Algebra 2 and Trigonometry Honors - White Plains …...Day 1: Chapter 5-1: The Complex Numbers...

Algebra 2 and Trigonometry

Honors

Chapter 5: Complex Numbers

Name:______________________________

Teacher:____________________________

Pd: _______

COMPLEX NUMBERS

In this part of the unit we will:

Define the complex unit 𝑖 Simplify powers of 𝑖 Graphing complex numbers on the complex plane

Operate on complex numbers

Rationalize denominators involving 𝑖

Review, from previous packet, this time using 𝑖

Determine the nature of the roots of a quadratic

Determine the sum and the roots of a quadratic

Write the equation of a quadratic given the roots

Table of Contents

Day 1: Chapter 5-1: The Complex Numbers SWBAT: Simplify expressions involving complex numbers; simplify powers of i;

graph complex numbers

Pages in Packet #1-8

HW: Pages 9- 11

Day 2: Chapter 4-9: Operations with Complex Numbers

SWBAT: Add, Subtract, Multiply and Divide Complex Numbers

Pages in Packet #12-15

HW: Pages 16 - 17

Day 3: Chapter 5-2: Complex Roots of Quadratic Equations

SWBAT: Solve quadratic equations and higher order polynomials with imaginary

roots

Pages in Packet #18-21

HW: Pages 22 - 24

Day 4: Chapter 5-2: Nature of Roots

SWBAT: use the discriminant to describe the roots of a quadratic function and the

graph of the function

Pages in Packet #25-29

HW: Pages 30 -31

Day 5: Chapter 5-2: Sum and Product of Roots

SWBAT: Find the roots of higher order polynomials

Pages in Packet #32-36

HW: Page 37- 38 #3-11 , #18 -23, #33, #35, #38, #39 and #41

*Answer Keys – Pages #39 - 48

1

DAY 1: IMAGINARY NUMBERS

Recall that in the Real Number System, it is not possible to take the square root of a negative

quantity because whenever a real number is squared it is non-negative. This fact has a ramification

for finding the x-intercepts of a parabola, as Exercise #1 will illustrate.

Exercise #1: On the axes below, a sketch of 2y x is shown. Now, consider the parabola whose

equation is given in function notation as 2 1f x x .

Since we cannot solve this equation using Real Numbers, we introduce a new number, called i, the

basis of imaginary numbers. Its definition allows us to now have a result when finding the square

root of a negative real number. Its definition is given below.

y

x

(c) What can be said about the x-intercepts of the

function ?

(d) Algebraically, show that these intercepts do

not exist, in the Real Number System, by

solving the incomplete quadratic .

(a) How is the graph of shifted to produce

the graph of ?

(b) Create a quick sketch of on the axes

below.

THE DEFINITION OF THE IMAGINARY NUMBER i

2

Examples:

Simplifying radicals √−75 = √−𝟏 ∙ 𝟐𝟓 ∙ 3 = 5√−𝟏 ∙ √3 = 5𝒊√𝟑

Adding (or subtracting like radicals) 2𝑖√3 + 5𝑖√3 = (2𝑖 + 7𝑖)√3 = 7𝑖√3

In order to simplify negative square roots, do it exactly as you would regular radicals, but

have one of the factors be -1. √−1 Simplifies to i

Example:

Positive Square Root

√75

√25 ∙ √3

5√3

Negative Square Roots

√−75

√25 ∙ √−1 ∙ √3

5𝒊√3

Exercise #2: Simplify each of the following square roots in terms of i.

(a) 9 (b) 100 (c) 32 (d) 18

Exercise #3: Solve each of the following incomplete quadratics. Place your answers in simplest

radical form.

(a) 25 8 12x (b) 2120 2

2x (c) 22 10 36x

Exercise #4: Which of the following is equivalent to 5 6i i ?

(1) 30i (3) 30

(2) 11i (4) 11

3

Simplifying Powers of i

𝒊𝟎 = 𝟏 𝒊𝟏 = 𝒊 𝒊𝟐 = −𝟏 𝒊𝟑 = −𝒊

You may not leave powers of 𝒊 in your

answer.

After 𝒊𝟑, the pattern start repeating,

meaning that 𝑖4 = 𝑖0, 𝑖5 = 𝑖1… etc.

To reduce powers of 𝑖 divide the power by 4, and the remainder is your new power of 𝑖.

