Download Name - Wantagh School

COMPLEX
NUMBERS
Algebra 2 & Trigonometry
Mrs. Martin
Room 227
Period 5
Name:____________________________
Topic
Pages
Day 1 – Imaginary Numbers / Powers of i
3–4
Day 2 – Graphing and Operations with Complex Numbers
8 – 13
Day 3 – Dividing Complex Numbers
14 – 17
Day 4 – Complex Roots of Quadratic Equations
18 – 23
Day 5 – Nature of the Roots (1)
24 – 27
Day 6 – Nature of the Roots (2)
28 – 32
Day 7– Review
33 – 35
2
Day 1 – Imaginary Numbers / Powers of i
Steps to Evaluate Radicals with a Negative Radicand
An equation, such as x   9 , has no solution in the real number system. However, a solution does exist in
the system of imaginary numbers. By definition,  1 is defined as i, the imaginary unit. Since
 9  9  1 , and 1  i , we can simplify  9 as 3i.
1. Remove the negative sign from under the radical.
2. Place an i in front of the radical sign.
3. Simplify the radical.
Directions: Simplify each number and express in terms of i.
4.) 5  12
1.)  25
2.)  100
3.)  18
Important Powers of i
Remember: i = 1
i0 = 1
i1 = i
i2 = –1
3
i
= -i
Steps in the Calculator: MATH BUTTON
 NUM (Arrow over once)
3: iPart (option #3)
Directions: Write each power of i in simplest terms.
5.) i 10
6.) i 15
7.) i 44
3
8.) i 97
9.) What is the sum of 7i 7 and 15i 15 ?
10.) What is the product of 4i 20 and 6i 13 ?
Practice Problems
11.) The sum of 6i 6 and 13i 34 is
(1) -19i
(3) -19
(2) 19i
(4) 19

13.) Simplify: 10i 13
(1) -100i
(2) 100i
12.) Find the product of 10i 18 and 7i 33 .
(1) -70i
(3) -70
(2) 70i
(4) 70

14.) Which of the following is not equal to the other
three?
(1) i 19
(3) i 27
(2) i 9
(4) i 35
2
(3) -100
(4) 100
15.) When 6i 18 is multiplied by 8i 6 , the result is
(1) -48i
(3) -48
(2) 48i
(4) 48
16.) The expression
(1)  3 3
(2) 3i 3
4
3
 48 is equivalent to
4
(3)  3i 3
3
(4)  i 3
2
Day 1 – Imaginary Numbers / Powers of i
HOMEWORK
**Complete any Practice Problems from class work that have not been completed**
Directions: Simplify.
7
1.)
28
2.)
2a2b5
 10 x 4 y 9
3.) 
10


Ans: 2i 7
Ans: ab i 2b
Directions: Perform the indicated operation and express in simplest form of i.
6.)
 36   49   64
25  4
4.)
5.)
Ans: 
2
7 2 4
x y i 10 y
10
100  81

