Complex Numbers Worksheet - Appendix F, Cengage Page 4
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APPENDIX F
Complex Numbers
Complex Solutions of Quadratic Equations
When using the Quadratic Formula to solve a quadratic equation, you often obtain a
result such as
3,
which you know is not a real number. By factoring out
i
1,
you can write this number in standard form.
3
3
1
3
1
3i
The number
3i
is called the principal square root of
3.
Principal Square Root of a Negative Number
The definition of principal
STUDY TIP
square root uses the rule
If a is a positive number, the principal square root of the negative number
a
ab
a b
is defined as
for
a > 0
and
b < 0.
This rule is not
a
ai.
valid if both a and b are negative.
For example,
EXAMPLE 4
Writing Complex Numbers in Standard Form
5
5
5
1
5
1
5i 5i
a.
3
12
3i 12i
2
25i
2
36i
2
5i
5
6
1
whereas
6
5
5
25
5.
b.
To avoid problems with multiplying
48
27
48i
27i
square roots of negative numbers, be
4 3i
3 3i
sure to convert to standard form before
multiplying.
3i
2
2
c.
1
3
1
3i
2
2
2
1
2 3i
3
i
1
2 3i
3
1
2
2 3i
EXAMPLE 5
Complex Solutions of a Quadratic Equation
2
Solve
3x
2x
5
0.
Solution
±
2
2
2
4 3 5
x
Quadratic Formula
2 3
±
2
56
Simplify.
6
±
2
2 14i
Write in i-form.
6
1
14
±
i
Write in standard form.
3
3
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