Example:

Evaluate 𝑖21

𝐴𝑛𝑠𝑤𝑒𝑟: 21 ÷ 4 = 5.25, 𝑠𝑜 𝑖21 = 𝑖1 = 𝑖.

A cheat on reducing a power of 𝑖: Divide the power by 4. You will get some number, and that number will either have no decimal (no remainder), or a .25, .5, or .75.

.25 represents 1

4, which means a remainder of ____ when divided by 4.

. 5 represents ½, which means 2

4, which means a remainder of ____ when divided by 4.

.75 represents 3

4, which means a remainder of ____ when divided by 4.

The equivalent power of 𝒊 is 𝒊𝒓𝒆𝒎𝒂𝒊𝒏𝒅𝒆𝒓.

Example: 2i83

Exercise #5: Simplify each of the following powers of i.

(a) 38i (b) 21i (c) 83i (d) 40i

(e) = 5i2 + 2i4 (f) = 2i5 +7i7

i

1 -1

-i

4

Exercise #6: Which of the following is equivalent to 16 23 265 3i i i ?

(1) 8 2i (3) 5 4i

(2) 4 3i (4) 2 7i

Exercise #7: Write as a complex number in standard form: (2 +4i5) + (1-9i

6) – (3 + i

7).

Exercise #8: Determine whether each statement is true or false. If it is true, give an example.

If it is false, give a counterexample.

a) The sum of two imaginary numbers is an imaginary number.

b) The product of two pure imaginary numbers is a real number.

c) A pure imaginary number is an imaginary number.

d) A complex number is a real number.

5

Graphing Complex Numbers

Due to their unique nature, complex numbers cannot be represented on a normal set of

coordinate axes.

In 1806, J. R. Argand developed a method for displaying complex numbers graphically as

a point in a coordinate plane. His method, called the Argand diagram, establishes a

relationship between the x-axis (real axis) with real numbers and the y-axis (imaginary

axis) with imaginary numbers.

In the Argand diagram, a complex number a + bi is the point (a,b)

or the vector from the origin to the point (a,b).

Graph the complex numbers:

1. 3 + 4i

2. 2 - 3i

3. -4 + 2i

4. 3 (which is really means )

5. 4i (which is really means )

6

The Parallelogram Rule for Complex Addition The parallelogram rule for complex addition says that if you are adding two complex numbers, then the sum of can be represented by the diagonal of the parallelogram that can be drawn using the two original vectors as adjacent sides.

Add 3 + 3i and -4 + 2i graphically. Subtract 3 + 4i from -2 + 2i

4

2

-2

-4

-10 -5 5 10

(1+4i)+(5+i)=6+5i

1+4i

5+i

6+5i

7

Exercise #9:

a) Represent the complex number 2 + 3i graphically.

b) Add graphically: c) Graphically Subtract

(-2+4i) and (4+i) (-1+i) from (3+2i)

8

Challenge

Summary/Closure:

Exit Ticket:

9

DAY 1: HW

IMAGINARY NUMBERS

COMMON CORE ALGEBRA II HOMEWORK

FLUENCY

1. The imaginary number i is defined as

(1) 1 (3) 4

(2) 1 (4) 2

1

2. Which of the following is equivalent to 128 ?

(1) 8 2 (3) 8 2

(2) 8i (4) 8 2i

3. The sum 9 16 is equal to

(1) 5 (3) 7i

(2) 5i (4) 7

4. Which of the following powers of i is not equal to one?

(1) 16i (3) 32i

(2) 26i (4) 48i

5. Which of the following represents all solutions to the equation 2110 7

3x ?

(1) 3x i (3) x i

(2) 5x i (4) 2x i

6. Solve each of the following incomplete quadratics. Express your answers in simplest radical form.

(a) 22 100 62x (b) 2220 2

3x

10

7. Which of the following represents the solution set of 2112 37

2x ?

