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Unit 5 – Complex Numbers
Introduction to imaginary numbers
Do Now:
1. Factor completely: 12a 2 x  2ax  24 x
Notes:
Imaginary numbers came about when there were negative numbers under the radical. Mathematicians had a
hard time accepting this, but in order to work with these numbers they let i  1 .
So now let’s evaluate the first four powers of i:
i1 
i2 
i3 
i4 
Examples:
For #1 – 12, perform the indicated operation and simplify.
2. i 22
1. i 7
8. 4i 20  6i13
3. i15
9.
18i12
6i 3
13. The expression
(1) 1
10.
4. 5i13
5.
 2i  5i 
11.
3 2
 3i 
3 5
6.
 4i 
8 3
15i120
25i11
7. 7i 7  15i15
12. 13i17  8i 25
is equal to
(2) -1
(3) i
(4) -i
HW on Introduction to Imaginary Numbers
In #1 – 8, perform the indicated operation and express the result in simplest terms.
1.
 3i 
3 2
2. 5i 91
3.
 4i   2i 
13
4. 3i 2011
16i 246
5.
8i 43
6.
9. Which expression is equivalent to
(1) 1
8.
(3) i
(4) -i
2 5
?
(2) -1
10. Expressed in simplest form,
(1) 1
 2i 
7.
is equivalent to
(2) -1
(3) i
(4) -i
11. Solve for x by completing the square. Place your answers in simplest radical form.
x( x  4)  6
Simplifying Radicals with Negative Radicands
Addition & Subtraction of Complex Numbers
Do Now:
1. If
, then
(1)
is equivalent to
(2)
(3)
(4)
Notes:

What is a complex number???
Examples:
Simplify each radical:
1.
36
3. 3 10
2. 2 49
4.
50
5.  175
For each expression below, perform the indicated operation and place your answer in simplest a  bi form.
6. (7  5i )  (8  3i)
7.
3 
 
64  10  25

8. 2 3  27  3 12
9.
10.
2  3
 
48  5  75

11. Solve for x:
HW on Simplifying Radicals/Adding & Subtracting Complex Numbers!
In # 1 – 5, perform the indicated operation and simplify.
1.
100
2. 4 49
4. 3 48
5.
1
2
3. 3 4  121
200  32
In #6 – 9, perform the indicated operation and place your answer in simplest a  bi form.
6. (5  2i )  (7  4i)  (12  8i)
8.
7.

3  2i   (8i  4)
10. If
11. Simplify:
 20 
 
28  4  7

 
9. 12  24  3  2 54

, find the value of a.
 2i 
5 3
12. Factor completely: 6 x 2  5 x  21
Multiplying & Dividing Complex Numbers
Inverses & Conjugates
Do Now:
1. Melissa and Joe are playing a game with complex numbers. If Melissa has a score of
score of
, what is their total score?
(1)
2. Simplify: 4 126
(2)
(3)
(4)
and Joe has a
Notes:

Multiplicative Inverse

Additive Inverse

Conjugate
Examples:
1. Find the reciprocal of 3  2i .
2. Find the additive inverse of 2  9i .
3. Find the conjugate of 1 13i .
4. Find the multiplicative inverse of 5i .
5. Find the sum of 4  9i and it’s conjugate.
Perform the indicated operation and place your answers in simplest a  bi form.
2i
6.
7. 5i  4i  3i 9 
8. 6  49 3  16
4i

4 
9.
12.

3 2  3

4
3  2

10.
4i
5  2i
11.
6  i
13.
 4  9i 3  2i 
14.
5
2i
HW on Operations with Complex Numbers
1. The product of
(1) 7
and i is
(2)
2. The expression
(1) -2
(3)
(4)
(3)
(4)
is equivalent to
(2)
2

3. The expression
(1)
is equivalent to
(2)
(3)
(4)
4. The relationship between voltage, E, current, I, and resistance, Z, is given by the equation
has a current
and a resistance
, what is the voltage of this circuit?
(1)
(2)
(3)
(4)
. If a circuit
In 5 - 8, perform the indicated operation and express your answer in simplest a  bi form.
5. (3  i )(2  i )
7.
2 
49

6.
2
2  3i
5i
8. 6i(10  2i)
9. What is the reciprocal of 6i ?
10. What’s the additive inverse of 4  5i ?
11. Find the product of 2  5i and it’s conjugate.
12. Find the multiplicative inverse of 2  i .
13. Solve for x:
5  3i    2  xi   3  7i
14. Factor completely: 10 x 2b  13xb  3b
y  2x  x2  3
15. Solve the system algebraically:
y  13  x
Graphing Complex Numbers &
Magnitude of Complex Numbers
Do Now:
1. Simplify: (3  2i)  (4  i)
2. Simplify:
3 

16 1  49

Notes:


To graph the complex number, a  bi , plot the point whose coordinates are (a, b), and then draw a ray
from the origin to that point.
Magnitude/Modulus/length/Absolute value of a complex number, a  bi , can be found by using the
formula
a 2  b2
Examples:
1. Graph the following complex numbers listed below. Find the magnitude of each complex number.
a) 3  5i
𝑏
b) 1  6i
𝑎 + 𝑏𝑖
𝑎
c) 5  2i
d) 2  4i
2. Evaluate each of the following. Place your answer in simplest radical form.
a) 5  144
b)
1  8
c) 6 10i
3. Which complex number is closest to the origin?
(1) 5  i
(2) 3  2i
(3) 4  5i
(4) 4  i
HW on Graphing and Finding the Magnitude of Complex Numbers
1. Graph each complex number on the set of axes below. Label each.
a) 3  i
b) 8  2i
c) 1  7i
d) 9  i
𝑎 + 𝑏𝑖
𝑏
𝑎
2. Let
z1  5  4i
z2  3  6i
. Find and graph on the set of axes below:
a) the sum of z1 and z2 .
b)
 z1  z2 
𝑏
𝑎
In # 3 – 5, evaluate each expression. Place your answer in simplest radical form.
4. 5  12
3. 3  4i
5. (5  3i)  (7  6i)
6. When graphed, which complex number is closest to the origin?
(1) 3  4i
(2) 2  4i
(3) 3  3i
(4) 4  4i
7. If  3  4i   (c  di)  1  2i , find c and d.
8. If f ( x)  x 2  5 x3 , find f (i ) and place your answer in simplest a + bi form.
9. Find the reciprocal of 3  5i .
11. Simplify:
10. Find the product of 2  6i and it’s conjugate.
175
12. Factor completely: 18  2x 2
Solving Quadratic Equations with Imaginary Roots
Do Now:
Simplify:
1. (9  9i )  ( 7  2i)
2.
5 
32

2
3. 5  25
Practice!
1. Solve for x by using the quadratic formula. Place your answer in simplest a  bi form.
3x 2  10 x  3  0
2. Solve for x by completing the square. Place your answer in simplest a  bi form.
x 2  4 x  10  0
3. Solve for x. Place your answer in simplest a  bi form.
4x
6

2 x2
HW on Solving Quadratic Equations with Imaginary Roots
For # 1 – 4, solve for x and place your answers in simplest a  bi form.
1. x 2  7  4 x
3. 2 x 2  6 x  5  0
2. x  6 
4.
13
x
9 x2
 3x  1
2
5. Find the magnitude of 5 12i
7. Place in simplest radical form:
6. Find the product of 9  i and its conjugate.
27  2 49  363
Discriminant and Describing the Nature of the Roots
Do Now: Solve for x by completing the square.
2 x 2  12 x  10  0
In # 1 – 3, find the discriminant and describe the nature of the roots for each quadratic equation.
If the discriminant is…
Then the nature of the
roots for the quadratic
equation will be…
The graph will look like..
a positive perfect square
A positive non-perfect square
A negative number
zero
1. 2 x 2  7 x  4
2. x 2  1  0
3. x( x  6)  9
4. Find all value(s) for k that makes the roots to the quadratic equation real, rational and equal.
x 2  kx  9  0
5. Find the smallest integer value of k that will make the roots imaginary.
kx 2  5 x  3  0
6. Find all values for k that make the roots to the following quadratic equation real.
3x 2  2 x  k  0
7. The roots of the equation 4 x 2  x  1  0 are
(1) real, rational and equal
(2) real, rational and unequal
(3) imaginary
(4) real and irrational
HW on Discriminant and Describing the Nature of the Roots
In # 1 – 3, find the discriminant and describe the nature of the roots for each quadratic equation.
1. 4 x  1 
9
x
2.
1 2
x  x6  0
3
3. 4 x 
12 x  9
x
4. Which is the smallest value of a that would make the roots to the equation ax 2  6 x  8  0 imaginary?
(1) 1
(2) 2
(3) 3
(4) 4
(3) x 2  10 x  25  0
(4) x 2  7 x  13  0
5. Which quadratic equation has equal roots?
(1) x 2  5 x  6  0
(2) x 2  9  0
6. Which quadratic equation has real, rational and unequal roots?
(1) x 2  2 x  1  0
(2) x 2  5 x  7  0
(3) x 2  36  0
(4) x 2  36  0
7. The roots of the equation 3x 2  2 x  7 are
(1) real, rational and equal
(2) real, rational and unequal
(3) imaginary
(4) real and irrational
8. Find the largest integer value for k that make the roots to the equation 2 x 2  7 x  k  0 real.
10. Find the magnitude of 4  5i .
9. Find the multiplicative inverse of 6i.
11. Simplify:
4 
 
