Download Sec1.3 - Gordon State College

Sullivan Algebra and
Trigonometry: Section 1.3
Quadratic Equations in the
Complex Number System
Objectives
• Add, Subtract, Multiply, and Divide Complex Numbers
• Solve Quadratic Equations in the Complex Number
System
The equation x2 = - 1 has no real number
solution.
To remedy this situation, we define a new
number that solves this equation, called the
imaginary unit, which is not a real number.
The solution to x2 = - 1 is the imaginary
unit I where i2 = - 1, or i  1
Complex numbers are numbers of the form
a + bi, where a and b are real numbers. The
real number a is called the real part of the
number a + bi; the real number b is called the
imaginary part of a + bi.
Equality of Complex Numbers
a + bi = c + di if and only if a = c and b = d
In other words, complex numbers are equal if
and only if there real and imaginary parts are
equal.
Addition with Complex Numbers
(a + bi) + (c + di) = (a + c) + (b + d)i
Example:
(2 + 4i) + (-1 + 6i) = (2 - 1) + (4 + 6)i
= 1 + 10i
Subtraction with Complex Numbers
(a + bi) - (c + di) = (a - c) + (b - d)i
Example
(3 + i) - (1 - 2i) = (3 - 1) + (1 - (-2))i
= 2 + 3i
Multiplication with Complex Numbers
a  bi   c  di   ac  bd   ad  bci
Example: Multiply using the distributive property
 4  3i   1  4i 
 4( 1)  4( 4i )  3i ( 1)  3i ( 4i )
 4  16i  3i  12i 2
 4  19i  12( 1)
 4  19i  12
 8  19i
If z = a + bi is a complex number, then its
conjugate, denoted by z , is defined as
z  a bi  a bi
Theorem
The product of a complex number and its
conjugate is a nonnegative real number.
Thus, if z = a + bi, then
zz  a  b
2
2
Division with Complex Numbers
To divide by a complex number, multiply the
dividend (numerator) and divisor (denominator) by
the conjugate of the divisor.
1  4i
1  4i 3  i
Example:


3i
3i 3i
3  i  12i  4i

9 1
7  11i

10
7 11
  i
10 10
2
In the complex number system, the solutions
2
of the quadratic equation ax  bx  c  0
where a, b, and c are real numbers and a  0,
are given by the formula
 b  b  4ac
x
2a
2
Since we now have a way of evaluating the square
root of a negative number, there are now no
restrictions placed on the quadratic formula.
2
2
x
 2x 1  0
Find all solutions to the equation
real or complex.
a = 2, b = -2, c = 1
b  4 ac  ( 2 )  4 ( 2 )(1)  4  8  4
2
2
1 1
 ( 2)   4 2  2i
  i

x
2 2
4
2( 2)