(1) 7i (3) 5 2i

(2) 7 2i (4) 3 2i

8. Simplify each of the following powers of i into either 1,1, , or i i .

(a) 2i (b) 3i (c) 4i (d) 11i

(e) 41i (f) 30i (g) 25i (h) 36i

(i) 51i (j) 45i (k) 80i (l) 70i

9. Which of the following is equivalent to 7 8 9 10i i i i ?

(1) 1 (3) 1 i

(2) 2 i (4) 0

10. When simplified the sum 18 25 28 435 7 2 6i i i i is equal to

(1) 2 4i (3) 5 7i

(2) 3 i (4) 8 i

11. The product 6 2 4 3i i can be written as

(1) 24 6i (3) 2 5i

(2) 18 10i (4) 30 10i

11

12. Graph the sum of (2 + 3i) and (6 -2i) graphically.

13. Subtract (-3 +2i) from (5+ 4i), graphically.

12

DAY 2: COMPLEX NUMBERS

Warm - Up: Express each expression in terms of i and simplify.

1) 2√−75 2) √−45 + √−5 3) i49 4) i246 All numbers fall into a very broad category known as complex numbers. Complex numbers can always be

thought of as a combination of a real number with an imaginary number and will have the form:

where and a bi a b are real numbers

We say that a is the real part of the number and bi is the imaginary part of the number. These two parts, the real

and imaginary, cannot be combined. Like real numbers, complex numbers may be added and subtracted. The

key to these operations is that real components can combine with real components and imaginary with

imaginary.

Exercise #1: Find each of the following sums and differences.

(a) 2 7 6 2i i (b) 8 4 12i i (c) 5 3 2 7i i (d) 3 5 8 2i i

Exercise #2: Which of the following represents the sum of 6 2 and 8 5i i ?

(1) 5i (3) 2 3i

(2) 2 3i (4) 5i

13

Adding and subtracting complex numbers is straightforward because the process is similar to combining

algebraic expressions that have like terms. The complex numbers are closed under addition and subtraction,

i.e. when you add or subtract two complex numbers the results is a complex number as well. But, is

multiplication closed?

Exercise #3 Find the following products. Write each of your answers as a complex number in the form a bi .

(a) 3 5 7 2i i (b) 2 6 3 2i i (c) 4 5 3i i

Exercise #4: Consider the more general product a bi c di where constants a, b, c and d are real numbers.

(c) Under what conditions will the product of two complex numbers always be a purely imaginary number?

Check by generating a pair of complex numbers that have this type of product.

Exercise #5: Determine the result of the calculation below in simplest a bi form.

5 2 3 4 2 3i i i i

(a) Show that the real component of this product

will always be ac bd .

(b) Show that the product of 2 3i and 4 6i

results in a purely real number.

14

Exercise #6: Which of the following products would be a purely real number?

(1) 4 2 3i i (3) 5 2 5 2i i

(2) 3 2 4i i (4) 6 3 6 3i i

When dividing complex numbers, we want an answer with a real denominator. To do this, we multiply by the complex conjugate of the denominator (similar to rationalizing a denominator in Chapter 3). Don’t forget to simplify your answer.

Example 1: 2+3𝑖

1+2𝑖

2+3𝑖

1+2𝑖 (

1−2𝑖

1−2𝑖) =

Exercise #7: Find the quotient of i

i

2

23. Exercise #8: Find the quotient:

8+𝑖

2−𝑖.

Exercise #9: Find the quotient: 2+𝑖

4𝑖

We do not like an

imaginary denominator!

Multiply by the complex

conjugate of the

denominator.

15

Multiplicative Inverses of Complex Numbers

The multiplicative inverse of a real number n is 1

𝑛, where n ≠ 0 because:

n ∙ 1

𝑛 = 1 and

1

𝑛 ∙ n = 1

The multiplicative inverse of a complex number a + bi is 1

𝑎+𝑏𝑖 , where

a+bi ≠ 0 + 0i because:

(a + bi) ∙ 1

𝑎+𝑏𝑖 = 1 and

1

𝑎+𝑏𝑖 ∙ (a + bi) = 1

Example: Write the multiplicative inverse of 2 + 4i.

Answer: 1

2+4𝑖 But we don’t like our answer with an imaginary denominator!

So...you must rationalize!

Answer (in correct form): 2−4𝑖

20 =

1−2𝑖

10

Exercise #10: Write the multiplicative inverse of 3 – 2i.

Exercise #11: Write the multiplicative inverse of -2 + 3i.

