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Unit 2 – 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.
1. i 7
2. i 22
5.
 3i 
6.
9.
18i12
6i 3
10.
3 5
 4i 
8 3
 2i3  5i 
13. The expression
(1) 1
2
3. i15
4. 5i13
7. 7i 7  15i15
8. 4i 20  6i13
11.
15i120
25i11
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 
2. 5i 91
3.
5.
16i 246
8i 43
6.
7.
8.
(3) i
(4) -i
3 2
9. Which expression is equivalent to
(1) 1
13
4. 3i 2011
 2i 
?
(2) -1
10. Expressed in simplest form,
(1) 1
 4i   2i 
is equivalent to
(2) -1
(3) i
11. Solve for x by completing the square. Place your answers in simplest radical form.
x( x  4)  6
(4) -i
2 5
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.
9.
10.
11. Solve for x:
3 
 
64  10  25
2  3
 

48  5  75
8. 2 3  27  3 12

HW on Simplifying Radicals/Adding & Subtracting Complex Numbers!
In # 1 – 5, perform the indicated operation and simplify.
1.
100
3. 3 4  121
2. 4 49
4. 3 48
5.
1
2
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.
11. Simplify:
 20 

3  2i   (8i  4)
10. If
7.
 
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)
(3)
and Joe has a
(4)
2. Simplify: 4 126
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.
6.
2i
4i
9.
4 
12.

3 2  3
4
3  2

7. 5i  4i  3i 9 
8.
6 
10.
4i
5  2i
11.
6  i
13.
 4  9i 3  2i 
14.
5
2i

49 3  16
2

HW on Operations with Complex Numbers
1. The product of
(1) 7
and i is
(2)
2. The expression
(1) -2
(4)
(3)
(4)
(3)
(4)
is equivalent to
(2)
3. The expression
(1)
(3)
is equivalent to
(2)
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)
In 5 - 8, perform the indicated operation and express your answer in simplest a  bi form.
5. (3  i )(2  i )
7.
2 
49

2
6.
2  3i
5i
8. 6i (10  2i )
. If a circuit
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
15. Solve the system algebraically:
y  2x  x2  3
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/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) 1  6i
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
2. Let
z1  5  4i
z2  3  6i
b) 8  2i
c) 1  7i
d) 9  i
. Find and graph on the set of axes below:
a) the sum of z1 and z2 .
b)
 z1  z2 
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:
175
10. Find the product of 2  6i and it’s conjugate.
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.
5. Find the magnitude of 5 12i
7. Place in simplest radical form:
13
x
9x2
 3x  1
2
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
If the discriminant is…
Then the nature of the roots The graph will look like..
for the quadratic equation
will be…
a positive perfect square
A positive non-perfect square
A negative number
zero
In # 1 – 3, find the discriminant and describe the nature of the roots for each quadratic equation.
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.
9. What is the reciprocal of 12 – 3i?
Review Sheet: Complex Numbers
1.
The complex number 5i 3  2i 2 is equivalent to:
(1) 2  5i
2. The expression
(1) 8 3
(2) 2  5i
(3) 2  5i
(4) 2  5i
192 is equivalent to:
(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
2
9. (3  4i )
4. (1  2 12)  (8  5 48)
6. (2  9)(3  16)
2
2
4
8. 4i (6  8i  5i  3i )
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?
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?
15. What is the additive inverse of:
a) 3  4i
b) 2  i
16. What is the multiplicative inverse of:
a) 12  3i
b) 6  i
17. Find the magnitude of the complex number z  3  6i . Leave your answer in simplest radical form.
18. Evaluate: 5  12i
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
20. Solve for x in simplest a  bi form: 3 x 2  12 x  21
21. Solve for x in simplest a  bi form: x 2  6 x  34
22. What is the sum and the product of the roots of the equation 2 x 2  4 x  1  0 ?
23. If the sum of the roots of x 2  4 x  6  0 is subtracted from the product of its roots, the result is
(1) 2
(2) -2
(3) 10
(4) -10
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
27. If the equation 9 x 2  12 x  k  0 has equal roots, find the value of k.
28. For which value of k will the roots of 2 x 2  kx  1  0 be real?
(1) 1
(2) 2
(3) 3
(4) 0
29. 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 .
30. Which quadratic equation has roots 3  i and 3  i ?
(1)
(2)
(3)
(4)
x 2  6 x  10  0
x 2  6 x  10  0
x2  6x  8  0
x2  6x  8  0
Unit 2: Complex Numbers – Chapter Summary
Powers of i:
i1  i
Simplify using the i – chart: 2
i  1
i  i
3
i4  1
Adding or Subtracting Complex Numbers:
Use the calculator!..OR add Reals with
Reals and i’s with i’s.
Multiplying Complex Numbers:
Use the calculator!..OR distribute the
terms (“FOIL” technique.)
Dividing Complex Numbers:
Use the calculator (change back to
fraction!)..OR multiply top & bottom by the
conjugate of the denominator.
Multiplicative Inverses:
Reciprocate the complex number
(“1/(a+bi)”)
Use the calculator (change back to
fraction!)..OR multiply top & bottom by the
conjugate of the denominator.
Using the Discriminant to Describe the
Roots (Nature of the Roots):
Use the b2 – 4ac part of the Quadratic Fmla
Graphing Complex Numbers:
Graph (a + bi) just like the coordinate (a, b)
Ex. (-3 + 5i)  (-3, 5). Draw an arrow from the origin to
the point. (Also be aware of what quadrant it is in)
Magnitude of a Complex Number (Absolute Value):
Use the distance formula for the length of the arrow.
a  bi  a2  b2
2
2
Ex. 3  5i  (3)  5
Solving for Roots of a Quadratic in a+bi form::
Use the Quadratic Formula
Reduce the resulting expression AND SEPARATE the
terms into the “a”term + the “bi” term.
Sum and Product of the Roots:
Sum = -b/a
Product = c/a
Writing an Equation Knowing the Sum and Product:
x2  ? x  ?  0
x2  5x  6  0
Take the Sum, change the sign and make it the “b” term.
Take the Product, keep it the same sign and make it the
“c” term.