48  2  3 12

12. In what quadrant does the difference of (3  2i )
and ( 1  2i ) lie?
13. Solve for x by completing the square. Place your answer in simplest a  bi form.
x 2  6 x  10  0
Sum and Product of the Roots
Do Now:
If a quadratic equation has real, rational, and equal roots, the graph of the parabola:
(1) intersects the x-axis at two distinct points
(2) is tangent to the x-axis
(3) lies entirely above the x-axis
(4) lies entirely below the x-axis
Solve for x in each equation by factoring:
2
1) x  x  12  0
2) 2 x 2  3x  1  0
For each example, answer the following questions:
What is the sum of the roots?
What is the product of the roots?
IN GENERAL, for any quadratic equation in the form 𝑎𝑥 2 + 𝑏𝑥 + 𝑐 = 0 , the:
SUM of the roots =
PRODUCT of the roots =
Let’s Practice! Find the sum and product for each quadratic equation below:
1. 3𝑥(𝑥 − 2) = 9
2. 𝑥 2 − 49 = 0
How are we going to use the formulas for the sum and product of the roots to HELP us write
QUADRATIC EQUATIONS?
Ex1: Write a quadratic equation if the sum of the roots is 5 and the product is 6. What are the roots to this
equation?
Ex2: Write a quadratic equation if the sum of the roots is -3 and the product is -10. What are the roots to this
equation?
Ex 3: If one root is 1 + 2i , find the other root. Write a quadratic equation with those roots. (Hint: Complex
roots always come in CONJUGATE pairs!)
Ex4: Find the second root and the value of k for each equation below.
a) 𝑥 2 − 𝑥 + 𝑘 = 0; 𝑟1 = −4
b) 𝑥 2 + 𝑘𝑥 + 18 = 0; 𝑟1 = 6
HW on Sum and Product of the Roots
1. Find the sum and product of the roots of the equation 2𝑥 2 − 6𝑥 + 10 = 0.
2. If one root of a quadratic equation is 6 + 2i, find the other root and the equation.
3. For which equation does the sum of the roots equal the product of the roots?
(1) 3𝑥 2 − 3𝑥 + 1 = 0
(3) 𝑥 2 + 13 = 13𝑥
(2) 𝑥 2 − 13 = 13𝑥
(4) 2𝑥 2 + 2𝑥 + 2 = 0
4. If the product or the roots of 4𝑥 2 − 20 = 8𝑥 is subtracted from the sum of the roots, the result is
(1) -7
(2) -4
(3) 7
(4) 9
5. Describe the nature of the roots of the equation 3𝑥 2 − 𝑥 + 2 = 5
6. Simplify the expression 2𝑖 6 − 3𝑖 2 .
7. In which quadrant would you find the sum of (2 − √−4) + (−5 + √−36) ?
8. Express the roots of the equation 𝑥 2 + 5𝑥 = 3𝑥 − 3 in simplest a + bi form.
Review Sheet: Complex Numbers
1.
3
2
The complex number 5i  2i is equivalent to:
(1) 2  5i
2. The expression
(2) 2  5i
(3) 2  5i
(4) 2  5i
192 is equivalent to:
(1) 8 3
(2) 3 8
(3) 8i 3
(4) 3i 8
Perform the indicated operations and express your answer in simplest a  bi form.
3.
(6  49)  (3  64)
5. (6  2i )  (4  5i)
7.
1  4
2  9
9. (3  4i )
2
4. (1  2 12)  (8  5 48)
6. (2  9)(3  16)
8. 4i (6  8i  5i  3i )
2
2
10.
6  7i
2i
11. Express the product of (5  6i ) and (3  5i ) in simplest a  bi form.
12. What is the product of 2  5i and its conjugate?
4
13. In which quadrant will the sum of (7  3i ) and (5  8i ) lie?
14. In which quadrant will the difference (5  11i )  (2  7i) lie?
19. If Z1  5  2i and Z 2  3  5i ,
a) Graph Z1 and Z 2
b) Graph the sum of Z1 and Z 2
2
20. Solve for x in simplest a  bi form: 3x  12 x  21
2
21. Solve for x in simplest a  bi form: x  6 x  34
2
22. What is the sum and the product of the roots of the equation 2 x  4 x  1  0 ?
24. The roots of the equation 3x 2  5 x  4 are
(1) real, rational, and unequal
(2) real, irrational, and unequal
(3) real, rational, and equal
(4) imaginary
25. The roots of the equation x 2  4 x  13  0 are
(1) real, rational, and unequal
(2) real, irrational, and unequal
(3) real, rational, and equal
(4) imaginary
26. The roots of a quadratic equation are real, rational, and equal when the discriminant is
(1) -2
(2) 2
(3) 0
(4) 4
28. The roots of a quadratic equation are r1  4  2i and .r2  4  2i
a) Find the sum of the roots.
b) Find the product of the roots.
c) Write a quadratic equation with roots r1 and r2 .