16

DAY 2: HW

COMPLEX NUMBERS

COMMON CORE ALGEBRA II HOMEWORK

FLUENCY

1. Find each of the following sum or difference.

(a) 6 3 2 9i i (b) 7 3 5i i (c) 10 3 6 8i i

(d) 2 7 15 6i i (e) 15 2 5 5i i (f) 1 5 6i i

2. Which of the following is equivalent to 3 5 2 2 3 6i i ?

(1) 9 18i (3) 9 6i

(2) 21 8i (4) 21 2i

3. Find each of the following products in simplest a bi form.

(a) 5 2 1 7i i (b) 3 9 2 4i i (c) 4 2 6i i

4. Complex conjugates are two complex numbers that have the form a bi and a bi . Find the following

products of complex conjugates:

(a) 5 7 5 7i i (b) 10 10i i (c) 3 8 3 8i i

(d) What's true about the product of two complex conjugates?

17

5. Show that the product of a bi and a bi is the purely real number 2 2a b .

6. The product of 8 2i and its conjugate is equal to

(1) 64 4i (3) 68

(2) 60 (4) 60 4i

7. The complex computation 6 2 6 2 3 4 3 4i i i i can be simplified to

(1) 15 (3) 10

(2) 39 (4) 35

8. Perform the following complex calculation. Express your answer in simplest a bi form.

8 5 3 2 4 4i i i i

9. Perform the following complex calculation. Express your answer in simplest a bi form.

7 3 5 4 2 6 7i i i

10. Simplify the following complex expression. Write your answer in simplest a bi form.

2 2

5 2 2i i

11. Find the quotient: 1−3𝑖

2−7𝑖

12. Find the multiplicative inverse: 5 +2i.

18

DAY 3: SOLVING QUADRATIC EQUATIONS WITH COMPLEX SOLUTIONS

COMMON CORE ALGEBRA II

As we saw in the last unit, the roots or zeroes of any quadratic equation can be found using the quadratic

formula:

2 4

2

b b acx

a

Since this formula contains a square root, it is fair to investigate solutions to quadratic equations now when the

quantity 2 4b ac , known as the discriminant, is negative. Up to this point, we would have concluded that if

the discriminant was negative, the quadratic had no (real) solutions. But, now it can have complex solutions.

Exercise #1: Use the quadratic formula to find all solutions to the following equation. Express your answers in

simplest a bi form. 2 4 29 0x x

As long as our solutions can include complex numbers, then any quadratic equation can be solved for two roots.

Exercise #2: Solve each of the following quadratic equations. Express your answers in simplest a bi form.

(a) 2 5 30 7 10x x x (b)

2 16 15 10 4x x x

19

There is an interesting connection between the x-intercepts (zeroes) of a parabola and complex roots with non-

zero imaginary parts. The next exercise illustrates this important concept.

Exercise #3: Consider the parabola whose equation is 2 6 13y x x .

Exercise #4: Use the discriminant of each of the following quadratics to determine whether it has x-intercepts.

(a) 2 3 10y x x (b)

2 6 10y x x (c) 22 3 5y x x

Exercise #5: Which of the following quadratic functions, when graphed, would not cross the x-axis?

(1) 22 5 3y x x (3)

24 4 5y x x

(2) 2 6y x x (4)

23 13 4y x x

Exercise #6: Solve for x:

(a) Algebraically find the x-intercepts of this

parabola. Express your answers in simplest

a bi form.

(b) Using your calculator, sketch a graph of the

parabola on the axes below. Use the window

indicated.

(c) From your answers to (a) and (b), what can be

said about parabolas whose zeroes are

complex roots with non-zero imaginary parts?

20

Solving Higher Degree Polynomial Equations

How many roots should we expect to find? A polynomial of degree n will have n roots, some of

which may be multiple roots (they repeat). For example, is a

polynomial of degree 3 (highest power) and as such will have 3 roots. This equation is really

giving solutions of x = 1 and x = 4 (repeated).

Exercise #7: Find the roots of f(x) = 𝑥3 − 3𝑥2 − 4𝑥 + 12

Exercise #8: Find the roots of f(x) = 𝑥3 + 5𝑥2 + 9𝑥 + 45

Exercise #9: Find the roots of f(x) = 𝑥3 − 2𝑥2 + 10𝑥 Exercise #10: Find the roots of (x) = 𝑥4 − 12𝑥2 + 64

Exercise #11: Find the roots of (x) = 𝑥5 + 10𝑥3 + 9𝑥

21

Challenge Summary/Closure:

Exit Ticket:

22

DAY 3: HW

SOLVING QUADRATIC EQUATIONS WITH COMPLEX SOLUTIONS

COMMON CORE ALGEBRA II HOMEWORK

FLUENCY

1. Solve each of the following quadratic equations. Express your solutions in simplest a bi form. Check.

(a) 2 4 20 12 5x x x (b)

2 1x x

(c) 22 25 27 15 10x x x (d)

28 36 24 12 5x x x

(e) 2 6 15 8 2x x x (f)

24 38 50 10 35x x x

23

2. Which of the following represents the solution set to the equation 2 2 2 0x x ?

(1) 1 or 2x (3) 2x i

(2) 1 2x i (4) 1x i

3. The solutions to the equation 2 6 11 0x x are

(1) 3 2x i (3) 6 11x i

(2) 3 2 2x i (4) 6 2 11x i

4. Using the discriminant, 2 4b ac , determine whether each of the following quadratics has real or imaginary

zeroes.

(a) 22 7 6y x x (b)

23 2 1y x x (c) 2 8 14y x x

(d) 22 12 26y x x (e)

22 6 5y x x (f) 24 4 1y x x

5. Which of the following quadratics, if graphed, would lie entirely above the x-axis? Try to use the

discriminant to solve this problem and then graph to check.

(1) 22 21y x x (3)

2 4 7y x x

(2) 2 6y x x (4)

2 10 16y x x

24

REASONING

6. For what values of c will the quadratic 2 6y x x c have no real zeroes? Set up and solve an inequality

for this problem.

7. Find the roots of f(x) = 𝑥3 − 8𝑥2 + 24𝑥.

8. Find the roots of f(x) = 𝑥4 − 13𝑥2 + 36.

9. Find the roots of f(x) = 𝑥5 + 12𝑥3 + 32𝑥.

25

DAY 4: THE DISCRIMINANT OF A QUADRATIC

Complete the table below. “Nature” of the Roots

Equation ROOTS?

(x-intercepts)

Roots are:

Real or Imaginary?

Roots are:

Rat’l or Irr’l?

Roots are:

= or

1. 𝑦 = 𝑥2 − 2𝑥 − 3

2. 𝑦 = 𝑥2 − 6𝑥 + 7

3. 𝑦 = 𝑥2 − 4𝑥 + 4

4. 𝑦 = 𝑥2 − 4𝑥 + 5

26

Since the roots of a quadratic can be found using 2 4

2

b b acx

a

, if , , and a b c are all rational numbers, the

quantity under the square root, 2 4b ac , truly dictates what type of numbers the roots of a quadratic (and its x-

intercepts) turn out to be. It reduces down to four cases which will be explored in Exercise #1.

Exercise #1: For each of the following quadratics, calculate its discriminant, its roots, and state the number and

nature (whether they are rational, irrational or imaginary) of the roots.

(a) Case I – The Discriminant is a Perfect Square - 2 3 10 0x x .

2 4D b ac Roots: Number and Nature:

(b) Case II – The Discriminant is Not a Perfect Square - 2 6 7 0x x .

2 4D b ac Roots: Number and Nature:

(c) Case III – The Discriminant is Equal to Zero - 2 10 25 0x x .

2 4D b ac Roots: Number and Nature:

(d) Case IV – The Discriminant is Less than Zero - 2 8 20 0x x

2 4D b ac Roots: Number and Nature:

27

In the last lesson, we explored Case IV extensively. In the case where the discriminant is negative, the roots of

the quadratic are imaginary and it does not have x-intercepts (i.e. it does not cross the x-axis).

Exercise #2: By using only the discriminant, determine the number and nature of the roots of each of the

following quadratics.

(a) 22 7 4 0x x (b)

2 8 25 0x x (c) 24 4 1 0x x

(d) 2 6 15 0x x (e)

24 4 7 0x x (f) 23 7 2 0x x

Exercise #3: Consider the quadratic function whose equation is 2 4 4y x x . Determine the number of x-

intercepts of this quadratic from the value of its discriminant. Then, sketch its graph on the axes given. We say

that this parabola is tangent to the x-axis.

Exercise #4: Which of the following parabolas has two unequal, rational x-intercepts?

(1) 2 2 1y x x (3) 2 8 16y x x

(2) 2 2 15y x x (4) 2 3 5y x x

Exercise #5: For what values of a will the parabola 2 4 2y ax x not cross the x-axis?