Ans: 21i
7.)
2 5  125
8.)
Ans: 3i
12  27  75

Ans: 19i
9.)
144  1

Ans: 0
Ans: 7i 5
Directions: Write each power of i in simplest terms.
10.) i 301
11.) i 431
Ans: 13i
12.) i14
Ans: –i
Ans: i
5
Ans: –1
13.) 4i 6  3i11
14.) 32i 32  40i10
Ans: 12i
15.) 2i  7i
5
Ans: 72
7
 20i
10i 16
23
16.)
Ans: –5i
Ans: 2i
Review
17.) Solve the system of equations algebraically:
x 2  14  y
y 1  2x
Ans: (5, 11), (-3, -5)
6
18.) One of the students in class was absent the day the class learned the technique of completing the
square. Using the technique of completing the square, write an explanation of how to solve the
following equation that you could give to the student who had been absent:
x2 + 8x – 3 = 0
Ans: x  4  19
19.) If one root of a quadratic equation is 1 and the equation is x  bx  3  0 , what is the other root?
2
Ans: 3
7
Day 2 – Graphing and Operations with Complex Numbers
Do Now: (Questions 1 & 2)
1.)
Which expression shows the simplified form of
i 34 ?
(1) –1
(3) i
(2) 1
(4) –i
2.)
Simplify: 16i  13i
5
23
Steps to Graph Complex Numbers:
Complex Numbers are always of the form a + bi, where a and b are real numbers and i is the imaginary unit.
1.)
The “a” is the x-value, and the “b” is the y-value. Graph the point (a, b).
2.)
The “i” tells you that it is not only a point, but a vector. You must draw an arrow from the origin
(0, 0) to the point that you graphed.
3.)
Label the vector a + bi.
Directions: Graph each complex number.
3.) Graph the complex number 3  4i .
4.) Graph the complex number 4  5i .
5.)
Graph the complex number 6  7i .
6.)
8
Graph the complex number 9  i .
Steps to Adding/Subtracting Complex Numbers:
Complex Numbers are always of the form a + bi, where a and b are real numbers and i is the imaginary unit.
1.)
Combine like terms. You can only combine imaginary numbers with other imaginary numbers.
2.)
Remember, if you are subtracting, you must distribute the negative.
Directions: Perform the indicated operation and express in simplest a + bi form.
7.)
(6 + 7i) + (1 + 2i)
8.)
(3 – 5i) + (2 + i)
9.)
(–3 + 3i) – (1 + 5i)
10.)
(2  8i)  (2  8i)
Steps to Finding the Additive Inverse of a Complex Number:
Change all signs, on a and on b.
Ex) a  bi  a  bi
Directions: Find the additive inverse of each expression.
11.)
12.) 1  3i
6i
14.)
2  8i
Steps to Finding the Conjugate of a Complex Number:
Change the middle sign, on b only. Ex) a  bi  a  bi
Directions: Find the conjugate of each expression.
15.)
16.)
6i
1  3i
13.)
17.)
1  9i
1  9i
18.)
9
2  8i
Steps to Multiplying Complex Numbers:
1.)
FOIL (or calculator!)
2.)
Remember, i  1 , so i 2  1 .
Directions: Perform the indicated operation and express the answer in simplest a + bi form.
(5  2i )(3  4i )
(3  5i )(2  i )
19.)
20.)
21.)
(3  2i)(3  3i)
22.)
(2  6i) 2
23.)
(9  3i)(9  3i)
24.)
(1  9i)(1  9i)
10
Day 2 – Graphing and Operations with Complex Numbers
HOMEWORK
Directions (Questions 1 – 3): Graph each complex number.
1.)
