solutions
b 2  4ac  0 the equation has a repeated real solution
b 2  4ac  0
the equation has two complex solutions that
are not real and are conjugates of each other
A.6 Complex
Numbers & Solving
Equations
Homework #9:
p.1008
#55 – 71 EOO, 73
STUDY GUIDE DUE
MONDAY!!
Challenge Problem
Due Tuesday
It is our choices…who show who we really are, far more
than our abilities.
-Albus Dumbledore