28

Challenge

SUMMARY

EXIT TICKET

29

Summary:

The discriminant is the name given to the expression that appears under

the square root (radical) sign in the quadratic formula.

Quadratic Formula

Discriminant

The discriminant tells you about the "nature" of the roots of a quadratic equation given that a,

band c are rational numbers. It quickly tells you the number of real roots, or in other words, the

number of x-intercepts, associated with a quadratic equation.

There are four situations:

If the discriminant is…..

(b2 – 4ac)

The nature of the roots is:

Inequality Number of Roots

Negative Imaginary b2 – 4ac < 0 2 Zero Real, Rational, Equal b2 – 4ac = 0 1 Positive, Perfect Square

Real, Rational, Unequal

b2 – 4ac > 0, perfect square

2

Positive, Non-perfect Square

Real, Irrational, Unequal

b2 – 4ac >0, non-perfect square

2

**

30

DAY 4: HW

THE DISCRIMINANT OF A QUADRATIC

COMMON CORE ALGEBRA II HOMEWORK

SKILLS

1. For each of the following quadratic equations, determine the number and the nature of the roots by first

calculating the quadratic’s discriminant.

(a) 22 4 5 0x x (b)

29 12 4 0x x (c) 24 13 3 0x x

(d) 2 8 11 0x x (e)

24 4 7 0x x (f) 236 12 1 0x x

(g) 23 4 8 0x x (h)

23 8 4 0x x (i) 2 8 41 0x x

2. The roots of 2 4 7 0x x are

(1) unequal and rational (3) unequal and irrational

(2) unequal and imaginary (4) equal and rational

3. Which of the following quadratics has imaginary roots?

(1) 2 3 5 0x x (3)

22 3 1 0x x

(2) 2 6 10 0x x (4)

2 7 10 0x x

4. Which of the following quadratic, when graphed, would touch the x-axis exactly once?

(1) 2 2 3y x x (3) 2 5 2y x x

(2) 22 3 7y x x (4) 2 12 36y x x

31

5. If graphed, which of the following parabolas would lie entirely below the x-axis?

(1) 2 5 10y x x (3) 22 6 5y x x

(2) 22 5 3y x x (4) 2 6 9y x x

6. Which parabola below, when graphed, would cross the x-axis at two unequal irrational locations?

(1) 22 11 12y x x (3) 29 6 1y x x

(2) 2 2 4y x x (4) 22 4 9y x x

REASONING

7. Determine all values of a that will cause the parabola 2 10 1y ax x to cross the x-axis at two distinct

locations.

8. Consider the parabola whose equation is 2 4y x x and the line whose equation is 2y x b , where b is

some unknown constant. Determine the value of b such that the line and the parabola will intersect at

exactly one location. Then, sketch the system of equations on the axes below. Label their intersection

point.

32

Day 5 - Sum and Product of the Roots of a Quadratic Equation

SWBAT: 1) find the sum and product of the roots from a quadratic equation, 2) use the sum and product of the

roots to write a quadratic equation, 3) using the sum and product to find the missing value of the quadratic

equation.

Warm - Up:

1) Find the roots of the quadratic equation by factoring:

x2 - 5x - 14 = 0

What is the sum of the roots?

What is the product of the roots?

2) Find the roots of the quadratic equation by factoring: x2 – 6x + 8 = 0

What is the sum of the roots?

What is the product of the roots?

What conclusion can you draw about the sum of the roots of and product of the roots of a

quadratic equation in the form ax2+bx + c = 0?

33

The sum of the roots of any quadratic

Sum = −𝒃

𝒂

The product of the roots of any quadratic

Product = 𝒄

𝒂

Concept 1: Calculating the sum and product of roots given a quadratic equation

Example 1: What is the sum and product of the roots of 2𝑥2 − 3𝑥 + 5?

Concept 2: Writing an equation of a quadratic given the sum and roots

Example 2: Write a quadratic equation whose roots have the indicated sum and product.

Sum = 4, product = 3

Concept 3: Writing an equation of a quadratic given the roots

x2 – (sum)x + Product = 0

Example 3: Write a quadratic equation with integer coefficients that has roots of 8 and -3.

** Always assume a = 1 **

Example 4: Write a quadratic equation with integer coefficients that has roots of −7 and -2.

Format: x2 – (sum)x + Product = 0

34

Example 5: Write a quadratic equation that has 2 − 3√5 as a root.

