# Integration Unit 5 Quadratic Toolbox 1: Working with Square Roots. Using your examples above, answer the following:

Size: px
Start display at page:

## Transcription

1 Integration Unit 5 Quadratic Toolbox 1: Working with Square Roots Name Period Objective 1: Understanding Square roots Defining a SQUARE ROOT: Square roots are like a division problem but both factors must be identical. The answer is given as the single factor. Answer the following division problems to find a square root of 16: Using your examples above, answer the following: 16 = Defining a PERFECT SQUARE: If the number under the radical sign has identical factors, it is called a perfect square. Exercise: Finding Square roots of Perfect Squares Find the square roots of the following perfect squares: a) 100 = b) 36 = c) 0 = d) 9 25 = d) 25 = f) 121 = g) ± 49 = h) 3 2 = i) ( 4 ) 2 = j) 8 2 = k) ( x ) 2 = l) x 2 = Types of real numbers: Rational numbers: Decimals/numbers that STOP or have a repeating pattern. Irrational Numbers: Decimals/numbers that go on forever without a pattern Exercise: Identifying rational and irrational numbers: Use a calculator to take the square root of the following then identify the number as either rational or irrational (circle one) a) 2 = rational or irrational b) 5 = rational or irrational c) 4 25 = rational or irrational d) 9 = rational or irrational 16 4 e) = rational or irrational f) 0.49 = rational or irrational 5 g) 5.2 rational or irrational h) rational or irrational i) 5 rational or irrational j) 2 rational or irrational 5 Page 1

3 Objective 2 Practice: Simplifying real Square Roots Page 3

4 Objective 3: Imaginary Numbers and Square Roots Definining an Imaginary Number Although 25 = 5, we can t simplify 25 because no two numbers that are the same will multiply to a negative number. Whenever there is a negative under the square root sign, the problem will appear impossible to simplify. Since 1 is the impossible part, we will make the following definition of 1 : Definition: 1 = i where i represents the imaginary answer to 1 By using the letter i we can work with our answers, knowing they don t really exist. We call these types of square roots imaginary numbers. Simplifying square roots with imaginary answers Negative Perfect Squares: If the number under a square root sign is a negative perfect square number (e.g. - 25), factor the ONE square root into two parts: First part: largest perfect square factor (rational root) Second part: 1 Simplify the perfect square root and write the letter i next to it on the right. This will be your simplified answer. Example: 25 = 1i 25 = 1 i 25 = 5i Negative Non- Perfect Squares: If the square root is NOT a perfect square (e.g. - 24) we will need to simplify the square roots into three parts, one of which is 1. First part: largest perfect square factor (rational root) Second part: a non- perfect square factor (irrational root) Third part: 1 Simplify the perfect square root and write the letter i next to it on the right. Leave the irrational square root under the radical sign. Example: 24 = 1i 4 i 6 = 1 i 4 i 6 = 2i ± 15 6) ) ± 100 Page 4

5 Objective 2 & 3 Practice: Simplifying Real and Imaginary Square Roots Page 5

6 Objectives 2 & 3 Practice: Real and Imaginary Square Roots Level 2 Simplify the following square roots COMPLETELY. Use i in your answer if it is imaginary. 1) 250 2) 0 3) 4 4) 6 4 5) 48 6) 48 7) ) 28 9) ) ) ) 3 ± 50 13) ) ) 3 ± 49 16) 5 ± 81 Answers (scrambled): i 7 3 ± 5i i i 3 4 and 14 3 ± 7i 3 ± 5i 2 5i Page 6

7 Objective 4: Solving Equations Using Square roots (TYPE 1) Solving Quadratic Equations: Any equation containing an x 2 is called quadratic. Depending on the difficulty of the equation, different techniques can be used to solve for x (to get x alone). Inverse operations Method: If there is only an x 2 in the equation (no x), then the problem can be solved using normal solving techniques of inverse operations: o Solve by getting rid of the numbers on the same side of the = sign as the x. Do the order of operations backwards: Add or subtract (inverse operation) Then multiply or divide (inverse operation) Then square root to remove the squared (inverse operation)- don t forget to ± See ** for details. Simplify all radicals answers as completely as possible. (objectives #1-3) ** WEIRD MATH FACT: The highest exponent on x in a solve problem tells you how many answers there will be. After taking the square root of both sides, you must ± the side without the variable to get your TWO solutions. So make sure you put the ± sign to get BOTH answers! Example 1: Solve x 2 = 4 Why are there TWO answers to this equation? Example 2: Solve 4x 2 = 100 Example 3: Solve 1 5 x2 = 100 Example 4: Solve 5x 2 = 100 Example 5: Solve x 2 = 30 Page 7