6 + 7i
2.)
–2 – 3i
3.)
3 – 4i
Directions (Questions 4 – 9): Find each sum or difference of the complex numbers in a + bi form.
(1  9i)  (1  2i)
(10  12i)  (12  7i)
4.)
5.)
6.)
(5  6i)  (4  2i)
8.)
1 1
1 3
(  i)  (  i)
2 2
4 4
Ans: 0  7i
7.)
(4  12i)  (4  2i)
9.)
3i  (1  4i)
Ans: 9  4i
3 1
 i
4 4
Directions (Questions 10 – 17): Find each product in simplest a + bi.
(1  9i )(1  2i )
(10  12i)(12  7i)
10.)
11.)
Ans: .75  .25i or
Ans: 17 11i
11
Ans: 2 19i
Ans: 0 10i
Ans: 1  7i
Ans: 204  74i
12.)
14.)
16.)
(3  5i )(2  i)
(4  4i)(4  i)
(2  i )(1  2i )
13.)
Ans: 1  13i
15.)
3i (1  4i)
(12  2i )(12  2i )
Ans: 20  12i
Ans: 12  3i
Ans: –148 + 0i
17.)
3 1
(3  i )(  i )
10 10
Ans: 0  5i
Ans: 1 + 0i
Directions (Questions 18-21): For each question, find a) the additive inverse and b) the conjugate.
18.) 3 + i
19.)
1  2i
Ans: a) 3  i , b) 3  i
20.)
21.)
1 1
 i
2 4
1 1
1 1
Ans: a)   i , b)  i
2 4
2 4
12
  2i
Ans: a) 1  2i , b) 1  2i
Ans: a)   2i , b)   2i
Review
22.)
One of the roots is given. Find the other root and the value of c.
2x2 + 3x + c = 0; one root = 1
Ans: x  2.5, c  5
13
Day 3 – Dividing Complex Numbers
Do Now: Questions 1 & 2
2.) The product of 1  i  and 1 2i  is equal to?
(1) 1  i
(3) 3  i
(2) 3  i
(4) 1 4i
1.) The expression 3  7i  is equivalent to
(1)  40  0i
(3)  40  42i
(2) 58  0i
(4) 58  42i
2
Steps to Finding the Multiplicative Inverse of a Complex Number:
Complex Numbers are always of the form a + bi, where a and b are real numbers and i is the imaginary unit.
1.) The multiplicative inverse is found by flipping the fraction. (i.e., put a “1” in the numerator)
2.) Similar to radicals, you can’t have an imaginary number in the denominator, so you must
rationalize the denominator by multiplying both the numerator and denominator by the conjugate of
the denominator.
3.) Simplify the resulting fraction to simplest a + bi form.
Directions: For each problem, find the multiplicative inverse in simplest a + bi form.
1.) 2 – 4i
2.) 3 + 4i
14
Steps to Dividing Complex Numbers:
Binomial Denominators
1.) Similar to radicals, you can’t have an
imaginary number in the denominator, so
Multiply top and bottom by the identical term.
you must _____________ the denominator
by multiplying both the numerator and
denominator by the ______________ of
the denominator.
2.) FOIL top and bottom separately.
(calculator i-parts)
3.) Simplify the resulting fraction to simplest
a + bi form.
Directions: For each problem, divide and express the answer in simplest a + bi form.
12  3i
3.)
3i
Monomial Denominator
4.)
2  3i
1  2i
Practice Problems
Directions: For each problem, divide and express the answer in simplest a + bi form.
5  2i
5.)
5  2i
6.)
10  5i
1  2i
15
Day 3 – Dividing Complex Numbers
HOMEWORK
**Complete any Practice Problems from class work that have not been completed**
Directions: For each problem, divide and express the answer in simplest a + bi form.
2i
1.)
3i
Ans:
2.)
2  9i
1 i
1 1
 i
2 2