Note** Irrational roots come in conjugate pairs, so if 2 − 3√5 is one root, then the other root must be

Example 6: Write a quadratic equation that has 2 + 𝑖 as a root.

Note** Complex roots come in conjugate pairs, so if 2 + 𝑖 is one root, then the other root must be

Concept 4: Finding “a, b, or c” and a root given one of the two roots

Example 7: If one root of x2 – 6x + k = 0 is 4, find the other root.

Try These!

Example 1: Write a quadratic equation with the following sum and product:

Sum = 8

Product = 25

Example 2: Given: 3x2 + 6x – 3

a) Find the sum of the roots. b) Find the product of the roots

Example 3: Given the quadratic equation x2 - 8x + k = 0 and r1= 5, find r2.

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Example 4: Write a quadratic equation with roots 5 and 7.

Example 5: Write a quadratic equation with roots 3 + 2i and 3 – 2i.

Example 6: Write a quadratic equation with roots 42

1and .

Example 7:

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Summary/Closure:

Exit Ticket:

37

DAY 5: HW

Sum and Product of Roots

Homework #5 page 37- 38 #3-11, #18 -23, #33, #35, #38, #39 and #41

38

39

ANSWER KEYS

Answer Key

Day 1

IMAGINARY NUMBERS

COMMON CORE ALGEBRA II HOMEWORK

FLUENCY

1. The imaginary number i is defined as

(1) 1 (3) 4

(2) 1 (4) 2

1

2. Which of the following is equivalent to 128 ?

(1) 8 2 (3) 8 2

(2) 8i (4) 8 2i

3. The sum 9 16 is equal to

(1) 5 (3) 7i

(2) 5i (4) 7

4. Which of the following powers of i is not equal to one?

(1) 16i (3) 32i

(2) 26i (4) 48i

5. Which of the following represents all solutions to the equation 2110 7

3x ?

(1) 3x i (3) x i

(2) 5x i (4) 2x i

6. Solve each of the following incomplete quadratics. Express your answers in simplest radical form.

(a) 22 100 62x (b) 2220 2

3x

(2)

(4)

(2) Since 26 is not a multiple of 4 this power of i is

not equal to 1.

(1)

(3)

40

7. Which of the following represents the solution set of 2112 37

2x ?

(1) 7i (3) 5 2i

(2) 7 2i (4) 3 2i

8. Simplify each of the following powers of i into either 1,1, , or i i .

(a) 2i (b) 3i (c) 4i (d) 11i

(e) 41i (f) 30i (g) 25i (h) 36i

(i) 51i (j) 45i (k) 80i (l) 70i

9. Which of the following is equivalent to 7 8 9 10i i i i ?

(1) 1 (3) 1 i

(2) 2 i (4) 0

10. When simplified the sum 18 25 28 435 7 2 6i i i i is equal to

(1) 2 4i (3) 5 7i

(2) 3 i (4) 8 i

11. The product 6 2 4 3i i can be written as

(1) 24 6i (3) 2 5i

(2) 18 10i (4) 30 10i

(3)

(4)

(2)

(4)

41

12.

13.

Day 2:

COMPLEX NUMBERS

COMMON CORE ALGEBRA II HOMEWORK

FLUENCY

1. Find each of the following sum or difference.

(a) 6 3 2 9i i (b) 7 3 5i i (c) 10 3 6 8i i

(d) 2 7 15 6i i (e) 15 2 5 5i i (f) 1 5 6i i

2. Which of the following is equivalent to 3 5 2 2 3 6i i ?

(1) 9 18i (3) 9 6i

(2) 21 8i (4) 21 2i

(1)

42

3. Find each of the following products in simplest a bi form.

(a) 5 2 1 7i i (b) 3 9 2 4i i (c) 4 2 6i i

4. Complex conjugates are two complex numbers that have the form a bi and a bi . Find the following

products of complex conjugates:

(a) 5 7 5 7i i (b) 10 10i i (c) 3 8 3 8i i

(d) What's true about the product of two complex conjugates?

5. Show that the product of a bi and a bi is the purely real number 2 2a b .

6. The product of 8 2i and its conjugate is equal to

(1) 64 4i (3) 68

(2) 60 (4) 60 4i

7. The complex computation 6 2 6 2 3 4 3 4i i i i can be simplified to

(1) 15 (3) 10

(2) 39 (4) 35

8. Perform the following complex calculation. Express your answer in simplest a bi form.

8 5 3 2 4 4i i i i

The product of two complex conjugates always results in a purely real number.

