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Class Notes and Examples
Complex Numbers and Roots
An imaginary number is the square root of a negative number.
Use the definition 1  i to simplify square roots.
Simplify.
25
 25  1
Factor out 1.
 25 
Separate roots.
1
5 1
Simplify.
5i
Express in terms of i.
 48

 48  1
Factor out 1.

 48 
Separate roots.
1
 16 3 1
Factor the perfect square.
4 3 1
Simplify.
4i 3
Express in terms of i.
Real
Imaginary
Complex numbers are numbers that can be written in the form a  bi.
The complex conjugate of a  bi is a  bi.
The complex conjugate of 5i is 5i.
Write as a  bi
Find 0  5i  5i
Express each number in terms of i.
1.
72
 36  2 1
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4. 5 54
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2. 4 45
4
3.
100
 9  5  1
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5. 2 64
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6.  98
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Find each complex conjugate.
7. 9i
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8. 1  4i
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9. 12  i
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Class Notes and Examples
Complex Numbers and Roots (continued)
You can use the square root property and
imaginary solutions.
1  i to solve quadratic equations with
Solve x2  64.
x 2   64
Take the square root of both sides.
x  8i
Express in terms of i.
2
Check each root: 8i  64i  641  64
2
Remember:


1
2
 i2  1
8i2  64i 2  641  64
Solve 5x2  80  0.
5x2  80
Subtract 80 from both sides.
x2  16
Divide both sides by 5.
x 2   16
Take the square root of both sides.
x   4i
Express in terms of i.
Check each root:
54i 2  80
54i 2  80
516i 2  80
516i 2  80
801  80
801  80
0
0
Solve each equation.
10. x2  18  0
x2  18
x
 9  2 1
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13. x  100  0
2
11. 6x2  24  0
12. x2  49  0
6x2  24
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14. 3x  108  0
2
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15. x2  12  0
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Class Notes and Examples
Operations with Complex Numbers (continued)
To add or subtract complex numbers, add the real parts and then add the
imaginary parts.
First, group to add the real parts and
3  2i   4  5i 
the imaginary parts. This is similar to
3  4  2i  5i 
adding like terms.
7  3i
Remember to distribute when
subtracting. Then group to add the
real parts and the imaginary parts.
4  i   2  6i 
4  i   2  6i
4  2  i  6i 
6  7i
Use the Distributive Property to multiply complex numbers.
Remember that i 2  1.
3i 2  i 
6i  3i 2
Distribute.
6i  31
Use i 2  1.
3  6i
Write in the form a  bi.
4  2i 5  i 
20  4i  10i  2i 2
Multiply.
20  6i  21
Combine imaginary parts and use i 2  1.
22  6i
Combine real parts.
Add, subtract, or multiply. Write the result in the form a  bi.
12. 6  i   3  2i 
6  3  i  2i 
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15. 2  4i   1  4i 
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18. 6  5i   5i  6
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13. 9  3i   2  i 
9  3i   2  i 
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16. 1  7i   1  5i 
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19. 2  i 3i  2
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14. 3  i 2  2i 
6  6i  2i  2i2
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17. 5i 4  3i 
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20. 2  4i 2
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