Download Math 060 Chapters 9 and 10 Notes and Homework 9.1: Square

Chapters 9 and 10 - Roots, Radicals, and
Quadratic Equations
Math 060 Chapters 9 and 10
Notes and Homework
Roots, Radicals, and
Quadratic Equations
9.1: Square Roots
►
►
►
If
b2

(a is a nonnegative real number and b is a real number)
= a, then b is called a square root of a.
Ex: Find all square roots of the following numbers.
1. 64
3. 9/16
2. 0
4. -4
The positive square root is written with the symbol
√
, called a radical sign.
5. √100
8. √9 + 16
6. -√49
9. √9 + √16
7. √1/4
Vocabulary of Radicals
► √a,
the positive square root of a, is the positive
number we square to get a.
 Also called the principal square root.
 a is called the radicand
 The entire symbol √a is called a radical
► Ex:
Identify the radical, radicand, and value of
√121
Math 060 - Spring 2009
1
Chapters 9 and 10 - Roots, Radicals, and
Quadratic Equations
Modeling Example
►
►
The formula
C  0.5 p 2  17
describes the average level of carbon monoxide (C,
measured in parts per million) when the population is p
thousand people.
How many parts per million of carbon monoxide exist in a
community inhabited by 4000 people?
Roots Bigger Than Square Root
►
3
8 = 2 because 23 = 8.
►
n
a  b if bn = a
 n is called the index
►
Examples. Find each root:
a)
3
d)
27
 4 16
b)
e)
3
4
1
c)
4
16
 16
f)
5
 32
No Real Square Roots of -1
► The
square of a real number can never be
negative
 If a is a negative number, then √a is not a real number.
 Examples
(a) √25
►
(b) -√25
(c) √-25
Can you take the nth root of a negative number if n is
 Even?
 Odd?
Math 060 - Spring 2009
2
Chapters 9 and 10 - Roots, Radicals, and
Quadratic Equations
Study Schedule for Final Exam
 4 Weeks Before Finals
 Rework Chapters 2 and 4 Reviews
 Rework Chapters 2 and 4 Exams
 Cumulative Reviews
Suggestions for Studying:
1. Try to understand the
mistakes you made (so you
won’t make them again)
2. Bring questions to office
hours
3. Don’t be satisfied until you
can get 100% correct on all
of the review exercises!
 Page 319 #1-8, 10-12, 14-16, 18-20, 22-24, 26-27
►
3 Weeks Before Finals



Rework Chapters 5 and 6 Reviews
Rework Chapters 5 and 6 Exams
Cumulative Reviews
►
►
►
►



Rework Chapters 7 and 8 Reviews
Rework Chapters 7 and 8 Exams
Cumulative Reviews
►
►
►
►
Page 319 #28-30
Page 369 #1-7, 10-16, 18-25, 27-30
Page 442 #1, 2-4, 6-23, 25, 27-30
2 Weeks Before Finals
Pages 506-507 #2-4, 7-17, 19-23, 25-30
Page 245 #2-5, 7-16, 18-24, 26, 28
Pages 586-587 #2, 3, 5-12, 14, 16-30
1 Week Before Finals


Rework Chapters 9 and 10 Reviews and the Chapter 9/10 Exam
Cumulative Review

Review Problems Covering Entire Book page A4 (Appendix B)

Get a good night’s sleep!
►
►
Page 644 #1, 3-11, 13-28
Page A4-A10 #1-10, 13-56, 59-60, 64, 67-69, 72, 74, 77, 81-90, 92-95, 97-119, 126-128, 130-140
9.2: Product Rule for Radicals
Examine √25  √4 and √25  4
►
Product Rule
►

►
√x  √y = √xy
Ex: Use the product rule to find each product:
1. √2  √5
3. √7  √7
2. √7  √11y
Product Rule for Radicals
►
Ex: Use the product rule to simplify:
1. √18
3. √500
2. √75
4. √17
Math 060 - Spring 2009
3
Chapters 9 and 10 - Roots, Radicals, and
Quadratic Equations
Steps: Multiplying Radicals
1.
2.
3.
4.
►
Simplify the radicals
Multiply the coefficients of radicals.
Multiply the radicals using √x  √y = √xy.
If possible, simplify radical answer.
Ex: Multiply:
1. √18  √27
2. √12  √32
Square Root of x2
►
►
If x is a real number, then √x2 = |x|.
If x is a nonnegative real number, then √x2 = x
 For the rest of the semester, assume that variables under radical
signs represent nonnegative real numbers.
►
Ex: Simplify:
1. √72x2
2. √10x4 √5x3
The Quotient Rule
► Examine:
√64/4 and √64 / √4
► The Quotient Rule for Radicals:
► Ex:
2.
9
1.
x
y
Simplify the following radicals:
1. 100
► Ex:
x