(3)

(1)

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9. Perform the following complex calculation. Express your answer in simplest a bi form.

7 3 5 4 2 6 7i i i

10. Simplify the following complex expression. Write your answer in simplest a bi form.

2 2

5 2 2i i

11. 53

23 i 12.

29

25 i

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DAY 3:

SOLVING QUADRATIC EQUATIONS WITH COMPLEX SOLUTIONS

COMMON CORE ALGEBRA II HOMEWORK

FLUENCY

1. Solve each of the following quadratic equations. Express your solutions in simplest a bi form. Check.

(a) 2 4 20 12 5x x x (b) 2 1x x

(c) 22 25 27 15 10x x x (d) 28 36 24 12 5x x x

(e) 2 6 15 8 2x x x (f)

24 38 50 10 35x x x

45

2. Which of the following represents the solution set to the equation 2 2 2 0x x ?

(1) 1 or 2x (3) 2x i

(2) 1 2x i (4) 1x i

3. The solutions to the equation 2 6 11 0x x are

(1) 3 2x i (3) 6 11x i

(2) 3 2 2x i (4) 6 2 11x i

4. Using the determinant, 2 4b ac , determine whether each of the following quadratics has real or imaginary

zeroes.

(a) 22 7 6y x x (b)

23 2 1y x x (c) 2 8 14y x x

(d) 22 12 26y x x (e)

22 6 5y x x (f) 24 4 1y x x

5. Which of the following quadratics, if graphed, would lie entirely above the x-axis? Try to use the

discriminant to solve this problem and then graph to check.

(1) 22 21y x x (3)

2 4 7y x x

(2) 2 6y x x (4)

2 10 16y x x

REASONING

6. 7. {0, 224 i } 8. {3,-3,2,-2} 9. 22,2 i

(4)

(1)

REAL

IMAGINARY

REAL

IMAGINARY

IMAGINARY

REAL

(3)

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Day 4:

THE DISCRIMINANT OF A QUADRATIC

COMMON CORE ALGEBRA II HOMEWORK

SKILLS

1. For each of the following quadratic equations, determine the number and the nature of the roots by first

calculating the quadratic’s discriminant.

(a) 22 4 5 0x x (b) 29 12 4 0x x (c) 24 13 3 0x x

(d) 2 8 11 0x x (e) 24 4 7 0x x (f) 236 12 1 0x x

(g) 23 4 8 0x x (h) 23 8 4 0x x (i) 2 8 41 0x x

2. The roots of 2 4 7 0x x are

(1) unequal and rational (3) unequal and irrational

(2) unequal and imaginary (4) equal and rational

3. Which of the following quadratics has imaginary roots?

(1) 2 3 5 0x x (3) 22 3 1 0x x

(2) 2 6 10 0x x (4)

2 7 10 0x x

4. Which of the following quadratic, when graphed, would touch the x-axis exactly once?

(1) 2 2 3y x x (3)

2 5 2y x x

(2) 22 3 7y x x (4)

2 12 36y x x

Two unequal, imaginary roots.

One double, rational root.

Two unequal, rational roots.

Two unequal, irrational roots.

Two unequal, irrational roots.

One double, rational root.

Two unequal, imaginary roots.

Two unequal, rational roots.

Two unequal, imaginary roots.

Two unequal, irrational roots.

(3)

Two unequal, irrational

(2)

(4)

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5. If graphed, which of the following parabolas would lie entirely below the x-axis?

(1) 2 5 10y x x (3)

22 6 5y x x

(2) 22 5 3y x x (4)

2 6 9y x x

6. Which parabola below, when graphed, would cross the x-axis at two unequal irrational locations?

(1) 22 11 12y x x (3)

29 6 1y x x

(2) 2 2 4y x x (4)

22 4 9y x x

REASONING

7. Determine all values of a that will cause the parabola 2 10 1y ax x to cross the x-axis at two distinct

locations.

8. Consider the parabola whose equation is 2 4y x x and the line whose equation is 2y x b , where b is

some unknown constant. Determine the value of b such that the line and the parabola will intersect at

exactly one location. Then, sketch the system of equations on the axes below. Label their intersection

point.

(3)

(2)

y

x

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Day 5: