Secondary 2
Chapter 14/15
Secondary II
Chapter 14 – Solving Quadratic Equations & Inequalities
Chapter 15 – Real Number Systems
2014/2015
Date
Section Assignment
Concept
A: 2/27
B: 3/2
15.1/15.2
Number System
Properties of Real Numbers
- Worksheet 15.1 & 15.2
ACT- Juniors
3/3
A: 3/4
B: 3/5
15.3/15.4
- Worksheet 15.3 & 15.4
Imaginary and Complex Numbers
Complex Number Operations
A: 3/6
B: 3/9
14.1/15.5
- Worksheet 14.1 & 15.5
Quadratic Formula with real and
Imaginary Solutions
A: 3/10
B: 3/11
14.3
- Worksheet 14.3
Solving Quadratic Inequalities
A: 3/12
B: 3/13
14.4
- Worksheet 14.4
Systems of Quadratic Equations
A: 3/16
B: 3/17
Review
A: 3/18
B: 3/19
Review
Chapter 14 & 15 TEST
Late and absent work will be due on the day of the review (absences must be excused). The review
assignment must be turned in on test day. All required work must be complete to get the curve on the test.
Remember, you are still required to take the test on the scheduled day even if you miss the review, so come
prepared. If you are absent on test day, you will be required to take the test in class the day you return. You
will not receive the curve on the test if you are absent on test day unless you take the test prior to your
absence.
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Secondary 2
Chapter 14/15
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Secondary 2
Chapter 14/15
Chapter 14 – Solving Quadratic Equations & Inequalities
Chapter 15 – Real Number Systems
15.1 - The Numbers of the Real Number System
(Standards: A.APR.1, N.RN.3)
Define the following number sets, then answer the following questions
Natural Numbers:
Whole Numbers:
Integers:
Rational Numbers:
Irrational Numbers:
Example1: Identify whether each given number set is closed or not closed under the operations
addition, subtraction, multiplication, and division.
a) The set of Natural numbers
b) The set of rational number
c)
How many natural numbers are between -1 and 1? List them
d) How many whole numbers are between -1 and 1? List them
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Chapter 14/15
Example 2: Draw a Venn diagram using the following numbers: natural numbers, rational numbers,
whole numbers, irrational numbers and integers
Example 3: Consider the following equations
A: 3x = 9
B: x + 1 = 5
C: 8x = 4
D: 𝑥 2 = 2
E: x + 5 = 1
F: 𝑥 2 = 4
a) Suppose you can only use natural numbers to evaluate the equations. Which equations could
you solve?
b) Suppose you can only use integers to evaluate the equations. Which equations could you solve?
c) Suppose you can only use rational numbers to evaluate the equations. Which equations could
you solve?
All irrational numbers have an infinite number of nonrepeating decimal places. Any infinite decimal that
repeats single digits or blocks of digits can be written as a fraction, therefore it is a rational number.
You can use a graphing calculator to evaluate repeating decimals.
1. Enter the decimal to the last space on your screen.
2. Press MATH, then 1: > 𝐅𝐫𝐚𝐜 .
Example 4: Represent each decimal as a fraction
a) .2222….
b) .512512….
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Chapter 14/15
15.2 - Real Number Properties
(Standards: N.CN.1, N.CN.2)
Table of Properties
Let a, b, and c be real numbers, variables, or algebraic expressions.
(These are properties you need to know.)
Property
1.
Commutative Property of Addition
a+b=b+a
2.
Commutative Property of Multiplication
a•b=b•a
3.
Associative Property of Addition
a+(b+c)=(a+b)+c
4.
Associative Property of Multiplication
a•(b•c)=(a•b)•c
5.
Distributive Property
a•(b+c)=a•b+a•c
6.
Additive Identity Property
a+0=a
7.
Multiplicative Identity Property
a• 1=a
8.
Additive Inverse Property
a + ( -a ) = 0
9.