y
3
25
3. 23 , x  0
6
x
Find each quotient:
75
3
2. 30 10
Math 060 - Spring 2009
5 2
3.
48 x 5
3x
4
Chapters 9 and 10 - Roots, Radicals, and
Quadratic Equations
Product and Quotient Rules
n
x  n y  n xy
n
n
►
x
x
n
y
y
Ex: Simplify:
1.
3
3.
4
2.
24
81
16
4.
4
3
8 4 4
4x2  3 8x
9.3: Adding and Subtracting Radicals
►
Combine like radicals
Like Terms:
3x  5 x 
7 x  8x 
Like Radicals: 3 2  5 2 
►
7 2 8 2 
Example:
3 7 4 6 3 5 2 7  6
Simplify and Add Examples
1.
44  99  11  27
Math 060 - Spring 2009
2.
2 3 24  5 3 81
5
Chapters 9 and 10 - Roots, Radicals, and
Quadratic Equations
9.5: Rationalizing Denominators
► What
number makes 2 a perfect square?
► Rationalize
a Denominator – Eliminate the Radical
in the Denominator
► Ex:
Rationalize the denominator.
5
2
Rationalize Denominator Examples
1.
12
8
2.
1 7

6 8
10.1: Square Root Property of Eqns
Ex: Solve x2 – 9 = 0 by factoring.
►
Square Root Property of Equations
►

►
If x2 = d (where d > 0), then x = √d or x = –√d
Ex: Solve:
1. x2 – 9 = 0
Math 060 - Spring 2009
2. 3x2 – 2 = 2(x2 + 3)
6
Chapters 9 and 10 - Roots, Radicals, and
Quadratic Equations
Square Root Property of Equations
Ex: Solve
►
1. 2y2 – 5 = 0
3. (2y + 4)2 = 8
2. (y – 5)2 = 16
Study Schedule for Final Exam
 4 Weeks Before Finals
 Rework Chapters 2 and 4 Reviews
 Rework Chapters 2 and 4 Exams
 Cumulative Reviews
 Page 319 #1-8, 10-12, 14-16, 18-20, 22-24, 26-27
 3 Weeks Before Finals
 Rework Chapters 5 and 6 Reviews
 Rework Chapters 5 and 6 Exams
 Cumulative Reviews
 Page 319 #28-30
 Page 369 #1-7, 10-16, 18-25, 27-30
 Page 442 #1, 2-4, 6-23, 25, 27-30
►
2 Weeks Before Finals



Rework Chapters 7 and 8 Reviews
Rework Chapters 7 and 8 Exams
Cumulative Reviews
►
►
►
►
Suggestions for Studying:
1. Try to understand the
mistakes you made (so you
won’t make them again)
2. Bring questions to office
hours
3. Don’t be satisfied until you
can get 100% correct on all
of the review exercises!
Pages 506-507 #2-4, 7-17, 19-23, 25-30
Page 245 #2-5, 7-16, 18-24, 26, 28
Pages 586-587 #2, 3, 5-12, 14, 16-30
1 Week Before Finals


Rework Chapters 9 and 10 Reviews and the Chapter 9/10 Exam
Cumulative Review

Review Problems Covering Entire Book page A4 (Appendix B)

Get a good night’s sleep!
►
►
Page 644 #1, 3-11, 13-28
Page A4-A10 #1-10, 13-56, 59-60, 64, 67-69, 72, 74, 77, 81-90, 92-95, 97-119, 126-128, 130-140
10.2: Completing the Square
►
►
►
►
Recall: Solving (x – 1)2 = 2.
Expressions that can be written as (x + d)2

x2 + 6x + 9

x2 + 8x + 16

x2 – 10x + 25

x2 – 12x + 36
What is the coefficient of x2 on each example?
How does the constant term relate to the coefficient of x?
Math 060 - Spring 2009
7
Chapters 9 and 10 - Roots, Radicals, and
Quadratic Equations
Making Perfect Square Trinomials
►
►
x2 + bx

What constant must be added to make a perfect square?

How does it then factor?
Write the term that must be added to each of the
following so that they become perfect square trinomials:
1. x2 + 8x
2. x2 – 14x
3. x2 + 5x
Solving By Completing the Square
►
Steps: Solving By Completing the Square
1.
2.
3.
4.
5.
►
Isolate variable terms on one side
Divide (if necessary) to make the coefficient of x2 one.
Complete the square by adding (b/2)2 to both sides.
Rewrite the perfect square trinomial as a binomial squared.
Solve using the square root property.
Solve by completing the square
1.
x2 – 6x + 2 = 0
Solving By Completing the Square
2.
2x2 + 3x = 2
3. 2x2 + 5x – 4 = 0
If solutions are rational, could
have been solved by factoring!
Math 060 - Spring 2009
8
Chapters 9 and 10 - Roots, Radicals, and
Quadratic Equations
Completing the Square Example
1. 4y2 + 12y + 7 = 0
10.3: The Quadratic Formula
►
If ax2 + bx + c = 0, where a ≠ 0, solve for x by completing
the square.
Using The Quadratic Formula
b
►
If ax2 + bx + c = 0, then
►
Steps: Using Quadratic Formula
1.
►
b 2  4 ac
2a
MEMORIZE!!!
Write in standard form: ax2 + bx + c = 0.

