name: 3 unit complex numbers
TRANSCRIPT
Introduction to imaginary numbers
Do Now:
1. Factor completely: 212 2 24a x ax 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 1i .
So now let’s evaluate the first four powers of i:
Examples:
For #1 – 12, perform the indicated operation and simplify.
71. i 2.
22i 3. 15i 4. 135i
5. 5
33i 6. 3
84i 7. 7 157 15i i 8. 20 134 6i i
9. 12
3
18
6
i
i 10.
232 5i i 11.
120
11
15
25
i
i 12. 17 2513 8i i
13. The expression is equal to
(1) 1 (2) -1 (3) i (4) -i
1
2
3
4
i
i
i
i
Unit 3 – Complex Numbers Name:_________________
HW on Introduction to Imaginary Numbers
In #1 – 8, perform the indicated operation and express the result in simplest terms.
1. 2
33i 2. 915i 3. 134 2i i 4. 20113i
5. 246
43
16
8
i
i 6. 7. 8.
522i
9. Which expression is equivalent to ?
(1) 1 (2) -1 (3) i (4) -i
10. Expressed in simplest form, is equivalent to
(1) 1 (2) -1 (3) i (4) -i
11. Solve for x by completing the square. Place your answers in simplest radical form.
( 4) 6x x
Simplifying Radicals with Negative Radicands
Addition & Subtraction of Complex Numbers
Do Now:
1. If , then is equivalent to
(1) (2) (3) (4)
Notes:
What is a complex number???
Examples:
Simplify each radical:
1. 36 2. 2 49 3. 3 10 4. 50 5. 175
For each expression below, perform the indicated operation and place your answer in simplest a bi form.
6. (7 5 ) (8 3 )i i 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 3. 3 4 121
4. 3 48 5. 12
200 32
In #6 – 9, perform the indicated operation and place your answer in simplest a bi form.
6. (5 2 ) (7 4 ) (12 8 )i i i 7. 20 28 4 7
8. 3 2 (8 4)i i 9. 12 24 3 2 54
10. If , find the value of a.
11. Simplify: 3
52i 12. Factor completely: 26 5 21x x
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 and Joe has a
score of , what is their total score?
(1) (2) (3) (4)
2. Simplify: 4 126
Notes:
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.
Multiplicative Inverse
Additive Inverse
Conjugate
Perform the indicated operation and place your answers in simplest a bi form.
6. 2
4
i
i 7. 95 4 3i i i 8. 6 49 3 16
9. 4 3 2 3 10. 4
5 2
i
i
11.
26 i
12. 4
3 2 13. 4 9 3 2i i 14.
5
2i
HW on Operations with Complex Numbers
1. The product of and i is
(1) 7 (2) (3) (4)
2. The expression is equivalent to
(1) -2 (2) (3) (4)
3. The expression is equivalent to
(1) (2)
(3) (4)
4. The relationship between voltage, E, current, I, and resistance, Z, is given by the equation . If a circuit
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 )(2 )i i 6. 2 3
5
i
i
7. 2
2 49 8. 6 (10 2 )i i
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 3 2 3 7i xi i 14. Factor completely: 210 13 3x b xb b
15. Solve the system algebraically:
22 3
13
y x x
y x
Graphing Complex Numbers &
Magnitude of Complex Numbers
Do Now:
1. Simplify: (3 2 ) (4 )i 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 2 2a b
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 b) 8 2i c) 1 7i d) 9 i
2. Let 1
2
5 4
3 6
z i
z i
. Find and graph on the set of axes below:
a) the sum of 1z and 2z .
b) 1 2z z
In # 3 – 5, evaluate each expression. Place your answer in simplest radical form.
3. 3 4i 4. 5 12 5. (5 3 ) (7 6 )i i
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 4 ( ) 1 2i c di i , find c and d.
8. If 2 3( ) 5 ,f x x x find ( )f i and place your answer in simplest a + bi form.
9. Find the reciprocal of 3 5i . 10. Find the product of 2 6i and it’s conjugate.
11. Simplify: 175 12. Factor completely: 218 2x
Solving Quadratic Equations with Imaginary Roots
Do Now:
Simplify:
1. (9 9 ) ( 7 2 )i i 2. 2
5 32 3. 5 25
Practice!
1. Solve for x by using the quadratic formula. Place your answer in simplest a bi form.
23 10 3 0x x
2. Solve for x by completing the square. Place your answer in simplest a bi form.
2 4 10 0x x
3. Solve for x. Place your answer in simplest a bi form.
4 6
2 2
x
x
HW on Solving Quadratic Equations with Imaginary Roots
For # 1 – 4, solve for x and place your answers in simplest a bi form.
1. 2 7 4x x
2. 13
6xx
3. 22 6 5 0x x 4.
293 1
2
xx
5. Find the magnitude of 5 12i 6. Find the product of 9 i and its conjugate.
7. Place in simplest radical form: 27 2 49 363
Discriminant and Describing the Nature of the Roots
Do Now: Solve for x by completing the square.
22 12 10 0x x
In # 1 – 3, find the discriminant and describe the nature of the roots for each quadratic equation.
1. 22 7 4x x 2. 2 1 0x 3. ( 6) 9x x
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
4. Find all value(s) for k that makes the roots to the quadratic equation real, rational and equal.
2 9 0x kx
5. Find the smallest integer value of k that will make the roots imaginary.
2 5 3 0kx x
6. Find all values for k that make the roots to the following quadratic equation real.
23 2 0x x k
7. The roots of the equation 24 1 0x x are
(1) real, rational and equal (3) imaginary
(2) real, rational and unequal (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. 9
4 1xx
2. 21
6 03
x x 3. 12 9
4x
xx
4. Which is the smallest value of a that would make the roots to the equation 2 6 8 0ax x imaginary?
(1) 1 (2) 2 (3) 3 (4) 4
5. Which quadratic equation has equal roots?
(1) 2 5 6 0x x (2) 2 9 0x (3) 2 10 25 0x x (4) 2 7 13 0x x
6. Which quadratic equation has real, rational and unequal roots?
(1) 2 2 1 0x x (2) 2 5 7 0x x (3) 2 36 0x (4) 2 36 0x
7. The roots of the equation 23 2 7x x are
(1) real, rational and equal (3) imaginary
(2) real, rational and unequal (4) real and irrational
8. Find the largest integer value for k that make the roots to the equation 22 7 0x x k real.
9. Find the multiplicative inverse of 6i. 10. Find the magnitude of 4 5i .
11. Simplify: 4 48 2 3 12 12. In what quadrant does the difference of (3 2 )i
and ( 1 2 )i lie?
13. Solve for x by completing the square. Place your answer in simplest a bi form.
2 6 10 0x x
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:
1) 2)
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 𝑎𝑥2 + 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. 𝑥2 − 49 = 0
2 12 0x x 22 3 1 0x x
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) 𝑥2 − 𝑥 + 𝑘 = 0; 𝑟1 = −4 b) 𝑥2 + 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 2𝑥2 − 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) 3𝑥2 − 3𝑥 + 1 = 0 (3) 𝑥2 + 13 = 13𝑥
(2) 𝑥2 − 13 = 13𝑥 (4) 2𝑥2 + 2𝑥 + 2 = 0
4. If the product or the roots of 4𝑥2 − 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 3𝑥2 − 𝑥 + 2 = 5
6. Simplify the expression 2𝑖6 − 3𝑖2.
7. In which quadrant would you find the sum of (2 − √−4) + (−5 + √−36 ?
8. Express the roots of the equation 𝑥2 + 5𝑥 = 3𝑥 − 3 in simplest a + bi form.
9. What is the reciprocal of 12 – 3i?
Review Sheet: Complex Numbers
1. The complex number 3 25 2i i is equivalent to:
(1) 2 5i (2) 2 5i (3) 2 5i (4) 2 5i
2. The expression 192 is equivalent to:
(1) 8 3 (2) 3 8 (3) 8 3i (4) 3 8i
Perform the indicated operations and express your answer in simplest a bi form.
3. (6 49) (3 64) 4. ( 1 2 12) (8 5 48)
5. (6 2 ) ( 4 5 )i i 6. (2 9)(3 16)
7. 1 4
2 9
8.
2 2 44 (6 8 5 3 )i i i i
9. 2(3 4 )i 10.
6 7
2
i
i
11. Express the product of (5 6 )i and (3 5 )i 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 3 )i and (5 8 )i lie?
14. In which quadrant will the difference ( 5 11 ) ( 2 7 )i i 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 3 6z i . Leave your answer in simplest radical form.
18. Evaluate: 5 12i
19. If 1 5 2Z i and 2 3 5Z i ,
a) Graph 1Z and 2Z
b) Graph the sum of 1Z and 2Z
20. Solve for x in simplest a bi form: 23 12 21x x
21. Solve for x in simplest a bi form: 2 6 34x x
22. What is the sum and the product of the roots of the equation 22 4 1 0x x ?
23. If the sum of the roots of 2 4 6 0x x 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 23 5 4x x are
(1) real, rational, and unequal
(2) real, irrational, and unequal
(3) real, rational, and equal
(4) imaginary
25. The roots of the equation 2 4 13 0x x 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
1 4 2r i 2 4 2r i
27. If the equation 29 12 0x x k has equal roots, find the value of k.
28. For which value of k will the roots of 22 1 0x kx be real?
(1) 1 (2) 2 (3) 3 (4) 0
29. The roots of a quadratic equation are and .
a) Find the sum of the roots.
b) Find the product of the roots.
c) Write a quadratic equation with roots 1r and 2r .
30. Which quadratic equation has roots 3 i and 3 i ?
(1) 2 6 10 0x x
(2) 2 6 10 0x x
(3) 2 6 8 0x x
(4) 2 6 8 0x x
Negative
Zero Positive,
Perfect Square
Real, Rational, and
EQUAL
Imaginary
Positive, Non-Perfect Square
Real, Rational, and
UNEQUAL
Real, Irrational, and
UNEQUAL
Unit 2: Complex Numbers – Chapter Summary
1
1
4
3
2
1
i
ii
i
iiPowers of i: Simplify using the i – chart:
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.
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. Ex.
2 2a bi a b 2 23 5 ( 3) 5i
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.
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.
Using the Discriminant to Describe the Roots (Nature of the Roots): Use the b2 – 4ac part of the Quadratic Fmla
Sum and Product of the Roots: Sum = -b/a Product = c/a Writing an Equation Knowing the Sum and Product:
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.
2 ? ? 0x x 2 5 6 0x x