Multiplicative Inverse Property
Example
Note: a cannot = 0
10. Zero Property
a•0=0
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Chapter 14/15
You may also write these in set notation for example the commutative property of addition would be
written
∀ 𝑎, 𝑏 ∈ ℝ, 𝑎 + 𝑏 = 𝑏 + 𝑎
The symbol ∀ is read as “for all”. The symbol ∈ is read as “is an element of” or “are elements of”. The
symbol ℝ is the set of all real numbers”.
Therefore the statement would be read as:
For all numbers a and b that are elements of the set of real numbers a plus b equals b + a.
Example 5: Each expression has been simplified or solved one step at a time. Next to each step, identify
the property, transformation or simplification used in the step.
a) 4 + 5(x + 7)
4 + (5x + 35)
_____________________________________
5x + 4 + 35
_____________________________________
5x + (4 + 35)
_____________________________________
5x + 39
_____________________________________
b) 7x + 4 -3(2x – 7)
7x + 4 – 6x + 21
______________________________________
7x – 6x + 4 + 21
______________________________________
(7x – 6x) + (4 + 21)
______________________________________
X + 25
______________________________________
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Chapter 14/15
Additional Notes
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Chapter 14/15
15.3 & 15.4 - Imaginary and Complex Numbers, and Operations
(Standards: A.REI.4)
Exponentiation:
Example 1: Simplify the expression if possible
a. 𝟒𝟐 =
𝟏
c. (−𝟒)𝟐 =
b. 𝟒𝟐 =
𝟏
d. −(𝟒)𝟐 =
e. (−𝟒)𝟐 =
Looking at part e. Is the set of real numbers closed for all real number exponents?
In order for the real numbers to be closed for all real number exponents, there must be some way to
calculate the square root of a negative number. Mathematicians defined this with the number i.
𝑖 = √−𝟏
𝑡ℎ𝑒𝑟𝑒𝑓𝑜𝑟𝑒
𝑖 2 = −1
Example 2: Find the values of i?
i =
i2 =
i3=
i4 =
i5 =
i6 =
i7 =
i8 =
a. i 13 =
b. i 20 =
c. i 47 =
d. i 34 =
e. 𝑖 50 =
f. 𝑖 104 =
How would you do 𝑖 −2 𝑜𝑟 𝑖 −400
?
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Secondary 2
Chapter 14/15
Now you can simplify expressions involving negative roots by using i.
Example 3: Simplify each expression
b. √−12
c. 5 + √−50
e. 𝑥𝑖 + 𝑥𝑖
f. 𝑥𝑖 − 𝑥𝑖
g. 3𝑥 + 5𝑖 − 2 − 3𝑖 − 𝑥𝑖 + 𝑥
h. (𝑥 + 𝑖)2
i. (2𝑥 − 𝑖)(𝑥 − 3𝑖)
j. √64 − √−63
k. Xi + XY
a. √−4
d.
6−√−8
2
l. (𝑥 + 𝑖)(𝑥 + 3) + (𝑥 + 3𝑖)(𝑥 + 3)
The set of imaginary numbers is the set of all numbers written in the form a + bi, where a is the real
part and bi is the imaginary part. The set of complex numbers is the set written in the form a + bi.
Example 4: Identify whether each number is a complex number. Explain your reasoning
a) i
b) 3
c) 𝜋 + 3.2𝑖
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Secondary 2
Chapter 14/15
Example 5: Create a diagram to show the relationship between each set of the numbers shown.
Complex numbers, imaginary numbers, integers, irrational numbers, natural numbers,
rational numbers, real numbers, whole numbers
Define the following and give an example
-
Complex conjugates:
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Secondary 2
Chapter 14/15
Additional Notes
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Secondary 2
Chapter 14/15
14.1 & 15.5 - The Quadratic Formula and Solving Quadratics with complex solutions
When you encounter a quadratic equation of function that you cannot factor/or is difficult to factor you
can always use the QUADRATIC FORMULA
The Quadratic Formula:
b  b2  4ac
x
2a
Remember: 𝑦 = 𝑎𝑥 2 + 𝑏𝑥 + 𝑐 = 0
Where does this Formula come from??
