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


 Peter Austin
 9 months ago
 Views:
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
2 Objective 2: Simplifying real Square Roots Perfect Squares: If the number under a square root sign is a perfect square number (e.g. 25), o There is only 1 part. Take the square root and write the answer without the radical sign. This is the simplified answer. Example: 25 = 5 Non Perfect Squares: If the square root is NOT a perfect square (e.g. 24) simplify the square roots into two parts. First part: largest perfect square factor (rational root) Second part: a non perfect square factor (irrational root) Simplify the perfect square root. Put that number OUTSIDE the radical sign. Note: it is STILL multiplied so it should be immediately to the left of the radical sign without any symbol after it. Leave the irrational square root under the radical sign. Example: 24 = 4 i 6 = 4 i 6 = 2 6 Optional check: Enter your ORIGINAL square root problem in your calculator and see the decimal. Enter your simplified radical answer into the calculator. The two decimal answers should match! Simplify the following square roots: 1) 125 2) 32 3) ) ) 28 6) 27 7) ) ± ) 3 ± 50 10) 1± 25 11) ) 3 ± 100 Page 2
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 #13) ** 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 #13) 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 #13) 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 1624
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
More information5.4 The Quadratic Formula
Section 5.4 The Quadratic Formula 481 5.4 The Quadratic Formula Consider the general quadratic function f(x) = ax + bx + c. In the previous section, we learned that we can find the zeros of this function
More information2. Simplify. College Algebra Student SelfAssessment of Mathematics (SSAM) Answer Key. Use the distributive property to remove the parentheses
College Algebra Student SelfAssessment 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
More informationBasic 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:
More informationAlgebra 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,
More informationZero: 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
More informationAlgebra 12. A. Identify and translate variables and expressions.
St. Mary's College High School Algebra 12 The Language of Algebra What is a variable? A. Identify and translate variables and expressions. The following apply to all the skills How is a variable used
More informationWhat 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
More informationPrentice 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
More informationALGEBRA 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
More informationObjectives. 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
More informationMethods to Solve Quadratic Equations
Methods to Solve Quadratic Equations We have been learning how to factor epressions. Now we will apply factoring to another skill you must learn solving quadratic equations. a b c 0 is a seconddegree
More informationStep 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
More informationThis 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
More informationFind 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 TI30XS MultiView rational
More informationAlgebra 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
More informationModuMath Algebra Lessons
ModuMath Algebra Lessons Program Title 1 Getting Acquainted With Algebra 2 Order of Operations 3 Adding & Subtracting Algebraic Expressions 4 Multiplying Polynomials 5 Laws of Algebra 6 Solving Equations
More information2.3. 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
More informationHigher 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
More informationLesson 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 yvalues. b. The verte in the third quadrant and no intercepts. c. The verte
More informationDevelopmental Math Course Outcomes and Objectives
Developmental Math Course Outcomes and Objectives I. Math 0910 Basic Arithmetic/PreAlgebra Upon satisfactory completion of this course, the student should be able to perform the following outcomes and
More informationCORRELATED TO THE SOUTH CAROLINA COLLEGE AND CAREERREADY FOUNDATIONS IN ALGEBRA
We Can Early Learning Curriculum PreK Grades 8 12 INSIDE ALGEBRA, GRADES 8 12 CORRELATED TO THE SOUTH CAROLINA COLLEGE AND CAREERREADY FOUNDATIONS IN ALGEBRA April 2016 www.voyagersopris.com Mathematical
More informationActually, if you have a graphing calculator this technique can be used to find solutions to any equation, not just quadratics. All you need to do is
QUADRATIC EQUATIONS Definition ax 2 + bx + c = 0 a, b, c are constants (generally integers) Roots Synonyms: Solutions or Zeros Can have 0, 1, or 2 real roots Consider the graph of quadratic equations.
More informationALGEBRA I / ALGEBRA I SUPPORT
Suggested Sequence: CONCEPT MAP ALGEBRA I / ALGEBRA I SUPPORT August 2011 1. Foundations for Algebra 2. Solving Equations 3. Solving Inequalities 4. An Introduction to Functions 5. Linear Functions 6.
More informationMyMathLab 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
More informationSolving Quadratic Equations by Completing the Square
9. Solving Quadratic Equations by Completing the Square 9. OBJECTIVES 1. Solve a quadratic equation by the square root method. Solve a quadratic equation by completing the square. Solve a geometric application
More information4.1. 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
More informationPolynomial 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
More informationMATH 65 NOTEBOOK CERTIFICATIONS
MATH 65 NOTEBOOK CERTIFICATIONS Review Material from Math 60 2.5 4.3 4.4a Chapter #8: Systems of Linear Equations 8.1 8.2 8.3 Chapter #5: Exponents and Polynomials 5.1 5.2a 5.2b 5.3 5.4 5.5 5.6a 5.7a 1
More informationQuadratic Functions and Parabolas
MATH 11 Quadratic Functions and Parabolas A quadratic function has the form Dr. Neal, Fall 2008 f () = a 2 + b + c where a 0. The graph of the function is a parabola that opens upward if a > 0, and opens
More informationLarson, R. and Boswell, L. (2016). Big Ideas Math, Algebra 2. Erie, PA: Big Ideas Learning, LLC. ISBN
ALG B Algebra II, Second Semester #PR0, BK04 (v.4.0) To the Student: After your registration is complete and your proctor has been approved, you may take the Credit by Examination for ALG B. WHAT TO
More informationAlgebra 2 YearataGlance Leander ISD 200708. 1st Six Weeks 2nd Six Weeks 3rd Six Weeks 4th Six Weeks 5th Six Weeks 6th Six Weeks
Algebra 2 YearataGlance Leander ISD 200708 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
More informationa) 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
More information2.1 QUADRATIC FUNCTIONS AND MODELS. Copyright Cengage Learning. All rights reserved.
2.1 QUADRATIC FUNCTIONS AND MODELS Copyright Cengage Learning. All rights reserved. What You Should Learn Analyze graphs of quadratic functions. Write quadratic functions in standard form and use the results
More informationAlgebra 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
More informationPrentice 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
More informationSimplify 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 249x + 6 x  6 A) 1, x 6 B) 8x  1, x 6 x 
More informationVocabulary 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
More informationFACTORING 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.
More informationAlgebra 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 reallife problem. algebraic expression  An expression with variables.
More information( 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
More informationARE 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).
More informationMTH124: 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,
More informationSolving 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
More informationExpression. 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
More informationSOLVING 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"
More informationMath 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
More informationChapter 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
More informationA 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
More informationThis 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
More information0.8 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
More informationNorwalk 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 K7. Students must demonstrate
More informationPolynomial 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.
More informationChapter 13: Polynomials
Chapter 13: Polynomials We will not cover all there is to know about polynomials for the math competency exam. We will go over the addition, subtraction, and multiplication of polynomials. We will not
More informationRational 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
More informationPre Cal 2 1 Lesson with notes 1st.notebook. January 22, Operations with Complex Numbers
0 2 Operations with Complex Numbers Objectives: To perform operations with pure imaginary numbers and complex numbers To use complex conjugates to write quotients of complex numbers in standard form Complex
More informationMSLC 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
More informationEquations 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.
More informationMath 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
More informationMATH 108 REVIEW TOPIC 10 Quadratic Equations. B. Solving Quadratics by Completing the Square
Math 108 T10Review 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
More information5.1 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
More informationMath 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
More informationALGEBRA 2 CRA 2 REVIEW  Chapters 16 Answer Section
ALGEBRA 2 CRA 2 REVIEW  Chapters 16 Answer Section MULTIPLE CHOICE 1. ANS: C 2. ANS: A 3. ANS: A OBJ: 53.1 Using Vertex Form SHORT ANSWER 4. ANS: (x + 6)(x 2 6x + 36) OBJ: 64.2 Solving Equations by
More informationFlorida 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
More information7.1 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
More informationVariable. 1.1 Order of Operations. August 17, evaluating expressions ink.notebook. Standards. letter or symbol used to represent a number
1.1 evaluating expressions ink.notebook page 8 Unit 1 Basic Equations and Inequalities 1.1 Order of Operations page 9 Square Cube Variable Variable Expression Exponent page 10 page 11 1 Lesson Objectives
More informationAlgebra 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
More informationFunctions and Equations
Centre for Education in Mathematics and Computing Euclid eworkshop # Functions and Equations c 014 UNIVERSITY OF WATERLOO Euclid eworkshop # TOOLKIT Parabolas The quadratic f(x) = ax + bx + c (with a,b,c
More informationBookTOC.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
More informationSimplifying SquareRoot 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!
More informationEER#21 Graph parabolas and circles whose equations are given in general form by completing the square.
EER#1 Graph parabolas and circles whose equations are given in general form by completing the square. Circles A circle is a set of points that are equidistant from a fixed point. The distance is called
More information1.3 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,
More informationAlgebra 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
More information27 = 3 Example: 1 = 1
Radicals: Definition: A number r is a square root of another number a if r = a. is a square root of 9 since = 9 is also a square root of 9, since ) = 9 Notice that each positive number a has two square
More informationAnswer 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.
More informationPOLYNOMIAL 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
More informationExponents. 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,
More informationMATH 095, College Prep Mathematics: Unit Coverage Prealgebra topics (arithmetic skills) offered through BSE (Basic Skills Education)
MATH 095, College Prep Mathematics: Unit Coverage Prealgebra topics (arithmetic skills) offered through BSE (Basic Skills Education) Accurately add, subtract, multiply, and divide whole numbers, integers,
More informationAssessment 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
More informationQuadratics  Graphs of Quadratics
9.11 Quadratics  Graphs of Quadratics Objective: Graph quadratic equations using the vertex, xintercepts, and yintercept. Just as we drew pictures of the solutions for lines or linear equations, we
More informationCOLLEGE ALGEBRA. Paul Dawkins
COLLEGE ALGEBRA Paul Dawkins Table of Contents Preface... iii Outline... iv Preliminaries... Introduction... Integer Exponents... Rational Exponents... 9 Real Exponents...5 Radicals...6 Polynomials...5
More informationBrunswick High School has reinstated a summer math curriculum for students Algebra 1, Geometry, and Algebra 2 for the 20142015 school year.
Brunswick High School has reinstated a summer math curriculum for students Algebra 1, Geometry, and Algebra 2 for the 20142015 school year. Goal The goal of the summer math program is to help students
More informationQuadratic 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
More informationExponents, 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) =
More informationUnit 1 Review Part 1 3 combined Handout KEY.notebook. September 26, 2013
Math 10c Unit 1 Factors, Powers and Radicals Key Concepts 1.1 Determine the prime factors of a whole number. 650 3910 1.2 Explain why the numbers 0 and 1 have no prime factors. 0 and 1 have no prime factors
More informationCollege Algebra. Barnett, Raymond A., Michael R. Ziegler, and Karl E. Byleen. College Algebra, 8th edition, McGrawHill, 2008, ISBN: 9780072867381
College Algebra Course Text Barnett, Raymond A., Michael R. Ziegler, and Karl E. Byleen. College Algebra, 8th edition, McGrawHill, 2008, ISBN: 9780072867381 Course Description This course provides
More informationMethod 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
More informationThe 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
More informationMath 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
More informationAn Essay on the Quadratic Formula: Origins, Derivation, and Applications
An Essay on the Quadratic Formula: Origins, Derivation, and Applications Foreword: The quadratic equation is a formula that is used to solve equations in the form of quadratics. A quadratic is an equation
More information3.1. 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
More informationMBA 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
More informationCRLS 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
More informationIndiana 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
More information4.4 Concavity and Curve Sketching
Concavity and Curve Sketching Section Notes Page We can use the second derivative to tell us if a graph is concave up or concave down To see if something is concave down or concave up we need to look at
More informationSolving Quadratic Equations
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
More informationChapter 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
More informationYear 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
More informationQuadratic Modeling Business 10 Profits
Quadratic Modeling Business 10 Profits In this activity, we are going to look at modeling business profits. We will allow q to represent the number of items manufactured and assume that all items that
More informationSECTION 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
More information