UNIT 5 QUADRATIC FUNCTIONS Lesson 2: Creating and Solving Quadratic Equations in One Variable Instruction

Size: px
Start display at page:

Download "UNIT 5 QUADRATIC FUNCTIONS Lesson 2: Creating and Solving Quadratic Equations in One Variable Instruction"

Transcription

1 Prerequisite Skills This lesson requires the use of the following skills: simplifying radicals working with complex numbers Introduction You can determine how far a ladder will extend from the base of a wall by creating a quadratic equation and then taking the square root. To find this length, you only need to find the positive square root because a negative distance would not make sense in this situation. For certain types of quadratics, we can solve by taking the square root, but in most problems, we need to take both the positive and negative square root. Key Concepts The imaginary unit i represents the non-real value i = 1. i is the number whose square is 1. We define i so that i = 1 and i 2 = 1. An imaginary number is any number of the form bi, where b is a real number, i = 1, and b 0. A complex number is a number with a real component and an imaginary component. Complex numbers can be written in the form a + bi, where a and b are real numbers, and i is the imaginary unit. For example, 5 + 3i is a complex number. 5 is the real component and 3i is the imaginary component. Real numbers are the set of all rational and irrational numbers. Real numbers do not contain an imaginary component. Real numbers are rational numbers when they can be written as m, where both m and n are n integers and n 0. Rational numbers can also be written as a decimal that ends or repeats. The real number 0.4 is a rational number because it can be written as the fraction 2 5. Real numbers are irrational when they cannot be written as m, where m and n are integers n and n 0. Irrational numbers cannot be written as a decimal that ends or repeats. The real number 3 is an irrational number because it cannot be written as the ratio of two integers. Other examples of irrational numbers include 2 and π. U5-39

2 A quadratic equation is an equation that can be written in the form ax 2 + bx + c = 0, where a 0. Quadratic equations can have no real solutions, one real solution, or two real solutions. When a quadratic has no real solutions, it has two complex solutions. Quadratic equations that contain only a squared term and a constant can be solved by taking the square root of both sides. These equations can be written in the form x 2 = c, where c is a constant. When we take the square root of both sides, we need to remember that a number and its opposite have the same square. Therefore, rather than simply taking the positive square root, we need to take the positive and negative square root. For x 2 = c, we find that x=± c. We can use a similar method to solve quadratic equations in the form (ax + b) 2 = c. c tells us the number and type of solutions for the equation. c Number and type of solutions Negative Two complex solutions 0 One real, rational solution Positive and a perfect square Two real, rational solutions Positive and not a perfect square Two real, irrational solutions Common Errors/Misconceptions forgetting to use ± and therefore forgetting that there may be two solutions taking the square root before isolating the squared term U5-40

3 Guided Practice Example 1 Solve 2x 2 5 = 195 for x. 1. Isolate x 2. 2x 2 5 = 195 Original equation 2x 2 = 200 Add 5 to both sides. x 2 = 100 Divide both sides by Use a square root to find all possible solutions to the equation. Take the square root of both sides. Remember that both 10 2 and ( 10) 2 equal 100. x =± 100 =± 10 The equation 2x 2 5 = 195 has two solutions, 10 and 10. U5-41

4 Example 2 Solve (x 1) = 1 for x. 1. Isolate the squared binomial. (x 1) = 1 Original equation (x 1) 2 = 16 Subtract 15 from both sides. 2. Use a square root to isolate the binomial. Take the square root of both sides. Remember to use the ± sign. x 1=± Simplify the square root. There is a negative number under the radical, so the answer will be a complex number. x 1=± 16 Equation x 1=± ( 1)( 16) x 1=± 1 16 x 1 = ±4i Write 16 as a product of a perfect square and 1. Product Property of Square Roots Simplify. 4. Isolate x. x 1 = ±4i x = 1 ± 4i Equation Add 1 to both sides. The equation (x 1) = 1 has two solutions, 1 ± 4i. U5-42

5 Example 3 Solve 4(x + 3) 2 10 = 6 for x. 1. Isolate the squared binomial. 4(x + 3) 2 10 = 6 Original equation 4(x + 3) 2 = 4 Add 10 to both sides. (x + 3) 2 = 1 Divide both sides by 4. x + 3=± 1 Take the square root of both sides. x + 3 = ±1 2. Isolate x. x + 3 = ±1 x = 3 ± 1 Equation Subtract 3 from both sides. 3. Split the answer into two separate expressions and evaluate. x = = 2 x = 3 1 = 4 The equation 4(x + 3) 2 10 = 6 has two solutions, 2 and 4. U5-43

Solving Quadratic Equations by Formula

Solving Quadratic Equations by Formula Algebra Unit: 05 Lesson: 0 Complex Numbers All the quadratic equations solved to this point have had two real solutions or roots. In some cases, solutions involved a double root, but there were always

More information

UNIT 5 QUADRATIC FUNCTIONS Lesson 2: Creating and Solving Quadratic Equations in One Variable Instruction

UNIT 5 QUADRATIC FUNCTIONS Lesson 2: Creating and Solving Quadratic Equations in One Variable Instruction Lesson : Creating and Solving Quadratic Equations in One Variale Prerequisite Skills This lesson requires the use of the following skills: understanding real numers and complex numers understanding rational

More information

UNIT 4 EXTENDING THE NUMBER SYSTEM Lesson 3: Operating with Complex Numbers Instruction

UNIT 4 EXTENDING THE NUMBER SYSTEM Lesson 3: Operating with Complex Numbers Instruction Prerequisite Skills This lesson requires the use of the following skills: simplifying expressions using properties of exponents finding quotients that include remainders understanding the real number system

More information

UNIT 5 QUADRATIC FUNCTIONS Lesson 1: Interpreting Structure in Expressions Instruction

UNIT 5 QUADRATIC FUNCTIONS Lesson 1: Interpreting Structure in Expressions Instruction Prerequisite Skills This lesson requires the use of the following skills: evaluating expressions using the order of operations evaluating expressions for a given value identifying parts of an expression

More information

CHAPTER EIGHT: SOLVING QUADRATIC EQUATIONS Review April 9 Test April 17 The most important equations at this level of mathematics are quadratic

CHAPTER EIGHT: SOLVING QUADRATIC EQUATIONS Review April 9 Test April 17 The most important equations at this level of mathematics are quadratic CHAPTER EIGHT: SOLVING QUADRATIC EQUATIONS Review April 9 Test April 17 The most important equations at this level of mathematics are quadratic equations. They can be solved using a graph, a perfect square,

More information

Skills Practice Skills Practice for Lesson 4.1

Skills Practice Skills Practice for Lesson 4.1 Skills Practice Skills Practice for Lesson.1 Name Date Thinking About Numbers Counting Numbers, Whole Numbers, Integers, Rational and Irrational Numbers Vocabulary Define each term in your own words. 1.

More information

To solve a radical equation, you must take both sides of an equation to a power.

To solve a radical equation, you must take both sides of an equation to a power. Topic 5 1 Radical Equations A radical equation is an equation with at least one radical expression. There are four types we will cover: x 35 3 4x x 1x 7 3 3 3 x 5 x 1 To solve a radical equation, you must

More information

UNIT 3 REASONING WITH EQUATIONS Lesson 2: Solving Systems of Equations Instruction

UNIT 3 REASONING WITH EQUATIONS Lesson 2: Solving Systems of Equations Instruction Prerequisite Skills This lesson requires the use of the following skills: graphing equations of lines using properties of equality to solve equations Introduction Two equations that are solved together

More information

Lesson 3.5 Exercises, pages

Lesson 3.5 Exercises, pages Lesson 3.5 Exercises, pages 232 238 A 4. Calculate the value of the discriminant for each quadratic equation. a) 5x 2-9x + 4 = 0 b) 3x 2 + 7x - 2 = 0 In b 2 4ac, substitute: In b 2 4ac, substitute: a 5,

More information

NAME DATE PERIOD. A negative exponent is the result of repeated division. Extending the pattern below shows that 4 1 = 1 4 or 1. Example: 6 4 = 1 6 4

NAME DATE PERIOD. A negative exponent is the result of repeated division. Extending the pattern below shows that 4 1 = 1 4 or 1. Example: 6 4 = 1 6 4 Lesson 4.1 Reteach Powers and Exponents A number that is expressed using an exponent is called a power. The base is the number that is multiplied. The exponent tells how many times the base is used as

More information

QUADRATIC EQUATIONS. + 6 = 0 This is a quadratic equation written in standard form. x x = 0 (standard form with c=0). 2 = 9

QUADRATIC EQUATIONS. + 6 = 0 This is a quadratic equation written in standard form. x x = 0 (standard form with c=0). 2 = 9 QUADRATIC EQUATIONS A quadratic equation is always written in the form of: a + b + c = where a The form a + b + c = is called the standard form of a quadratic equation. Eamples: 5 + 6 = This is a quadratic

More information

Unit 2-1: Factoring and Solving Quadratics. 0. I can add, subtract and multiply polynomial expressions

Unit 2-1: Factoring and Solving Quadratics. 0. I can add, subtract and multiply polynomial expressions CP Algebra Unit -1: Factoring and Solving Quadratics NOTE PACKET Name: Period Learning Targets: 0. I can add, subtract and multiply polynomial expressions 1. I can factor using GCF.. I can factor by grouping.

More information

Solving Linear Equations

Solving Linear Equations Solving Linear Equations Golden Rule of Algebra: Do unto one side of the equal sign as you will do to the other Whatever you do on one side of the equal sign, you MUST do the same exact thing on the other

More information

A2 HW Imaginary Numbers

A2 HW Imaginary Numbers Name: A2 HW Imaginary Numbers Rewrite the following in terms of i and in simplest form: 1) 100 2) 289 3) 15 4) 4 81 5) 5 12 6) -8 72 Rewrite the following as a radical: 7) 12i 8) 20i Solve for x in simplest

More information

Introduction. Adding and Subtracting Polynomials

Introduction. Adding and Subtracting Polynomials Introduction Polynomials can be added and subtracted like real numbers. Adding and subtracting polynomials is a way to simplify expressions. It can also allow us to find a shorter way to represent a sum

More information

CP Algebra 2. Unit 2-1 Factoring and Solving Quadratics

CP Algebra 2. Unit 2-1 Factoring and Solving Quadratics CP Algebra Unit -1 Factoring and Solving Quadratics Name: Period: 1 Unit -1 Factoring and Solving Quadratics Learning Targets: 1. I can factor using GCF.. I can factor by grouping. Factoring Quadratic

More information

SECTION Types of Real Numbers The natural numbers, positive integers, or counting numbers, are

SECTION Types of Real Numbers The natural numbers, positive integers, or counting numbers, are SECTION.-.3. Types of Real Numbers The natural numbers, positive integers, or counting numbers, are The negative integers are N = {, 2, 3,...}. {..., 4, 3, 2, } The integers are the positive integers,

More information

The greatest common factor, or GCF, is the largest factor that two or more terms share.

The greatest common factor, or GCF, is the largest factor that two or more terms share. Unit, Lesson Factoring Recall that a factor is one of two or more numbers or expressions that when multiplied produce a given product You can factor certain expressions by writing them as the product of

More information

Math 1302 Notes 2. How many solutions? What type of solution in the real number system? What kind of equation is it?

Math 1302 Notes 2. How many solutions? What type of solution in the real number system? What kind of equation is it? Math 1302 Notes 2 We know that x 2 + 4 = 0 has How many solutions? What type of solution in the real number system? What kind of equation is it? What happens if we enlarge our current system? Remember

More information

Algebra I Unit Report Summary

Algebra I Unit Report Summary Algebra I Unit Report Summary No. Objective Code NCTM Standards Objective Title Real Numbers and Variables Unit - ( Ascend Default unit) 1. A01_01_01 H-A-B.1 Word Phrases As Algebraic Expressions 2. A01_01_02

More information

P.1 Prerequisite skills Basic Algebra Skills

P.1 Prerequisite skills Basic Algebra Skills P.1 Prerequisite skills Basic Algebra Skills Topics: Evaluate an algebraic expression for given values of variables Combine like terms/simplify algebraic expressions Solve equations for a specified variable

More information

Perform the following operations. 1) (2x + 3) + (4x 5) 2) 2(x + 3) 3) 2x (x 4) 4) (2x + 3)(3x 5) 5) (x 4)(x 2 3x + 5)

Perform the following operations. 1) (2x + 3) + (4x 5) 2) 2(x + 3) 3) 2x (x 4) 4) (2x + 3)(3x 5) 5) (x 4)(x 2 3x + 5) 2/24 week Add subtract polynomials 13.1 Multiplying Polynomials 13.2 Radicals 13.6 Completing the square 13.7 Real numbers 15.1 and 15.2 Complex numbers 15.3 and 15.4 Perform the following operations 1)

More information

Order of Operations Practice: 1) =

Order of Operations Practice: 1) = Order of Operations Practice: 1) 24-12 3 + 6 = a) 6 b) 42 c) -6 d) 192 2) 36 + 3 3 (1/9) - 8 (12) = a) 130 b) 171 c) 183 d) 4,764 1 3) Evaluate: 12 2-4 2 ( - ½ ) + 2 (-3) 2 = 4) Evaluate 3y 2 + 8x =, when

More information

UNIT 4 EXTENDING THE NUMBER SYSTEM Lesson 3: Operating with Complex Numbers Instruction

UNIT 4 EXTENDING THE NUMBER SYSTEM Lesson 3: Operating with Complex Numbers Instruction Prerequisite Skills This lesson requires the use of the following skills: simplifying sums and differences of numeric and algebraic quantities using the Commutative Property to reorder sums and differences

More information

Algebra II Unit #2 4.6 NOTES: Solving Quadratic Equations (More Methods) Block:

Algebra II Unit #2 4.6 NOTES: Solving Quadratic Equations (More Methods) Block: Algebra II Unit # Name: 4.6 NOTES: Solving Quadratic Equations (More Methods) Block: (A) Background Skills - Simplifying Radicals To simplify a radical that is not a perfect square: 50 8 300 7 7 98 (B)

More information

Math-2 Lesson 2-4. Radicals

Math-2 Lesson 2-4. Radicals Math- Lesson - Radicals = What number is equivalent to the square root of? Square both sides of the equation ( ) ( ) = = = is an equivalent statement to = 1.7 1.71 1.70 1.701 1.7008... There is no equivalent

More information

Chapter 6 Complex Numbers

Chapter 6 Complex Numbers Chapter 6 Complex Numbers Lesson 1: Imaginary Numbers Lesson 2: Complex Numbers Lesson 3: Quadratic Formula Lesson 4: Discriminant This assignment is a teacher-modified version of Algebra 2 Common Core

More information

correlated to the Utah 2007 Secondary Math Core Curriculum Algebra 1

correlated to the Utah 2007 Secondary Math Core Curriculum Algebra 1 correlated to the Utah 2007 Secondary Math Core Curriculum Algebra 1 McDougal Littell Algebra 1 2007 correlated to the Utah 2007 Secondary Math Core Curriculum Algebra 1 The main goal of Algebra is to

More information

UNIT 4 EXTENDING THE NUMBER SYSTEM Lesson 3: Operating with Complex Numbers Instruction

UNIT 4 EXTENDING THE NUMBER SYSTEM Lesson 3: Operating with Complex Numbers Instruction Prerequisite Skills This lesson requires the use of the following skills: finding the product of two binomials simplifying powers of i adding two fractions with different denominators (for application

More information

3.9 My Irrational and Imaginary Friends A Solidify Understanding Task

3.9 My Irrational and Imaginary Friends A Solidify Understanding Task 3.9 My Irrational and Imaginary Friends A Solidify Understanding Task Part 1: Irrational numbers Find the perimeter of each of the following figures. Express your answer as simply as possible. 2013 www.flickr.com/photos/lel4nd

More information

Complex Numbers: Definition: A complex number is a number of the form: z = a + bi where a, b are real numbers and i is a symbol with the property: i

Complex Numbers: Definition: A complex number is a number of the form: z = a + bi where a, b are real numbers and i is a symbol with the property: i Complex Numbers: Definition: A complex number is a number of the form: z = a + bi where a, b are real numbers and i is a symbol with the property: i 2 = 1 Sometimes we like to think of i = 1 We can treat

More information

Roots are: Solving Quadratics. Graph: y = 2x 2 2 y = x 2 x 12 y = x 2 + 6x + 9 y = x 2 + 6x + 3. real, rational. real, rational. real, rational, equal

Roots are: Solving Quadratics. Graph: y = 2x 2 2 y = x 2 x 12 y = x 2 + 6x + 9 y = x 2 + 6x + 3. real, rational. real, rational. real, rational, equal Solving Quadratics Graph: y = 2x 2 2 y = x 2 x 12 y = x 2 + 6x + 9 y = x 2 + 6x + 3 Roots are: real, rational real, rational real, rational, equal real, irrational 1 To find the roots algebraically, make

More information

Find two positive factors of 24 whose sum is 10. Make an organized list.

Find two positive factors of 24 whose sum is 10. Make an organized list. 9.5 Study Guide For use with pages 582 589 GOAL Factor trinomials of the form x 2 1 bx 1 c. EXAMPLE 1 Factor when b and c are positive Factor x 2 1 10x 1 24. Find two positive factors of 24 whose sum is

More information

Midterm 3 Review. Terms. Formulas and Rules to Use. Math 1010, Fall 2011 Instructor: Marina Gresham. Odd Root ( n x where n is odd) Exponent

Midterm 3 Review. Terms. Formulas and Rules to Use. Math 1010, Fall 2011 Instructor: Marina Gresham. Odd Root ( n x where n is odd) Exponent Math 1010, Fall 2011 Instructor: Marina Gresham Terms Midterm 3 Review Exponent Polynomial - Monomial - Binomial - Trinomial - Standard Form - Degree - Leading Coefficient - Constant Term Difference of

More information

Pre-Calculus Summer Packet

Pre-Calculus Summer Packet 2013-2014 Pre-Calculus Summer Packet 1. Complete the attached summer packet, which is due on Friday, September 6, 2013. 2. The material will be reviewed in class on Friday, September 6 and Monday, September

More information

P.5 Solving Equations

P.5 Solving Equations PRC Ch P_5.notebook P.5 Solving Equations What you should learn How to solve linear equations How to solve quadratic equations equations How to solve polynomial equations of degree three or higher How

More information

Solving Quadratic Equations

Solving Quadratic Equations Solving Quadratic Equations MATH 101 College Algebra J. Robert Buchanan Department of Mathematics Summer 2012 Objectives In this lesson we will learn to: solve quadratic equations by factoring, solve quadratic

More information

Beginning Algebra. 1. Review of Pre-Algebra 1.1 Review of Integers 1.2 Review of Fractions

Beginning Algebra. 1. Review of Pre-Algebra 1.1 Review of Integers 1.2 Review of Fractions 1. Review of Pre-Algebra 1.1 Review of Integers 1.2 Review of Fractions Beginning Algebra 1.3 Review of Decimal Numbers and Square Roots 1.4 Review of Percents 1.5 Real Number System 1.6 Translations:

More information

x y x y ax bx c x Algebra I Course Standards Gap 1 Gap 2 Comments a. Set up and solve problems following the correct order of operations (including proportions, percent, and absolute value) with rational

More information

Lesson 7.1 Polynomial Degree and Finite Differences

Lesson 7.1 Polynomial Degree and Finite Differences Lesson 7.1 Polynomial Degree and Finite Differences 1. Identify the degree of each polynomial. a. 1 b. 0.2 1. 2 3.2 3 c. 20 16 2 20 2. Determine which of the epressions are polynomials. For each polynomial,

More information

Never leave a NEGATIVE EXPONENT or a ZERO EXPONENT in an answer in simplest form!!!!!

Never leave a NEGATIVE EXPONENT or a ZERO EXPONENT in an answer in simplest form!!!!! 1 ICM Unit 0 Algebra Rules Lesson 1 Rules of Exponents RULE EXAMPLE EXPLANANTION a m a n = a m+n A) x x 6 = B) x 4 y 8 x 3 yz = When multiplying with like bases, keep the base and add the exponents. a

More information

HONORS GEOMETRY Summer Skills Set

HONORS GEOMETRY Summer Skills Set HONORS GEOMETRY Summer Skills Set Algebra Concepts Adding and Subtracting Rational Numbers To add or subtract fractions with the same denominator, add or subtract the numerators and write the sum or difference

More information

Instructor Quick Check: Question Block 11

Instructor Quick Check: Question Block 11 Instructor Quick Check: Question Block 11 How to Administer the Quick Check: The Quick Check consists of two parts: an Instructor portion which includes solutions and a Student portion with problems for

More information

Solving Equations by Factoring. Solve the quadratic equation x 2 16 by factoring. We write the equation in standard form: x

Solving Equations by Factoring. Solve the quadratic equation x 2 16 by factoring. We write the equation in standard form: x 11.1 E x a m p l e 1 714SECTION 11.1 OBJECTIVES 1. Solve quadratic equations by using the square root method 2. Solve quadratic equations by completing the square Here, we factor the quadratic member of

More information

Lesson 7.1 Polynomial Degree and Finite Differences

Lesson 7.1 Polynomial Degree and Finite Differences Lesson 7.1 Polynomial Degree and Finite Differences 1. Identify the degree of each polynomial. a. 3x 4 2x 3 3x 2 x 7 b. x 1 c. 0.2x 1.x 2 3.2x 3 d. 20 16x 2 20x e. x x 2 x 3 x 4 x f. x 2 6x 2x 6 3x 4 8

More information

Math 101 Study Session Spring 2016 Test 4 Chapter 10, Chapter 11 Chapter 12 Section 1, and Chapter 12 Section 2

Math 101 Study Session Spring 2016 Test 4 Chapter 10, Chapter 11 Chapter 12 Section 1, and Chapter 12 Section 2 Math 101 Study Session Spring 2016 Test 4 Chapter 10, Chapter 11 Chapter 12 Section 1, and Chapter 12 Section 2 April 11, 2016 Chapter 10 Section 1: Addition and Subtraction of Polynomials A monomial is

More information

October 28, Complex Numbers.notebook. Discriminant

October 28, Complex Numbers.notebook. Discriminant OBJECTIVE Students will be able to utilize complex numbers to simplify roots of negative numbers. Students will be able to plot complex numbers on a complex coordinate plane. Students will be able to add

More information

Summer MA Lesson 11 Section 1.5 (part 1)

Summer MA Lesson 11 Section 1.5 (part 1) Summer MA 500 Lesson Section.5 (part ) The general form of a quadratic equation is a + b + c = 0, where a, b, and c are real numbers and a 0. This is a second degree equation. There are four ways to possibly

More information

5.1 Monomials. Algebra 2

5.1 Monomials. Algebra 2 . Monomials Algebra Goal : A..: Add, subtract, multiply, and simplify polynomials and rational expressions (e.g., multiply (x ) ( x + ); simplify 9x x. x Goal : Write numbers in scientific notation. Scientific

More information

Secondary Math 2H Unit 3 Notes: Factoring and Solving Quadratics

Secondary Math 2H Unit 3 Notes: Factoring and Solving Quadratics Secondary Math H Unit 3 Notes: Factoring and Solving Quadratics 3.1 Factoring out the Greatest Common Factor (GCF) Factoring: The reverse of multiplying. It means figuring out what you would multiply together

More information

Solving Quadratic Equations Using the Quadratic Formula

Solving Quadratic Equations Using the Quadratic Formula Section 9 : Solving Quadratic Equations Using the Quadratic Fmula Quadratic Equations are equations that have an x term as the highest powered term. They are also called Second Degree Equations. The Standard

More information

Lesson #33 Solving Incomplete Quadratics

Lesson #33 Solving Incomplete Quadratics Lesson # Solving Incomplete Quadratics A.A.4 Know and apply the technique of completing the square ~ 1 ~ We can also set up any quadratic to solve it in this way by completing the square, the technique

More information

TECHNIQUES IN FACTORISATION

TECHNIQUES IN FACTORISATION TECHNIQUES IN FACTORISATION The process where brackets are inserted into an equation is referred to as factorisation. Factorisation is the opposite process to epansion. METHOD: Epansion ( + )( 5) 15 Factorisation

More information

DON ROBERT B. ESTRELLA SR. NATIONAL HIGH SCHOOL Nagsaag, San Manuel, Pangasinan. (Effective Alternative Secondary Education) MATHEMATICS II

DON ROBERT B. ESTRELLA SR. NATIONAL HIGH SCHOOL Nagsaag, San Manuel, Pangasinan. (Effective Alternative Secondary Education) MATHEMATICS II DON ROBERT B. ESTRELLA SR. NATIONAL HIGH SCHOOL Nagsaag, San Manuel, Pangasinan. (Effective Alternative Secondary Education) MATHEMATICS II Y X MODULE 1 Quadratic Equations BUREAU OF SECONDARY EDUCATION

More information

Chapter 1A -- Real Numbers. iff. Math Symbols: Sets of Numbers

Chapter 1A -- Real Numbers. iff. Math Symbols: Sets of Numbers Fry Texas A&M University! Fall 2016! Math 150 Notes! Section 1A! Page 1 Chapter 1A -- Real Numbers Math Symbols: iff or Example: Let A = {2, 4, 6, 8, 10, 12, 14, 16,...} and let B = {3, 6, 9, 12, 15, 18,

More information

ALGEBRA I FORM I. Textbook: Algebra, Second Edition;Prentice Hall,2002

ALGEBRA I FORM I. Textbook: Algebra, Second Edition;Prentice Hall,2002 ALGEBRA I FORM I Textbook: Algebra, Second Edition;Prentice Hall,00 Prerequisites: Students are expected to have a knowledge of Pre Algebra and proficiency of basic math skills including: positive and

More information

Solving Quadratic Equations Review

Solving Quadratic Equations Review Math III Unit 2: Polynomials Notes 2-1 Quadratic Equations Solving Quadratic Equations Review Name: Date: Period: Some quadratic equations can be solved by. Others can be solved just by using. ANY quadratic

More information

Polynomial Form. Factored Form. Perfect Squares

Polynomial Form. Factored Form. Perfect Squares We ve seen how to solve quadratic equations (ax 2 + bx + c = 0) by factoring and by extracting square roots, but what if neither of those methods are an option? What do we do with a quadratic equation

More information

Algebra II Chapter 5: Polynomials and Polynomial Functions Part 1

Algebra II Chapter 5: Polynomials and Polynomial Functions Part 1 Algebra II Chapter 5: Polynomials and Polynomial Functions Part 1 Chapter 5 Lesson 1 Use Properties of Exponents Vocabulary Learn these! Love these! Know these! 1 Example 1: Evaluate Numerical Expressions

More information

Math 3 Variable Manipulation Part 4 Polynomials B COMPLEX NUMBERS A Complex Number is a combination of a Real Number and an Imaginary Number:

Math 3 Variable Manipulation Part 4 Polynomials B COMPLEX NUMBERS A Complex Number is a combination of a Real Number and an Imaginary Number: Math 3 Variable Manipulation Part 4 Polynomials B COMPLEX NUMBERS A Complex Number is a combination of a Real Number and an Imaginary Number: 1 Examples: 1 + i 39 + 3i 0.8.i + πi + i/ A Complex Number

More information

2.5 Operations With Complex Numbers in Rectangular Form

2.5 Operations With Complex Numbers in Rectangular Form 2.5 Operations With Complex Numbers in Rectangular Form The computer-generated image shown is called a fractal. Fractals are used in many ways, such as making realistic computer images for movies and squeezing

More information

May 16, Aim: To review for Quadratic Function Exam #2 Homework: Study Review Materials. Warm Up - Solve using factoring: 5x 2 + 7x + 2 = 0

May 16, Aim: To review for Quadratic Function Exam #2 Homework: Study Review Materials. Warm Up - Solve using factoring: 5x 2 + 7x + 2 = 0 Aim: To review for Quadratic Function Exam #2 Homework: Study Review Materials Warm Up - Solve using factoring: 5x 2 + 7x + 2 = 0 Review Topic Index 1. Consecutive Integer Word Problems 2. Pythagorean

More information

Herndon High School Geometry Honors Summer Assignment

Herndon High School Geometry Honors Summer Assignment Welcome to Geometry! This summer packet is for all students enrolled in Geometry Honors at Herndon High School for Fall 07. The packet contains prerequisite skills that you will need to be successful in

More information

A repeated root is a root that occurs more than once in a polynomial function.

A repeated root is a root that occurs more than once in a polynomial function. Unit 2A, Lesson 3.3 Finding Zeros Synthetic division, along with your knowledge of end behavior and turning points, can be used to identify the x-intercepts of a polynomial function. This information allows

More information

In a previous lesson, we solved certain quadratic equations by taking the square root of both sides of the equation.

In a previous lesson, we solved certain quadratic equations by taking the square root of both sides of the equation. In a previous lesson, we solved certain quadratic equations by taking the square root of both sides of the equation. x = 36 (x 3) = 8 x = ± 36 x 3 = ± 8 x = ±6 x = 3 ± Taking the square root of both sides

More information

A-2. Polynomials and Factoring. Section A-2 1

A-2. Polynomials and Factoring. Section A-2 1 A- Polynomials and Factoring Section A- 1 What you ll learn about Adding, Subtracting, and Multiplying Polynomials Special Products Factoring Polynomials Using Special Products Factoring Trinomials Factoring

More information

Online Courses for High School Students

Online Courses for High School Students Online Courses for High School Students 1-888-972-6237 Algebra I Course Description: Students explore the tools of algebra and learn to identify the structure and properties of the real number system;

More information

Lesson 21 Not So Dramatic Quadratics

Lesson 21 Not So Dramatic Quadratics STUDENT MANUAL ALGEBRA II / LESSON 21 Lesson 21 Not So Dramatic Quadratics Quadratic equations are probably one of the most popular types of equations that you ll see in algebra. A quadratic equation has

More information

1-1. Expressions and Formulas. Lesson 1-1. What You ll Learn. Active Vocabulary

1-1. Expressions and Formulas. Lesson 1-1. What You ll Learn. Active Vocabulary 1-1 Expressions and Formulas What You ll Learn Skim the lesson. Write two things you already know about expressions and formulas. 1. Active Vocabulary 2. Review Vocabulary Identify the four grouping symbols

More information

Algebra 1 FINAL EXAM REVIEW Spring Semester Material (by chapter)

Algebra 1 FINAL EXAM REVIEW Spring Semester Material (by chapter) Name: Per.: Date: Algebra 1 FINAL EXAM REVIEW Spring Semester Material (by chapter) Your Algebra 1 Final will be on at. You will need to bring your tetbook and number 2 pencils with you to the final eam.

More information

SUMMER MATH PACKET ALGEBRA TWO COURSE 229

SUMMER MATH PACKET ALGEBRA TWO COURSE 229 SUMMER MATH PACKET ALGEBRA TWO COURSE 9 MATH SUMMER PACKET INSTRUCTIONS MATH SUMMER PACKET INSTRUCTIONS Attached you will find a packet of exciting math problems for your enjoyment over the summer. The

More information

Equations and Inequalities. College Algebra

Equations and Inequalities. College Algebra Equations and Inequalities College Algebra Radical Equations Radical Equations: are equations that contain variables in the radicand How to Solve a Radical Equation: 1. Isolate the radical expression on

More information

Graphing Radicals Business 7

Graphing Radicals Business 7 Graphing Radicals Business 7 Radical functions have the form: The most frequently used radical is the square root; since it is the most frequently used we assume the number 2 is used and the square root

More information

Chapter 1: Foundations for Algebra

Chapter 1: Foundations for Algebra Chapter 1: Foundations for Algebra 1 Unit 1: Vocabulary 1) Natural Numbers 2) Whole Numbers 3) Integers 4) Rational Numbers 5) Irrational Numbers 6) Real Numbers 7) Terminating Decimal 8) Repeating Decimal

More information

Math-2A Lesson 2-1. Number Systems

Math-2A Lesson 2-1. Number Systems Math-A Lesson -1 Number Systems Natural Numbers Whole Numbers Lesson 1-1 Vocabulary Integers Rational Numbers Irrational Numbers Real Numbers Imaginary Numbers Complex Numbers Closure Why do we need numbers?

More information

2.1 Notes: Simplifying Radicals (Real and Imaginary)

2.1 Notes: Simplifying Radicals (Real and Imaginary) Chapter 2 Calendar Name: Day Date Assignment (Due the next class meeting) Friday Tuesday Wednesday Thursday Friday Monday Tuesday Wednesday Thursday Friday Monday Tuesday Wednesday Thursday 8/30/13 (A)

More information

CHAPTER 3: Quadratic Functions and Equations; Inequalities

CHAPTER 3: Quadratic Functions and Equations; Inequalities MAT 171 Precalculus Algebra Dr. Claude Moore Cape Fear Community College CHAPTER 3: Quadratic Functions and Equations; Inequalities 3.1 The Complex Numbers 3.2 Quadratic Equations, Functions, Zeros, and

More information

Math 110 (S & E) Textbook: Calculus Early Transcendentals by James Stewart, 7 th Edition

Math 110 (S & E) Textbook: Calculus Early Transcendentals by James Stewart, 7 th Edition Math 110 (S & E) Textbook: Calculus Early Transcendentals by James Stewart, 7 th Edition 1 Appendix A : Numbers, Inequalities, and Absolute Values Sets A set is a collection of objects with an important

More information

EX: Simplify the expression. EX: Simplify the expression. EX: Simplify the expression

EX: Simplify the expression. EX: Simplify the expression. EX: Simplify the expression SIMPLIFYING RADICALS EX: Simplify the expression 84x 4 y 3 1.) Start by creating a factor tree for the constant. In this case 84. Keep factoring until all of your nodes are prime. Two factor trees are

More information

Factor each expression. Remember, always find the GCF first. Then if applicable use the x-box method and also look for difference of squares.

Factor each expression. Remember, always find the GCF first. Then if applicable use the x-box method and also look for difference of squares. NOTES 11: RATIONAL EXPRESSIONS AND EQUATIONS Name: Date: Period: Mrs. Nguyen s Initial: LESSON 11.1 SIMPLIFYING RATIONAL EXPRESSIONS Lesson Preview Review Factoring Skills and Simplifying Fractions Factor

More information

Math 096--Quadratic Formula page 1

Math 096--Quadratic Formula page 1 Math 096--Quadratic Formula page 1 A Quadratic Formula. Use the quadratic formula to solve quadratic equations ax + bx + c = 0 when the equations can t be factored. To use the quadratic formula, the equation

More information

Math 0320 Final Exam Review

Math 0320 Final Exam Review Math 0320 Final Exam Review SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Factor out the GCF using the Distributive Property. 1) 6x 3 + 9x 1) Objective:

More information

Math-2 Section 1-1. Number Systems

Math-2 Section 1-1. Number Systems Math- Section 1-1 Number Systems Natural Numbers Whole Numbers Lesson 1-1 Vocabulary Integers Rational Numbers Irrational Numbers Real Numbers Imaginary Numbers Complex Numbers Closure Why do we need numbers?

More information

Algebra One As of: September 2014 Teacher Contact: Ms.Zinn (CVHS-NGC)

Algebra One As of: September 2014 Teacher Contact: Ms.Zinn (CVHS-NGC) Algebra One As of: September 2014 Teacher Contact: Ms.Zinn (CVHS-NGC) CCSS Unit Theme SKILLS ASSESSMENT & PRODUCTS Translate sentences into equations such as, The length of a rectangle is ten less than

More information

B. Complex number have a Real part and an Imaginary part. 1. written as a + bi some Examples: 2+3i; 7+0i; 0+5i

B. Complex number have a Real part and an Imaginary part. 1. written as a + bi some Examples: 2+3i; 7+0i; 0+5i Section 11.8 Complex Numbers I. The Complex Number system A. The number i = -1 1. 9 and 24 B. Complex number have a Real part and an Imaginary part II. Powers of i 1. written as a + bi some Examples: 2+3i;

More information

Note: In this section, the "undoing" or "reversing" of the squaring process will be introduced. What are the square roots of 16?

Note: In this section, the undoing or reversing of the squaring process will be introduced. What are the square roots of 16? Section 8.1 Video Guide Introduction to Square Roots Objectives: 1. Evaluate Square Roots 2. Determine Whether a Square Root is Rational, Irrational, or Not a Real Number 3. Find Square Roots of Variable

More information

Summer Mathematics Packet Say Hello to Algebra 2. For Students Entering Algebra 2

Summer Mathematics Packet Say Hello to Algebra 2. For Students Entering Algebra 2 Summer Math Packet Student Name: Say Hello to Algebra 2 For Students Entering Algebra 2 This summer math booklet was developed to provide students in middle school an opportunity to review grade level

More information

Honours Advanced Algebra Unit 2: Polynomial Functions Factors, Zeros, and Roots: Oh My! Learning Task (Task 5) Date: Period:

Honours Advanced Algebra Unit 2: Polynomial Functions Factors, Zeros, and Roots: Oh My! Learning Task (Task 5) Date: Period: Honours Advanced Algebra Name: Unit : Polynomial Functions Factors, Zeros, and Roots: Oh My! Learning Task (Task 5) Date: Period: Mathematical Goals Know and apply the Remainder Theorem Know and apply

More information

A Level Summer Work. Year 11 Year 12 Transition. Due: First lesson back after summer! Name:

A Level Summer Work. Year 11 Year 12 Transition. Due: First lesson back after summer! Name: A Level Summer Work Year 11 Year 12 Transition Due: First lesson back after summer! Name: This summer work is compulsory. Your maths teacher will ask to see your work (and method) in your first maths lesson,

More information

Polynomials: Adding, Subtracting, & Multiplying (5.1 & 5.2)

Polynomials: Adding, Subtracting, & Multiplying (5.1 & 5.2) Polynomials: Adding, Subtracting, & Multiplying (5.1 & 5.) Determine if the following functions are polynomials. If so, identify the degree, leading coefficient, and type of polynomial 5 3 1. f ( x) =

More information

Students will be able to simplify numerical expressions and evaluate algebraic expressions. (M)

Students will be able to simplify numerical expressions and evaluate algebraic expressions. (M) Morgan County School District Re-3 August What is algebra? This chapter develops some of the basic symbolism and terminology that students may have seen before but still need to master. The concepts of

More information

Polynomial Form. Factored Form. Perfect Squares

Polynomial Form. Factored Form. Perfect Squares We ve seen how to solve quadratic equations (ax 2 + bx + c = 0) by factoring and by extracting square roots, but what if neither of those methods are an option? What do we do with a quadratic equation

More information

MATH 190 KHAN ACADEMY VIDEOS

MATH 190 KHAN ACADEMY VIDEOS MATH 10 KHAN ACADEMY VIDEOS MATTHEW AUTH 11 Order of operations 1 The Real Numbers (11) Example 11 Worked example: Order of operations (PEMDAS) 7 2 + (7 + 3 (5 2)) 4 2 12 Rational + Irrational Example

More information

Natural Numbers Positive Integers. Rational Numbers

Natural Numbers Positive Integers. Rational Numbers Chapter A - - Real Numbers Types of Real Numbers, 2,, 4, Name(s) for the set Natural Numbers Positive Integers Symbol(s) for the set, -, - 2, - Negative integers 0,, 2,, 4, Non- negative integers, -, -

More information

Review for Mastery. Integer Exponents. Zero Exponents Negative Exponents Negative Exponents in the Denominator. Definition.

Review for Mastery. Integer Exponents. Zero Exponents Negative Exponents Negative Exponents in the Denominator. Definition. LESSON 6- Review for Mastery Integer Exponents Remember that means 8. The base is, the exponent is positive. Exponents can also be 0 or negative. Zero Exponents Negative Exponents Negative Exponents in

More information

Ch. 7.6 Squares, Squaring & Parabolas

Ch. 7.6 Squares, Squaring & Parabolas Ch. 7.6 Squares, Squaring & Parabolas Learning Intentions: Learn about the squaring & square root function. Graph parabolas. Compare the squaring function with other functions. Relate the squaring function

More information

Study Guide and Intervention. The Quadratic Formula and the Discriminant. Quadratic Formula. Replace a with 1, b with -5, and c with -14.

Study Guide and Intervention. The Quadratic Formula and the Discriminant. Quadratic Formula. Replace a with 1, b with -5, and c with -14. 4-6 Study Guide and Intervention Quadratic Formula The Quadratic Formula can be used to solve any quadratic equation once it is written in the form ax 2 + bx + c = 0. Quadratic Formula The solutions of

More information

Acquisition Lesson Planning Form Key Standards addressed in this Lesson: MM2A4b & MM2A4c Time allotted for this Lesson: 9 hours

Acquisition Lesson Planning Form Key Standards addressed in this Lesson: MM2A4b & MM2A4c Time allotted for this Lesson: 9 hours Acquisition Lesson Planning Form Key Standards addressed in this Lesson: MM2A4b & MM2A4c Time allotted for this Lesson: 9 hours Essential Question: LESSON 3 Solving Quadratic Equations and Inequalities

More information

Objective Mathematics

Objective Mathematics Multiple choice questions with ONE correct answer : ( Questions No. 1-5 ) 1. If the equation x n = (x + ) is having exactly three distinct real solutions, then exhaustive set of values of 'n' is given

More information

Quadratic Equations 6 QUESTIONS. Relatively Easy: Questions 1 to 2 Moderately Difficult: Questions 3 to 4 Difficult: Questions 5 to 6

Quadratic Equations 6 QUESTIONS. Relatively Easy: Questions 1 to 2 Moderately Difficult: Questions 3 to 4 Difficult: Questions 5 to 6 Quadratic Equations 6 QUESTIONS Relatively Easy: Questions 1 to 2 Moderately Difficult: Questions 3 to 4 Difficult: Questions 5 to 6 Questions www.tutornext.com Page 2 of 11 Q1. The factors of 2x² - 7x

More information