5.7 Complex Numbers Worksheet With Answer Key
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1
2
35.7
PLAN
AND
Complex Numbers
1
PREPARE
Warm-Up Exercises
GOAL
Understand and use complex numbers.
Key Words
Transparency Available
• imaginary unit i
}
}
1. Ï
2 Ï
52
13
2
• complex number
The square of any real number x is never negative, so the equation x
1 has
}
Ï
• imaginary number
3
15
no real number solution.
}
}
2.
}
• complex conjugates
5
Ï
15
To solve this equation, mathematicians created an expanded system of numbers
Solve the equation.
2
using the imaginary unit i, defined as a number such that i
1. Note that
}
1 8 5 23
6 Ï
Skills
2
Prerequisite
3. 3x
5
i
1 . The imaginary unit i can be used to write the square root of any
4. 4(n 1 7)
5 16
25, 29
2
negative number.
Simplify.
5. An object falls off a 1000 foot tall
8
1.
2 2
building. How long does it take
The Square Root of a Negative Number
20
2.
2 5
to hit the ground?
about 7.9 sec
PROPERTY
EXAMPLE
Find the product.
Notetaking Guide
i r .
i 5
1.
If r is a positive real number, then
r
5
2
(x
2)(x
2)
x
4
3.
2
2
2
Transparency Available
By Property (1), it follows that (
i r )
r.
(
i 5 )
i
5
5
2.
2
2x(3
x)
6x
2x
4.
Promotes interactive learning and
notetaking skills, pp. 107–109.
EXAMPLE
1
Solve a Quadratic Equation
Teaching Resources
Solve the equation.
For a complete list of lesson
resources, see page 218B.
2
2
7x
49
3x
5
29
a.
b.
Pacing
Solution
STUDENT HELP
R
A
EADING
LGEBRA
2
7x
49
Average: 2 days
a.
Write original equation.
The letter i represents a
Advanced: 2 days
2
x
7
Divide each side by 7.
number. It is a constant,
Block: 1 block
not a variable. Writing
x
7
Take the square root of each side.
i instead of
1 makes
• See Teaching Guide/Lesson Plan
writing complex numbers
x
i 7
Write in terms of
i
.
in Chapter 5 Resource Book,
simpler.
pp. 67–68.
2
3x
5
29
b.
Write original equation.
2
3x
24
Add 5 to each side.
FOCUS
AND
2
x
8
Divide each side by 3.
2
MOTIVATE
x
8
Take the square root of each side.
x
i 8
i
Write in terms of
.
Essential Question
x
2i 2
Simplify the radical.
How do you perform operations on
complex numbers?
Tell students
they will learn how to answer this
question by learning how to add,
Checkpoint
Solve a Quadratic Equation
subtract, multiply, and divide with
the imaginary unit i.
Solve the equation.
2
2
2
x
3
i 3 , i 3
x
7
i 7 , i 7
x
20
1.
2.
3.
Starting the Lesson
2i 5 , 2i 5
2
2
2
x
3
2
y
4
12
2x
7
17
4.
i 5 , i 5
5.
6.
Ask students what kinds of
2i 2 , 2i 2
2i 3 , 2i 3
numbers they used as young
5.7 Complex Numbers
261
children. Discuss how each time
they learned about new kinds of
numbers, they were able to solve
equations they could not solve
before.
261
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