8 Objective 5: Solving using Square Roots (Type 2) Inverse operations Method: o Solve by getting rid of the numbers on the same side of the = sign as the x. Do the order of operations backwards: Add or subtract then multiply or divide OUTSIDE of the parentheses Square root to remove the squared (inverse operation)- don t forget to ± If there are parentheses, remove the numbers inside of them LAST by adding or subtracting it. Put the added or subtracted number IN FRONT OF the ±! Simplify all radicals answers completely. (See Objectives #1-3) Example 1: Solve (x 10) 2 = 25 Example 2: 4(x 6) 2 = 64 Objective 4 Quick Practice: Solving using square roots (TYPE 1) Solve the following equations, showing all steps. Remember, each problems will have TWO solutions. CHECK YOUR ANSWERS AS YOU GO!! 1) 3n 2 = 75 2) x 2 12 = 20 3) 2x 2 8 = 192 4) 8x 2 = 128 Objective 5 Quick Practice: Solving using square roots (TYPE 2) 1) 4 = 1 4 (x 5)2 2) 64 = 4(x 5) 2 3) 25 = 5(2x 3) 2 4) 2(5x + 3) 2 = 50 Page 8

9 Objective 4 Practice: Solving Using Square Roots Type 1: Solve the equations showing proper solving steps. Make sure you use the ± sign when taking the square root! Use the puzzle on page 11 to check your answers. SIMPLIFY ALL YOUR ANSWERS COMPLETELY (see Objective #2) 1) x 2 = 81 8) b = 86 2) a 2 = 20 9) 2x 2 3 = 15 3) 3n 2 = 45 10) 5w = 58 4) 7x 2 = 84 11) 4x = 20 5) 2v 2 = 180 6) y 2 49 = 0 12) 7y = 4 7) x 2 16 = 8 13) 7y = 4 Page 9

10 Objective 5 Practice: Solving Using Square Roots Type 2: Solve the equations showing proper solving steps. Make sure you use the ± sign when taking the square root! Use the puzzle on page 11 to check your answers. SIMPLIFY ALL YOUR ANSWERS COMPLETELY (see Objective #2) 14) (a + 3) 2 = 25 18) 5(n +1) 2 = 40 15) (t 4) 2 = 7 19) (2x 3) 2 = 81 16) (x 2) 2 = 28 20) (4t +1) 2 = 49 17) 3(x 5) 2 = 12 Page 10

11 Page 11

12 Objective 6: Solving Quadratic Equations using Square Roots (TYPE 3) Solve by getting rid of the numbers on the same side of the = sign as the x. Do the order of operations backwards: o Add or subtract then multiply or divide OUTSIDE of the parentheses o Square root to remove the squared (inverse operation)- don t forget to ± o If there are parentheses, remove the numbers inside of them LAST by adding or subtracting it. Put the added or subtracted number IN FRONT OF the ±! o Simplify all radicals answers completely. (See Objectives #1-3) Example 1: Solve 5(x 2) = 24 Example 2) 2(x 3) 2 7 = 1 Objective 6 Quick Practice: Solving using square roots (TYPE 3) 1) 5 = 2(x 3) ) (x 10) = 12 3) (y +10) 2 49 = 0 4) 0 = 2(x + 5) ) 5 = 2(x 3) ) (5x + 4) = 40 Page 12

13 Practice ALL TYPES: Hidden Message Card 1 s Letter Copy and solve Card 5 s Letter Copy and solve Card 2 s Letter Copy and solve Card 6 s Letter Copy and solve Card 3 s Letter Copy and solve Card 7 s Letter Copy and solve Card 4 s Letter Copy and solve Card 8 s Letter Copy and solve Page 13

14 Card 9 s Letter Copy and solve Card 13 s Letter Copy and solve Card 10 s Letter Copy and solve Card 14 s Letter Copy and solve Card 11 s Letter Copy and solve Card 15 s Letter Copy and solve Card 12 s Letter Copy and solve Card 16 s Letter Hidden Message Page 14

15 Extension Objective 7: Applying Solving techniques to Quadratic Functions REVIEW: Vertex form of a parabola: y = a(x h) 2 + k where (h,k) is the vertex Vertex Form: y = a(x h) 2 + k is called vertex form because the parent function s main point is called the vertex. The vertex is the top or bottom of the parabola. The vertex is (h,k) and can be seen in the shifted equation y = a(x h) 2 + k. The line of symmetry is a vertical line that goes through the vertex of the parabola. The equation of the line of symmetry is x = h Zeros of a Quadratic Function (Roots or x- intercepts of a graph): The zeros of a function are the values of x when y = 0. On the graph, they are the x- values where the graph crosses the x- axis. Zeros are always found by substituting 0 in for y and solving for x. Example 1) y = (x + 6) Example 2) f (x) = 3(x 3) 2 27 a) Vertex: a) Vertex: b) Line of Symmetry: b) Line of Symmetry: c) Solve for roots: c) Solve for roots: Graph the equations above using the 5 key points AND the x- intercepts, if any. Page 15

16 Problem Set: 1) y = (x 5) 2 3 2) f (x) = 2(x + 4) 2 10 a) Vertex: a) Vertex: b) Line of Symmetry: b) Line of Symmetry: c) Solve for roots: c) Solve for roots: Exact answers Decimal values if any: Exact answers Decimal values if any: Graph each equation using the 5 key points AND the x- intercepts, if they exist. Page 16-24

17 3) y = (x + 4) 2 6 4) f (x) = 2(x 3) 2 8 a) Vertex: a) Vertex: b) Line of Symmetry: b) Line of Symmetry: c) Solve for roots: c) Solve for roots: Exact answers Decimal values if any: Exact answers Decimal values if any: Graph each equation using the 5 key points AND the x- intercepts, if they exist Page 17

18 Page 18

### Algebra 1 Course Title

Algebra 1 Course Title Course- wide 1. What patterns and methods are being used? Course- wide 1. Students will be adept at solving and graphing linear and quadratic equations 2. Students will be adept

### 2. Simplify. College Algebra Student Self-Assessment of Mathematics (SSAM) Answer Key. Use the distributive property to remove the parentheses

College Algebra Student Self-Assessment of Mathematics (SSAM) Answer Key 1. Multiply 2 3 5 1 Use the distributive property to remove the parentheses 2 3 5 1 2 25 21 3 35 31 2 10 2 3 15 3 2 13 2 15 3 2

### Basic Math Refresher A tutorial and assessment of basic math skills for students in PUBP704.

Basic Math Refresher A tutorial and assessment of basic math skills for students in PUBP704. The purpose of this Basic Math Refresher is to review basic math concepts so that students enrolled in PUBP704:

### Algebra 2/Trig Unit 2 Notes Packet Period: Quadratic Equations

Algebra 2/Trig Unit 2 Notes Packet Name: Date: Period: # Quadratic Equations (1) Page 253 #4 6 **Check on Graphing Calculator (GC)** (2) Page 253 254 #20, 26, 32**Check on GC** (3) Page 253 254 #10 12,

### Zero: If P is a polynomial and if c is a number such that P (c) = 0 then c is a zero of P.

MATH 11011 FINDING REAL ZEROS KSU OF A POLYNOMIAL Definitions: Polynomial: is a function of the form P (x) = a n x n + a n 1 x n 1 + + a x + a 1 x + a 0. The numbers a n, a n 1,..., a 1, a 0 are called

### What are the place values to the left of the decimal point and their associated powers of ten?

The verbal answers to all of the following questions should be memorized before completion of algebra. Answers that are not memorized will hinder your ability to succeed in geometry and algebra. (Everything

### Prentice Hall Mathematics: Algebra 2 2007 Correlated to: Utah Core Curriculum for Math, Intermediate Algebra (Secondary)

Core Standards of the Course Standard 1 Students will acquire number sense and perform operations with real and complex numbers. Objective 1.1 Compute fluently and make reasonable estimates. 1. Simplify

### ALGEBRA 2: 4.1 Graph Quadratic Functions in Standard Form

ALGEBRA 2: 4.1 Graph Quadratic Functions in Standard Form Goal Graph quadratic functions. VOCABULARY Quadratic function A function that can be written in the standard form y = ax 2 + bx+ c where a 0 Parabola

### Objectives. By the time the student is finished with this section of the workbook, he/she should be able

QUADRATIC FUNCTIONS Completing the Square..95 The Quadratic Formula....99 The Discriminant... 0 Equations in Quadratic Form.. 04 The Standard Form of a Parabola...06 Working with the Standard Form of a

### Step 1: Set the equation equal to zero if the function lacks. Step 2: Subtract the constant term from both sides:

In most situations the quadratic equations such as: x 2 + 8x + 5, can be solved (factored) through the quadratic formula if factoring it out seems too hard. However, some of these problems may be solved

### This is a square root. The number under the radical is 9. (An asterisk * means multiply.)

Page of Review of Radical Expressions and Equations Skills involving radicals can be divided into the following groups: Evaluate square roots or higher order roots. Simplify radical expressions. Rationalize

### 2.3. Finding polynomial functions. An Introduction:

2.3. Finding polynomial functions. An Introduction: As is usually the case when learning a new concept in mathematics, the new concept is the reverse of the previous one. Remember how you first learned

### Find the Square Root

verview Math Concepts Materials Students who understand the basic concept of square roots learn how to evaluate expressions and equations that have expressions and equations TI-30XS MultiView rational

### Algebra and Geometry Review (61 topics, no due date)

Course Name: Math 112 Credit Exam LA Tech University Course Code: ALEKS Course: Trigonometry Instructor: Course Dates: Course Content: 159 topics Algebra and Geometry Review (61 topics, no due date) Properties

### Higher Education Math Placement

Higher Education Math Placement Placement Assessment Problem Types 1. Whole Numbers, Fractions, and Decimals 1.1 Operations with Whole Numbers Addition with carry Subtraction with borrowing Multiplication

### Lesson 9.1 Solving Quadratic Equations

Lesson 9.1 Solving Quadratic Equations 1. Sketch the graph of a quadratic equation with a. One -intercept and all nonnegative y-values. b. The verte in the third quadrant and no -intercepts. c. The verte

### CORRELATED TO THE SOUTH CAROLINA COLLEGE AND CAREER-READY FOUNDATIONS IN ALGEBRA

We Can Early Learning Curriculum PreK Grades 8 12 INSIDE ALGEBRA, GRADES 8 12 CORRELATED TO THE SOUTH CAROLINA COLLEGE AND CAREER-READY FOUNDATIONS IN ALGEBRA April 2016 www.voyagersopris.com Mathematical

### MyMathLab ecourse for Developmental Mathematics

MyMathLab ecourse for Developmental Mathematics, North Shore Community College, University of New Orleans, Orange Coast College, Normandale Community College Table of Contents Module 1: Whole Numbers and

### Polynomial and Rational Functions

Polynomial and Rational Functions Quadratic Functions Overview of Objectives, students should be able to: 1. Recognize the characteristics of parabolas. 2. Find the intercepts a. x intercepts by solving

### 4.1. COMPLEX NUMBERS

4.1. COMPLEX NUMBERS What You Should Learn Use the imaginary unit i to write complex numbers. Add, subtract, and multiply complex numbers. Use complex conjugates to write the quotient of two complex numbers

### Algebra 2 Year-at-a-Glance Leander ISD 2007-08. 1st Six Weeks 2nd Six Weeks 3rd Six Weeks 4th Six Weeks 5th Six Weeks 6th Six Weeks

Algebra 2 Year-at-a-Glance Leander ISD 2007-08 1st Six Weeks 2nd Six Weeks 3rd Six Weeks 4th Six Weeks 5th Six Weeks 6th Six Weeks Essential Unit of Study 6 weeks 3 weeks 3 weeks 6 weeks 3 weeks 3 weeks

### a) x 2 8x = 25 x 2 8x + 16 = (x 4) 2 = 41 x = 4 ± 41 x + 1 = ± 6 e) x 2 = 5 c) 2x 2 + 2x 7 = 0 2x 2 + 2x = 7 x 2 + x = 7 2

Solving Quadratic Equations By Square Root Method Solving Quadratic Equations By Completing The Square Consider the equation x = a, which we now solve: x = a x a = 0 (x a)(x + a) = 0 x a = 0 x + a = 0

### Algebra 1 If you are okay with that placement then you have no further action to take Algebra 1 Portion of the Math Placement Test

Dear Parents, Based on the results of the High School Placement Test (HSPT), your child should forecast to take Algebra 1 this fall. If you are okay with that placement then you have no further action

### Prentice Hall Mathematics: Algebra 1 2007 Correlated to: Michigan Merit Curriculum for Algebra 1

STRAND 1: QUANTITATIVE LITERACY AND LOGIC STANDARD L1: REASONING ABOUT NUMBERS, SYSTEMS, AND QUANTITATIVE SITUATIONS Based on their knowledge of the properties of arithmetic, students understand and reason

### Simplify the rational expression. Find all numbers that must be excluded from the domain of the simplified rational expression.

MAC 1105 Final Review Simplify the rational expression. Find all numbers that must be excluded from the domain of the simplified rational expression. 1) 8x 2-49x + 6 x - 6 A) 1, x 6 B) 8x - 1, x 6 x -

### Vocabulary Words and Definitions for Algebra

Name: Period: Vocabulary Words and s for Algebra Absolute Value Additive Inverse Algebraic Expression Ascending Order Associative Property Axis of Symmetry Base Binomial Coefficient Combine Like Terms

### Algebra 2 Chapter 1 Vocabulary. identity - A statement that equates two equivalent expressions.

Chapter 1 Vocabulary identity - A statement that equates two equivalent expressions. verbal model- A word equation that represents a real-life problem. algebraic expression - An expression with variables.

### FACTORING QUADRATICS 8.1.1 and 8.1.2

FACTORING QUADRATICS 8.1.1 and 8.1.2 Chapter 8 introduces students to quadratic equations. These equations can be written in the form of y = ax 2 + bx + c and, when graphed, produce a curve called a parabola.

### ( yields. Combining the terms in the numerator you arrive at the answer:

Algebra Skillbuilder Solutions: 1. Starting with, you ll need to find a common denominator to add/subtract the fractions. If you choose the common denominator 15, you can multiply each fraction by one

### MTH124: Honors Algebra I

MTH124: Honors Algebra I This course prepares students for more advanced courses while they develop algebraic fluency, learn the skills needed to solve equations, and perform manipulations with numbers,

### ARE YOU A RADICAL OR JUST A SQUARE ROOT? EXAMPLES

ARE YOU A RADICAL OR JUST A SQUARE ROOT? EXAMPLES 1. Squaring a number means using that number as a factor two times. 8 8(8) 64 (-8) (-8)(-8) 64 Make sure students realize that x means (x ), not (-x).

### Expression. Variable Equation Polynomial Monomial Add. Area. Volume Surface Space Length Width. Probability. Chance Random Likely Possibility Odds

Isosceles Triangle Congruent Leg Side Expression Equation Polynomial Monomial Radical Square Root Check Times Itself Function Relation One Domain Range Area Volume Surface Space Length Width Quantitative

### Solving Logarithmic Equations

Solving Logarithmic Equations Deciding How to Solve Logarithmic Equation When asked to solve a logarithmic equation such as log (x + 7) = or log (7x + ) = log (x + 9), the first thing we need to decide

### SOLVING EQUATIONS BY COMPLETING THE SQUARE

SOLVING EQUATIONS BY COMPLETING THE SQUARE s There are several ways to solve an equation by completing the square. Two methods begin by dividing the equation by a number that will change the "lead coefficient"

### Chapter 7 - Roots, Radicals, and Complex Numbers

Math 233 - Spring 2009 Chapter 7 - Roots, Radicals, and Complex Numbers 7.1 Roots and Radicals 7.1.1 Notation and Terminology In the expression x the is called the radical sign. The expression under the

### A Quick Algebra Review

1. Simplifying Epressions. Solving Equations 3. Problem Solving 4. Inequalities 5. Absolute Values 6. Linear Equations 7. Systems of Equations 8. Laws of Eponents 9. Quadratics 10. Rationals 11. Radicals

### Math 0980 Chapter Objectives. Chapter 1: Introduction to Algebra: The Integers.

Math 0980 Chapter Objectives Chapter 1: Introduction to Algebra: The Integers. 1. Identify the place value of a digit. 2. Write a number in words or digits. 3. Write positive and negative numbers used

### 0.8 Rational Expressions and Equations

96 Prerequisites 0.8 Rational Expressions and Equations We now turn our attention to rational expressions - that is, algebraic fractions - and equations which contain them. The reader is encouraged to

This assignment will help you to prepare for Algebra 1 by reviewing some of the things you learned in Middle School. If you cannot remember how to complete a specific problem, there is an example at the

### Polynomial Operations and Factoring

Algebra 1, Quarter 4, Unit 4.1 Polynomial Operations and Factoring Overview Number of instructional days: 15 (1 day = 45 60 minutes) Content to be learned Identify terms, coefficients, and degree of polynomials.

### Norwalk La Mirada Unified School District. Algebra Scope and Sequence of Instruction

1 Algebra Scope and Sequence of Instruction Instructional Suggestions: Instructional strategies at this level should include connections back to prior learning activities from K-7. Students must demonstrate

### Rational Exponents. Squaring both sides of the equation yields. and to be consistent, we must have

8.6 Rational Exponents 8.6 OBJECTIVES 1. Define rational exponents 2. Simplify expressions containing rational exponents 3. Use a calculator to estimate the value of an expression containing rational exponents

### Equations and Inequalities

Rational Equations Overview of Objectives, students should be able to: 1. Solve rational equations with variables in the denominators.. Recognize identities, conditional equations, and inconsistent equations.

### Math 1050 Khan Academy Extra Credit Algebra Assignment

Math 1050 Khan Academy Extra Credit Algebra Assignment KhanAcademy.org offers over 2,700 instructional videos, including hundreds of videos teaching algebra concepts, and corresponding problem sets. In

### Math 1. Month Essential Questions Concepts/Skills/Standards Content Assessment Areas of Interaction

Binghamton High School Rev.9/21/05 Math 1 September What is the unknown? Model relationships by using Fundamental skills of 2005 variables as a shorthand way Algebra Why do we use variables? What is a

### MATH 108 REVIEW TOPIC 10 Quadratic Equations. B. Solving Quadratics by Completing the Square

Math 108 T10-Review Topic 10 Page 1 MATH 108 REVIEW TOPIC 10 Quadratic Equations I. Finding Roots of a Quadratic Equation A. Factoring B. Quadratic Formula C. Taking Roots II. III. Guidelines for Finding

### MSLC Workshop Series Math 1148 1150 Workshop: Polynomial & Rational Functions

MSLC Workshop Series Math 1148 1150 Workshop: Polynomial & Rational Functions The goal of this workshop is to familiarize you with similarities and differences in both the graphing and expression of polynomial

### 5.1 Radical Notation and Rational Exponents

Section 5.1 Radical Notation and Rational Exponents 1 5.1 Radical Notation and Rational Exponents We now review how exponents can be used to describe not only powers (such as 5 2 and 2 3 ), but also roots

### ALGEBRA 2 CRA 2 REVIEW - Chapters 1-6 Answer Section

ALGEBRA 2 CRA 2 REVIEW - Chapters 1-6 Answer Section MULTIPLE CHOICE 1. ANS: C 2. ANS: A 3. ANS: A OBJ: 5-3.1 Using Vertex Form SHORT ANSWER 4. ANS: (x + 6)(x 2 6x + 36) OBJ: 6-4.2 Solving Equations by

### Florida Math 0028. Correlation of the ALEKS course Florida Math 0028 to the Florida Mathematics Competencies - Upper

Florida Math 0028 Correlation of the ALEKS course Florida Math 0028 to the Florida Mathematics Competencies - Upper Exponents & Polynomials MDECU1: Applies the order of operations to evaluate algebraic

### 7.1 Graphs of Quadratic Functions in Vertex Form

7.1 Graphs of Quadratic Functions in Vertex Form Quadratic Function in Vertex Form A quadratic function in vertex form is a function that can be written in the form f (x) = a(x! h) 2 + k where a is called

### Algebra 2: Q1 & Q2 Review

Name: Class: Date: ID: A Algebra 2: Q1 & Q2 Review Multiple Choice Identify the choice that best completes the statement or answers the question. 1. Which is the graph of y = 2(x 2) 2 4? a. c. b. d. Short

### BookTOC.txt. 1. Functions, Graphs, and Models. Algebra Toolbox. Sets. The Real Numbers. Inequalities and Intervals on the Real Number Line

College Algebra in Context with Applications for the Managerial, Life, and Social Sciences, 3rd Edition Ronald J. Harshbarger, University of South Carolina - Beaufort Lisa S. Yocco, Georgia Southern University

### Simplifying Square-Root Radicals Containing Perfect Square Factors

DETAILED SOLUTIONS AND CONCEPTS - OPERATIONS ON IRRATIONAL NUMBERS Prepared by Ingrid Stewart, Ph.D., College of Southern Nevada Please Send Questions and Comments to ingrid.stewart@csn.edu. Thank you!

### 1.3 Algebraic Expressions

1.3 Algebraic Expressions A polynomial is an expression of the form: a n x n + a n 1 x n 1 +... + a 2 x 2 + a 1 x + a 0 The numbers a 1, a 2,..., a n are called coefficients. Each of the separate parts,

### Answer Key for California State Standards: Algebra I

Algebra I: Symbolic reasoning and calculations with symbols are central in algebra. Through the study of algebra, a student develops an understanding of the symbolic language of mathematics and the sciences.

### Algebra I Vocabulary Cards

Algebra I Vocabulary Cards Table of Contents Expressions and Operations Natural Numbers Whole Numbers Integers Rational Numbers Irrational Numbers Real Numbers Absolute Value Order of Operations Expression

### POLYNOMIAL FUNCTIONS

POLYNOMIAL FUNCTIONS Polynomial Division.. 314 The Rational Zero Test.....317 Descarte s Rule of Signs... 319 The Remainder Theorem.....31 Finding all Zeros of a Polynomial Function.......33 Writing a

### Exponents, Radicals, and Scientific Notation

General Exponent Rules: Exponents, Radicals, and Scientific Notation x m x n = x m+n Example 1: x 5 x = x 5+ = x 7 (x m ) n = x mn Example : (x 5 ) = x 5 = x 10 (x m y n ) p = x mp y np Example : (x) =

### Exponents. Exponents tell us how many times to multiply a base number by itself.

Exponents Exponents tell us how many times to multiply a base number by itself. Exponential form: 5 4 exponent base number Expanded form: 5 5 5 5 25 5 5 125 5 625 To use a calculator: put in the base number,

### MATH 095, College Prep Mathematics: Unit Coverage Pre-algebra topics (arithmetic skills) offered through BSE (Basic Skills Education)

MATH 095, College Prep Mathematics: Unit Coverage Pre-algebra topics (arithmetic skills) offered through BSE (Basic Skills Education) Accurately add, subtract, multiply, and divide whole numbers, integers,

### Assessment Schedule 2013

NCEA Level Mathematics (9161) 013 page 1 of 5 Assessment Schedule 013 Mathematics with Statistics: Apply algebraic methods in solving problems (9161) Evidence Statement ONE Expected Coverage Merit Excellence

### Quadratic and Square Root Functions. Square Roots & Quadratics: What s the Connection?

Activity: TEKS: Overview: Materials: Grouping: Time: Square Roots & Quadratics: What s the Connection? (2A.9) Quadratic and square root functions. The student formulates equations and inequalities based

### 3.1. RATIONAL EXPRESSIONS

3.1. RATIONAL EXPRESSIONS RATIONAL NUMBERS In previous courses you have learned how to operate (do addition, subtraction, multiplication, and division) on rational numbers (fractions). Rational numbers

### Brunswick High School has reinstated a summer math curriculum for students Algebra 1, Geometry, and Algebra 2 for the 2014-2015 school year.

Brunswick High School has reinstated a summer math curriculum for students Algebra 1, Geometry, and Algebra 2 for the 2014-2015 school year. Goal The goal of the summer math program is to help students

### College Algebra. Barnett, Raymond A., Michael R. Ziegler, and Karl E. Byleen. College Algebra, 8th edition, McGraw-Hill, 2008, ISBN: 978-0-07-286738-1

College Algebra Course Text Barnett, Raymond A., Michael R. Ziegler, and Karl E. Byleen. College Algebra, 8th edition, McGraw-Hill, 2008, ISBN: 978-0-07-286738-1 Course Description This course provides

### Method To Solve Linear, Polynomial, or Absolute Value Inequalities:

Solving Inequalities An inequality is the result of replacing the = sign in an equation with ,, or. For example, 3x 2 < 7 is a linear inequality. We call it linear because if the < were replaced with

### The program also provides supplemental modules on topics in geometry and probability and statistics.

Algebra 1 Course Overview Students develop algebraic fluency by learning the skills needed to solve equations and perform important manipulations with numbers, variables, equations, and inequalities. Students

### Math Common Core Sampler Test

High School Algebra Core Curriculum Math Test Math Common Core Sampler Test Our High School Algebra sampler covers the twenty most common questions that we see targeted for this level. For complete tests

### MBA Jump Start Program

MBA Jump Start Program Module 2: Mathematics Thomas Gilbert Mathematics Module Online Appendix: Basic Mathematical Concepts 2 1 The Number Spectrum Generally we depict numbers increasing from left to right

### CRLS Mathematics Department Algebra I Curriculum Map/Pacing Guide

Curriculum Map/Pacing Guide page 1 of 14 Quarter I start (CP & HN) 170 96 Unit 1: Number Sense and Operations 24 11 Totals Always Include 2 blocks for Review & Test Operating with Real Numbers: How are

### Indiana State Core Curriculum Standards updated 2009 Algebra I

Indiana State Core Curriculum Standards updated 2009 Algebra I Strand Description Boardworks High School Algebra presentations Operations With Real Numbers Linear Equations and A1.1 Students simplify and

9.3 Solving Quadratic Equations by Using the Quadratic Formula 9.3 OBJECTIVES 1. Solve a quadratic equation by using the quadratic formula 2. Determine the nature of the solutions of a quadratic equation

### Chapter 4 -- Decimals

Chapter 4 -- Decimals \$34.99 decimal notation ex. The cost of an object. ex. The balance of your bank account ex The amount owed ex. The tax on a purchase. Just like Whole Numbers Place Value - 1.23456789

### Algebra I Credit Recovery

Algebra I Credit Recovery COURSE DESCRIPTION: The purpose of this course is to allow the student to gain mastery in working with and evaluating mathematical expressions, equations, graphs, and other topics,

### SECTION 0.6: POLYNOMIAL, RATIONAL, AND ALGEBRAIC EXPRESSIONS

(Section 0.6: Polynomial, Rational, and Algebraic Expressions) 0.6.1 SECTION 0.6: POLYNOMIAL, RATIONAL, AND ALGEBRAIC EXPRESSIONS LEARNING OBJECTIVES Be able to identify polynomial, rational, and algebraic

### CONVERT QUADRATIC FUNCTIONS FROM ONE FORM TO ANOTHER (Standard Form <==> Intercept Form <==> Vertex Form) (By Nghi H Nguyen Dec 08, 2014)

CONVERT QUADRATIC FUNCTIONS FROM ONE FORM TO ANOTHER (Standard Form Intercept Form Vertex Form) (By Nghi H Nguyen Dec 08, 2014) 1. THE QUADRATIC FUNCTION IN INTERCEPT FORM The graph of the quadratic

### Section 1.1 Linear Equations: Slope and Equations of Lines

Section. Linear Equations: Slope and Equations of Lines Slope The measure of the steepness of a line is called the slope of the line. It is the amount of change in y, the rise, divided by the amount of

### Section 1.5 Exponents, Square Roots, and the Order of Operations

Section 1.5 Exponents, Square Roots, and the Order of Operations Objectives In this section, you will learn to: To successfully complete this section, you need to understand: Identify perfect squares.

### Year 9 set 1 Mathematics notes, to accompany the 9H book.

Part 1: Year 9 set 1 Mathematics notes, to accompany the 9H book. equations 1. (p.1), 1.6 (p. 44), 4.6 (p.196) sequences 3. (p.115) Pupils use the Elmwood Press Essential Maths book by David Raymer (9H

### Packet 1 for Unit 2 Intercept Form of a Quadratic Function. M2 Alg 2

Packet 1 for Unit Intercept Form of a Quadratic Function M Alg 1 Assignment A: Graphs of Quadratic Functions in Intercept Form (Section 4.) In this lesson, you will: Determine whether a function is linear

### Mth 95 Module 2 Spring 2014

Mth 95 Module Spring 014 Section 5.3 Polynomials and Polynomial Functions Vocabulary of Polynomials A term is a number, a variable, or a product of numbers and variables raised to powers. Terms in an expression

### Algebra 2 Chapter 5 Practice Test (Review)

Name: Class: Date: Algebra 2 Chapter 5 Practice Test (Review) Multiple Choice Identify the choice that best completes the statement or answers the question. Determine whether the function is linear or

9.5 Quadratics - Build Quadratics From Roots Objective: Find a quadratic equation that has given roots using reverse factoring and reverse completing the square. Up to this point we have found the solutions

### Solving Systems of Equations with Absolute Value, Polynomials, and Inequalities

Solving Systems of Equations with Absolute Value, Polynomials, and Inequalities Solving systems of equations with inequalities When solving systems of linear equations, we are looking for the ordered pair

### Indices and Surds. The Laws on Indices. 1. Multiplication: Mgr. ubomíra Tomková

Indices and Surds The term indices refers to the power to which a number is raised. Thus x is a number with an index of. People prefer the phrase "x to the power of ". Term surds is not often used, instead

### 1. The algebra of exponents 1.1. Natural Number Powers. It is easy to say what is meant by a n a (raised to) to the (power) n if n N.

CHAPTER 3: EXPONENTS AND POWER FUNCTIONS 1. The algebra of exponents 1.1. Natural Number Powers. It is easy to say what is meant by a n a (raised to) to the (power) n if n N. For example: In general, if

### SECTION 2.5: FINDING ZEROS OF POLYNOMIAL FUNCTIONS

SECTION 2.5: FINDING ZEROS OF POLYNOMIAL FUNCTIONS Assume f ( x) is a nonconstant polynomial with real coefficients written in standard form. PART A: TECHNIQUES WE HAVE ALREADY SEEN Refer to: Notes 1.31

### Prentice Hall Mathematics Algebra 1 2004 Correlated to: Alabama Course of Study: Mathematics (Grades 9-12)

Alabama Course of Study: Mathematics (Grades 9-12) NUMBER AND OPERATIONS 1. Simplify numerical expressions using properties of real numbers and order of operations, including those involving square roots,

### QUADRATIC, EXPONENTIAL AND LOGARITHMIC FUNCTIONS

QUADRATIC, EXPONENTIAL AND LOGARITHMIC FUNCTIONS Content 1. Parabolas... 1 1.1. Top of a parabola... 2 1.2. Orientation of a parabola... 2 1.3. Intercept of a parabola... 3 1.4. Roots (or zeros) of a parabola...

### Pre-Algebra 2008. Academic Content Standards Grade Eight Ohio. Number, Number Sense and Operations Standard. Number and Number Systems

Academic Content Standards Grade Eight Ohio Pre-Algebra 2008 STANDARDS Number, Number Sense and Operations Standard Number and Number Systems 1. Use scientific notation to express large numbers and small

### Zeros of a Polynomial Function

Zeros of a Polynomial Function An important consequence of the Factor Theorem is that finding the zeros of a polynomial is really the same thing as factoring it into linear factors. In this section we

### Supplemental Worksheet Problems To Accompany: The Pre-Algebra Tutor: Volume 1 Section 1 Real Numbers

Supplemental Worksheet Problems To Accompany: The Pre-Algebra Tutor: Volume 1 Please watch Section 1 of this DVD before working these problems. The DVD is located at: http://www.mathtutordvd.com/products/item66.cfm

### http://www.aleks.com Access Code: RVAE4-EGKVN Financial Aid Code: 6A9DB-DEE3B-74F51-57304

MATH 1340.04 College Algebra Location: MAGC 2.202 Meeting day(s): TR 7:45a 9:00a, Instructor Information Name: Virgil Pierce Email: piercevu@utpa.edu Phone: 665.3535 Teaching Assistant Name: Indalecio

### Algebra 1 Advanced Mrs. Crocker. Final Exam Review Spring 2014

Name: Mod: Algebra 1 Advanced Mrs. Crocker Final Exam Review Spring 2014 The exam will cover Chapters 6 10 You must bring a pencil, calculator, eraser, and exam review flip book to your exam. You may bring

### PREPARATION FOR MATH TESTING at CityLab Academy

PREPARATION FOR MATH TESTING at CityLab Academy compiled by Gloria Vachino, M.S. Refresh your math skills with a MATH REVIEW and find out if you are ready for the math entrance test by taking a PRE-TEST

### 3.2. Solving quadratic equations. Introduction. Prerequisites. Learning Outcomes. Learning Style

Solving quadratic equations 3.2 Introduction A quadratic equation is one which can be written in the form ax 2 + bx + c = 0 where a, b and c are numbers and x is the unknown whose value(s) we wish to find.

### Administrative - Master Syllabus COVER SHEET

Administrative - Master Syllabus COVER SHEET Purpose: It is the intention of this to provide a general description of the course, outline the required elements of the course and to lay the foundation for

### Section summaries. d = (x 2 x 1 ) 2 + (y 2 y 1 ) 2. 1 + y 2. x1 + x 2

Chapter 2 Graphs Section summaries Section 2.1 The Distance and Midpoint Formulas You need to know the distance formula d = (x 2 x 1 ) 2 + (y 2 y 1 ) 2 and the midpoint formula ( x1 + x 2, y ) 1 + y 2