Ans:
3.)
4  6i
5i
Ans:
4.)
7 11
 i
2 2
6 4
 i
5 5
1
2  5i
Ans:
16
2
5
 i
29 29
5.)
3  2i
2i
3
Ans: 1  i
2
6.)
3
2  2i
Ans:
Review
7.)
For what value(s) of x does
3 3
 i
4 4
x  3 2x  1

?
2
3
Ans: –11
8.)
Write
2 3
2 3
with a rational denominator.
Ans: 7  4 3
17
Day 4 – Complex Roots of Quadratic Equations
Do Now: (Questions 1-2)
1.)
2.)
What is the quotient when 2  5i  is divided by 1  i  ?
7  7i
(1)
(3) –3
2
 3  7i
(2)
(4) 7
2
Divide and simplify in simplest a + bi form:
10  5i
2  6i
Directions: Solve each quadratic equation by the method given.
3.)
Find all roots of the equation x 2  4 x  13  0 , using the method of completing the square.
18
4.)
Find the zeros of the equation 4 x 2  12 x  25  0 , using the quadratic formula.
5.)
Solve for x in simplest radical form: 2 x 2  9  8 x
19
Day 4 – Complex Roots of Quadratic Equations
HOMEWORK
Directions: Use the quadratic formula to find the imaginary roots, in simplest radical form if possible.
x2  4x  8  0
1.)
Ans: 2  2i
2.)
x  13  4 x
2
Ans: 2  3i
20
3.)
Solve the equation x 2  6 x  21 and express the roots in simplest a + bi form.
Ans: 3  2i 3
Directions: Use the method of completing the square to find the imaginary roots, in simplest radical form.
4.) x 2  6 x  10  0
Ans: 3  i
21
5.) x 2  2 x  10
Ans: 1  3i
6.) x  17  8 x
2
Ans: 4  i
22
Review
7.) Write in simplest radical form:
4
64 x 4  4 40 x 6
4
4 x4
Ans: 2  4 10x2
8.) Divide and simplify:
8  6i
2i
Ans: 3  4i
23
Day 5 – Nature of the Roots (1)
Do Now: QUIZ
The discriminant is an expression that determines the nature of the roots of a quadratic equation, and the
preferable method of solving the equation.
Ways to Describe the Roots of a Quadratic Equation
(1)
(2)
(3)
(4)
Real, Rational, Unequal
Real, Irrational, Unequal
Real, Rational, Equal
Imaginary
DISCRIMINANT
Case 1:
Case 4:
b  4ac
2
Case 3:
Case 2:
24
Directions: Describe the nature of the roots.
(1) Real, Rational, Unequal; (2) Real, Irrational, Unequal; (3) Real, Rational, Equal; (4) Imaginary
2.) x 2  4 x  3  0
3.) x 2  3 x  5  0
4.) 2 x 2  5  0
5.)
x2  4x  4  0
6.) x 2  2 x  5  0
7.)
2x2  x  4
25
Day 5 – Nature of the Roots (1)
HOMEWORK
Directions: Find the value of the discriminant and determine if the roots of the quadratic equation are
(1) Rational and unequal; (2) Rational and equal; (3) Irrational and unequal; or (4) Imaginary.
2
x  12 x  36  0
1.)
2.) 2 x 2  7 x  0
Ans: (2)
3.)
x  3x  1  0
5.)
4x  x  1
2
Ans: (1)
4.)
2x  8  0
6.)
3x  5 x  3
2
Ans: (3)
2
Ans: (3)
26
Ans: (1)
2
Ans: (4)
Review
7.)
Solve the quadratic equation by completing the square and express your answer in simplest radical
form, if possible.
x2  6x  4  0
8.)
Solve the quadratic equation 3 x  6 x  1 using the quadratic formula.
a. Express your answer in simplest radical form.
b. Write, to the nearest tenth, a rational approximation for the roots.
Ans: x  3  5
2
Ans: a)
27
3 2 3
2 3
or 1 
, b) {2.2, –0.2}
3
3
Day 6 – Nature of the Roots (2)
Do Now: (Questions 1 & 2)
1.)
x2  4x  4  0
2.)
2 x2  6 x  3  0
a) Find the value of the discriminant.
a) Find the value of the discriminant.
b) Describe the nature of the roots:
(1) rational and unequal
(2) rational and equal
(3) irrational and unequal
(4) imaginary
b) Describe the nature of the roots:
(1) rational and unequal
(2) rational and equal
(3) irrational and unequal
(4) imaginary
c) Describe the graph:
(1) Crosses in two distinct rational places
(2) Crosses in two distinct irrational places
(3) Tangent to the x-axis
(4) Lies entirely above the x-axis
(5) Lies entirely below the x-axis
c) Describe the graph:
(1) Crosses in two distinct rational places
(2) Crosses in two distinct irrational places
(3) Tangent to the x-axis
(4) Lies entirely above the x-axis
(5) Lies entirely below the x-axis
Equations
Steps to Find the Missing Value Given the Nature of the Roots:
1.) Find your a, b, and c values based on the standard form ax 2  bx  c  0 .
2.) Plug into the discriminant.
 If you are told your roots are equal, set the discriminant equal to zero and solve for the
missing value.
 If you are told your roots are real, set the discriminant greater than or equal to zero and solve
for the missing value.
 If you are told your roots are imaginary, set the discriminant less than zero and solve for the
missing value.
Equal
Real
Imaginary
2
2
b  4ac  0
b  4ac  0
b 2  4ac  0
Directions: Find the missing value.
3.) Find the value of k if the roots of the equation x 2  6 x  k  0 are equal.
28
4.)
Find the values of k which will make the roots
of the equation real: 2 x 2  8 x  k  0
5.)
Find the values of k for which the roots of the
equation will be imaginary: kx 2  12 x  4  0
Practice Problems
6.)
Find the value of k which will make the roots
of the equation equal: x 2  8 x  k
7.)
Find the value of k which will make the roots
of the equation real: kb 2  9b  4
8.)
Find the values of k for which the roots of the
equation will be imaginary:
kc 2  10c  12  0
9.)
Find the value of k which will make the roots
of the equation equal: y 2  ky  36  0
29
10.)
Find the values of k which will make the
roots of the equation real: x 2  8 x  2k  0
11.)
30
Find the value of k which will make the roots
of the equation equal: kx 2  4 x  1  0
Day 6 – Nature of the Roots (2)
HOMEWORK
Directions: Find the missing value.
1.)
For what value of k are the roots of
2 x 2  8 x  k  0 equal?
2.)
Find the values of k which will make the
roots of the equation real: x 2  12 x  k  0
3.)
Ans: k = 8
Find the values of k for which the roots of the 4.)
equation will be imaginary: y 2  k  8 y
Ans: k  36
Find the value of k which will make the roots
of the equation equal: kx 2  10 x  5  0
5.)
Ans: k > 16
Find the values of k which will make the
6.)
roots of the equation real: ky 2  8 y  4  0
Ans: k = 5
Find the values of k which will make the
roots of the equation imaginary:
x2  6 x  k  0
Ans: k  4
31
Ans: k > 9
Review
7.)
Write the expression in simplest form:
8.)
3 72 x 2 y  5 x 18 y
Ans: 3x 2 y
32
Solve for x in simplest a + bi form:
2 x2  4 x  4  0
Ans: 1  i
1.)
The roots of the quadratic equation
3x 2  x  3  x 2  5 x  7 are
(1)
(2)
(3)
(4)
3.)
real, rational, and equal
real, rational, and unequal
real, irrational, and unequal
imaginary
(1)
(2)
(3)
(4)
Which might be the value of a discriminant of 4.)
a quadratic equation whose graph lies entirely
above the x-axis?
(1)
(2)
(3)
(4)
5.)
2.)
Day 7 – Review
If a quadratic equation with real coefficients
has a discriminant whose value is 25, then the
two roots must be
real, rational, and equal
real, rational, and unequal
real, irrational, and unequal
imaginary
In simplest form, what is the sum of 4i 10 and
11i 38 ?
-10
0
6
9
In simplest form, what is the product of 2i 10
and 4i 23 ?
33
6.)
Simplify: 4  2  12  4  48   75  2.5
7.)
Graph the sum of  4  5i and 2  7i .
9.)
Find the roots in simplest a + bi form: 3 x 2  6 x  6  0
8.)
Find the values of k for which the roots of the
equation will be imaginary: x 2  2 x  k  0
Directions: Perform the indicated operation and express in simplest a + bi form.
10.) (3 + 5i) + (1 + 2i)
11.) (–2 + 5i) – (1 + 5i)
34
12.)
 2  3i 4  6i 
13.)
What is the multiplicative inverse of 8  3i ?
14.)
What is the additive inverse of 6 + 5i?
15.)
What is the conjugate of 4  2i ?
16.)
2  5i
6i
17.)
If one root of a quadratic equation is 6 + i,
find the equation.
35