You can derive the quadratic formula as follows:
The original equation
𝒂𝒙𝟐 + 𝒃𝒙 + 𝒄 = 𝟎
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Secondary 2
Chapter 14/15
Example 1: Use the quadratic equation to find the exact and the approximate zeros to 3 decimal
places. Beware there could be imaginary zeros!
a) 𝒇(𝒙) = −𝟐𝒙𝟐 − 𝟑𝒙 + 𝟕
c)
b) f(𝒙) = 𝒙𝟐 − 𝟕𝒙 + 𝟏𝟏
𝒇(𝒙) = 𝟐𝒙𝟐 + 𝟐
d) 𝒇(𝒙) = −𝟐𝒙𝟐 − 𝟖𝒙 − 𝟏𝟖
Let’s think about the zeros of quadratic functions
You can determine how many zeros there are by finding the discriminant.
The discriminant is equal to: 𝒃𝟐 − 𝟒𝒂𝒄
Finding the discriminant helps you know many real zeros does a quadratic function has.
Draw each and look at them graphically?
Number of real zeros
Discriminant
Graphically
0
1
2
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Chapter 14/15
Example 2: Use the discriminant to determine the number of root/zeros that each function has.
a) 𝒇(𝒙) = 𝟐𝒙𝟐 + 𝟏𝟐𝒙 + 𝟐𝟎
c)
𝒇(𝒙) = 𝟗𝒙𝟐 + 𝟏𝟐𝒙 + 𝟒
b) 𝒇(𝒙) = 𝟑𝒙𝟐 + 𝟕𝒙 − 𝟐𝟎
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Chapter 14/15
Additional Notes
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Chapter 14/15
14.3 - Solving Quadratic Inequalities
(Standards: A.REI.4)
Example 1: A firework is shot straight up into the air with an initial velocity of 500 feet per second from
5 feet of the ground.
a) Write a quadratic equation to represent this situation and identify the variables.
b) Sketch the graph. Hint: Windows [0, 40] by [0, 4500]
c) Determine when the firework will hit the ground.
d) Determine when the firework will be 2000 feet off the ground?
e) When will the firework be higher than 2000ft? Write in interval notation.
f)
When is the firework below 2000 ft? Write in interval notation.
g) Write a quadratic inequality that represents the times when the firework is below 2000ft.
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Chapter 14/15
Steps to solving a quadratic inequality:
1234-
Set equal to zero
Find the roots (solve) by factoring of the quadratic formula
Plot the roots on a number line and test points
Write answer in interval notation
Example 2: Solve
a)
𝒙𝟐 − 𝟕𝒙 + 𝟏𝟔 ≥ 𝟏𝟎
b) 𝒙𝟐 + 𝒙 − 𝟏𝟓 < 𝟒
Example 3: A water balloon is launched from a machine upward from a height of 10ft with an initial
velocity of 46 ft per second. Determine when the balloon is above 30 feet?
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Chapter 14/15
Additional Notes
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Chapter 14/15
14.4 - Systems of Quadratic Equations
A system of equations can also involve non-linear functions, such as quadratic equations. Luckily,
methods for solving non-linear are the same for solving linear.
Example1: Solve the system of equations then graph it to check.
𝑦 = 2𝑥 + 7
{
𝑦 = 𝑥2 + 4
Example 2: Think about the graphs of a linear equation and a quadratic equation. Describe/show
the different ways the two graphs can intersect.
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What about graphs of 2 quadratic equations?
Example 3: Solve the following system of equations algebraically and then verify it graphically
𝑦 = −2𝑥 + 4
a) {
𝑦 = 4𝑥 2 + 2𝑥 + 5
b) {
𝑦 = −4𝑥 − 7
𝑦 = 3𝑥 2 + 𝑥 − 3
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Secondary 2
𝑦 = 𝑥 2 + 2𝑥 + 1
c) {
𝑦 = 2𝑥 2 − 𝑥 − 3
Chapter 14/15
d) {
𝑦 = 2𝑥 2 − 7𝑥 + 6
𝑦 = −2𝑥 2 + 5𝑥 − 3
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Secondary 2
Chapter 14/15
Additional Notes
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