2.
3.
x 
Clear fractions
Identify a, b, and c; plug them into the quadratic formula
Simplify
Solve by the quadratic formula:
1.
2x2 + 9x – 5 = 0
Math 060 - Spring 2009
9
Chapters 9 and 10 - Roots, Radicals, and
Quadratic Equations
Examples
2. x2 = 2x + 16
x 
b
b 2  4 ac
2a
3. 4(y2 + 4) = 9 + 12y
The Discriminant
If ax2 + bx + c = 0, then
►
x 
b
b 2  4 ac
2a
What do the solutions look like if
►
b2 – 4ac is a perfect square (ie: 25)?

►
Can the equation be solved by factoring?
b2 – 4ac is not a perfect square (ie: 3)?

►
Can the equation be solved by factoring?
b2 – 4ac < 0 (ie: -4)?

►
Can the equation be solved?
b2 – 4ac = 0?

►
►
How many solutions are there?
Ex: Solve x2 – 6x + 9 = 0 by any method.
b2 – 4ac is the
discriminant
Which Method?
►
Different Methods for Solving Quadratic Equations:
1.
2.
If the equation is in the form x2 = d or (x + c)2 = d, use the
square root property.
If not, write the equation in standard form: ax2 + bx + c = 0.
a)
b)
►
Try to solve by factoring.
If prime, solve using the quadratic formula.
Examples. Which method should you use?
1.
6 – 5x + x2 = 0
2. (y + 3)2 – 25 = 0
3.
4w2 – 16 = 0
4. 2x2 + 4x + 1 = 0
Math 060 - Spring 2009
10
Chapters 9 and 10 - Roots, Radicals, and
Quadratic Equations
10.4: Application Examples
►
Ex 1. Numbers: When the sum of 6 and twice a positive
number is subtracted from the square of the number, zero
results. Find the number.
►
Ex 2. Geometry: The length of a rectangular flower bed
is 2 yards longer than the width. If the area is 10 square
yards, what are the exact values of the length and width of
the flower bed?
►
Ex 4: As indicated in the figure, a new road is to be built
from A to B and then from B to C. What is the exact
length of the road?
Geometry Example
C
6 miles
New
x
Road
B
A
x+1
New Road
Work Example
►
Working together, two people can do a job in 2 hours. Working alone,
one person takes 2 hours longer than the other to complete the job.
How long would it take each of them to do the job working alone?
Fractional Part of
Job Completed in
1 hour
Fractional Part
By Person 1
+

Time
Working
Together
Fractional Part
By Person 2
Math 060 - Spring 2009
=
=
Fractional Part of Job
Completed When
Working Together
One whole job
11
Chapters 9 and 10 - Roots, Radicals, and
Quadratic Equations
Chapters 9 & 10 Exam Topics
►
Radicals:




►
Simplifying
Multiplication/Division
Addition/Subtraction
Rationalize Denominator
Solving Quadratic Equations:




Square root property
Completing the square
Quadratic formula
Factoring
►
Applications to quadratic equations
►
Suggested Practice
 Page 642 #1-41, 52-56, 83, 88-89
 Page 705 #1-8, 12-31
Study Schedule for Final Exam
 4 Weeks Before Finals
 Rework Chapters 2 and 4 Reviews
 Rework Chapters 2 and 4 Exams
 Cumulative Reviews
 Page 319 #1-8, 10-12, 14-16, 18-20, 22-24, 26-27
 3 Weeks Before Finals
 Rework Chapters 5 and 6 Reviews
 Rework Chapters 5 and 6 Exams
 Cumulative Reviews
 Page 319 #28-30
 Page 369 #1-7, 10-16, 18-25, 27-30
 Page 442 #1, 2-4, 6-23, 25, 27-30
Suggestions for Studying:
1. Try to understand the
mistakes you made (so you
won’t make them again)
2. Bring questions to office
hours
3. Don’t be satisfied until you
can get 100% correct on all
of the review exercises!
 2 Weeks Before Finals
 Rework Chapters 7 and 8 Reviews
 Rework Chapters 7 and 8 Exams
 Cumulative Reviews
 Pages 506-507 #2-4, 7-17, 19-23, 25-30
 Page 245 #2-5, 7-16, 18-24, 26, 28
 Pages 586-587 #2, 3, 5-12, 14, 16-30
►
1 Week Before Finals


Rework Chapters 9 and 10 Reviews and the Chapter 9/10 Exam
Cumulative Review

Review Problems Covering Entire Book page A4 (Appendix B)

Get a good night’s sleep!








9.1: #1-81 eoo, 91, 92
9.2: #1-89 eoo, 93, 94
9.3: #1-49 odd, 98-100
9.5: #1-15 odd, 63-78 odd, 94-96
10.1: #1-41 eoo, 43, 59, 61, 65, 69, 70-72
10.2: #1-37 odd, 42-47
10.3: #1-17 odd, 21-47 eoo, 63, 65-66
10.4: #1, 3, 5, 9, 11, 13, 21, 23, 25, 31, 32
►
►
Page 644 #1, 3-11, 13-28
Page A4-A10 #1-10, 13-56, 59-60, 64, 67-69, 72, 74, 77, 81-90, 92-95, 97-119, 126-128, 130-140
Homework
Math 060 - Spring 2009
12