By Mary Jane Sterling - Softouch

Algebra IIFORDUMmIESby Mary Jane Sterling‰

Algebra II For Dummies Published byWiley Publishing, Inc.111 River St.Hoboken, NJ 07030-5774www.wiley.comCopyright 2006 by Wiley Publishing, Inc., Indianapolis, IndianaPublished simultaneously in CanadaNo part of this publication may be reproduced, stored in a retrieval system, or transmitted in any formor by any means, electronic, mechanical, photocopying, recording, scanning, or otherwise, except as permitted under Sections 107 or 108 of the 1976 United States Copyright Act, without either the prior writtenpermission of the Publisher, or authorization through payment of the appropriate per-copy fee to theCopyright Clearance Center, 222 Rosewood Drive, Danvers, MA 01923, 978-750-8400, fax 978-646-8600.Requests to the Publisher for permission should be addressed to the Legal Department, Wiley Publishing,Inc., 10475 Crosspoint Blvd., Indianapolis, IN 46256, 317-572-3447, fax 317-572-4355, or online at http://www.wiley.com/go/permissions.Trademarks: Wiley, the Wiley Publishing logo, For Dummies, the Dummies Man logo, A Reference for theRest of Us!, The Dummies Way, Dummies Daily, The Fun and Easy Way, Dummies.com and related tradedress are trademarks or registered trademarks of John Wiley & Sons, Inc. and/or its affiliates in the UnitedStates and other countries, and may not be used without written permission. All other trademarks are theproperty of their respective owners. Wiley Publishing, Inc., is not associated with any product or vendormentioned in this book.LIMIT OF LIABILITY/DISCLAIMER OF WARRANTY: THE PUBLISHER AND THE AUTHOR MAKE NO REPRESENTATIONS OR WARRANTIES WITH RESPECT TO THE ACCURACY OR COMPLETENESS OF THE CONTENTS OF THIS WORK AND SPECIFICALLY DISCLAIM ALL WARRANTIES, INCLUDING WITHOUTLIMITATION WARRANTIES OF FITNESS FOR A PARTICULAR PURPOSE. NO WARRANTY MAY BE CREATEDOR EXTENDED BY SALES OR PROMOTIONAL MATERIALS. THE ADVICE AND STRATEGIES CONTAINEDHEREIN MAY NOT BE SUITABLE FOR EVERY SITUATION. THIS WORK IS SOLD WITH THE UNDERSTANDING THAT THE PUBLISHER IS NOT ENGAGED IN RENDERING LEGAL, ACCOUNTING, OR OTHERPROFESSIONAL SERVICES. IF PROFESSIONAL ASSISTANCE IS REQUIRED, THE SERVICES OF A COMPETENT PROFESSIONAL PERSON SHOULD BE SOUGHT. NEITHER THE PUBLISHER NOR THE AUTHORSHALL BE LIABLE FOR DAMAGES ARISING HEREFROM. THE FACT THAT AN ORGANIZATION ORWEBSITE IS REFERRED TO IN THIS WORK AS A CITATION AND/OR A POTENTIAL SOURCE OF FURTHERINFORMATION DOES NOT MEAN THAT THE AUTHOR OR THE PUBLISHER ENDORSES THE INFORMATION THE ORGANIZATION OR WEBSITE MAY PROVIDE OR RECOMMENDATIONS IT MAY MAKE.FURTHER, READERS SHOULD BE AWARE THAT INTERNET WEBSITES LISTED IN THIS WORK MAY HAVECHANGED OR DISAPPEARED BETWEEN WHEN THIS WORK WAS WRITTEN AND WHEN IT IS READ.For general information on our other products and services, please contact our Customer CareDepartment within the U.S. at 800-762-2974, outside the U.S. at 317-572-3993, or fax 317-572-4002.For technical support, please visit www.wiley.com/techsupport.Wiley also publishes its books in a variety of electronic formats. Some content that appears in print maynot be available in electronic books.Library of Congress Control Number: 2006923792ISBN-13: 978-0-471-77581-2ISBN-10: 0-471-77581-9Manufactured in the United States of America10 9 8 7 6 5 4 3 2 11O/QV/QW/QW/IN

About the AuthorMary Jane Sterling has authored Algebra For Dummies, TrigonometryFor Dummies, Algebra Workbook For Dummies, Trigonometry Workbook ForDummies, Algebra I CliffsStudySolver, and Algebra II CliffsStudySolver. Shetaught junior high and high school math for many years before beginning hercurrent 25-year-and-counting career at Bradley University in Peoria, Illinois.Mary Jane enjoys working with her students both in the classroom and outside the classroom, where they do various community service projects.DedicationThe author dedicates this book to some of the men in her life. Her husband,Ted Sterling, is especially patient and understanding when her behaviorbecomes erratic while working on her various projects — his support isgreatly appreciated. Her brothers Tom, Don, and Doug knew her “backwhen.” Don, in particular, had an effect on her teaching career when he threwa pencil across the room during a tutoring session. It was then that sherethought her approach — and look what happened! And brother-in-law Jeffis an ongoing inspiration with his miracle comeback and continued recovery.Author’s AcknowledgmentsThe author wants to thank Mike Baker for being a great project editor — goodnatured (very important) and thorough. He took the many challenges withgrace and handled them with diplomacy. Also, thank you to Josh Dials, awonderful editor who straightened out her circuitous explanations and madethem understandable. A big thank you to the technical editor, Alexsis Venter,who helped her on an earlier project — and still agreed to sign on! Also,thanks to Kathy Cox for keeping the projects coming; she can be counted onto keep life interesting.

Publisher’s AcknowledgmentsWe’re proud of this book; please send us your comments through our Dummies online registrationform located at www.dummies.com/register/.Some of the people who helped bring this book to market include the following:Acquisitions, Editorial, andMedia DevelopmentProject Editor: Mike BakerComposition ServicesProject Coordinator: Jennifer TheriotAcquisitions Editor: Kathy CoxLayout and Graphics: Lauren Goddard,Denny Hager, Barry Offringa, Heather RyanCopy Editor: Josh DialsProofreader: Betty KishEditorial Program Coordinator: Hanna K. ScottIndexer: Slivoskey Indexing ServicesTechnical Editor: Alexsis VenterEditorial Manager: Christine Meloy BeckEditorial Assistants: Erin Calligan, David LuttonCover Photos: Wiley Publishing, Inc.Cartoons: Rich Tennant (www.the5thwave.com)Publishing and Editorial for Consumer DummiesDiane Graves Steele, Vice President and Publisher, Consumer DummiesJoyce Pepple, Acquisitions Director, Consumer DummiesKristin A. Cocks, Product Development Director, Consumer DummiesMichael Spring, Vice President and Publisher, TravelKelly Regan, Editorial Director, TravelPublishing for Technology DummiesAndy Cummings, Vice President and Publisher, Dummies Technology/General UserComposition ServicesGerry Fahey, Vice President of Production ServicesDebbie Stailey, Director of Composition Services

Contents at a GlanceIntroduction .1Part I: Homing in on Basic Solutions .7Chapter 1: Going Beyond Beginning Algebra.9Chapter 2: Toeing the Straight Line: Linear Equations.23Chapter 3: Cracking Quadratic Equations.37Chapter 4: Rooting Out the Rational, Radical, and Negative .57Chapter 5: Graphing Your Way to the Good Life .77Part II: Facing Off with Functions.97Chapter 6: Formulating Function Facts .99Chapter 7: Sketching and Interpreting Quadratic Functions .117Chapter 8: Staying Ahead of the Curves: Polynomials .133Chapter 9: Relying on Reason: Rational Functions .157Chapter 10: Exposing Exponential and Logarithmic Functions .177Part III: Conquering Conics and Systems of Equations .201Chapter 11: Cutting Up Conic Sections.203Chapter 12: Solving Systems of Linear Equations .225Chapter 13: Solving Systems of Nonlinear Equations and Inequalities .247Part IV: Shifting into High Gearwith Advanced Concepts .267Chapter 14: Simplifying Complex Numbers in a Complex World .269Chapter 15: Making Moves with Matrices .281Chapter 16: Making a List: Sequences and Series .303Chapter 17: Everything You Wanted to Know about Sets .323Part V: The Part of Tens .347Chapter 18: Ten Multiplication Tricks .349Chapter 19: Ten Special Types of Numbers .357Index .361

Table of ContentsIntroduction.1About This Book.1Conventions Used in This Book .2Foolish Assumptions .2How This Book Is Organized.3Part I: Homing in on Basic Solutions.3Part II: Facing Off with Functions .4Part III: Conquering Conics and Systems of Equations .4Part IV: Shifting into High Gear with Advanced Concepts .5Part V: The Part of Tens.5Icons Used in This Book.5Where to Go from Here.6Part I: Homing in on Basic Solutions.7Chapter 1: Going Beyond Beginning Algebra . . . . . . . . . . . . . . . . . . . . . .9Outlining Algebra Properties.10Keeping order with the commutative property .10Maintaining group harmony with the associative property .10Distributing a wealth of values .11Checking out an algebraic ID .12Singing along in-verses .13Ordering Your Operations.13Equipping Yourself with the Multiplication Property of Zero .14Expounding on Exponential Rules .15Multiplying and dividing exponents .15Getting to the roots of exponents .15Raising or lowering the roof with exponents.16Making nice with negative exponents.17Implementing Factoring Techniques .17Factoring two terms .17Taking on three terms.18Factoring four or more terms by grouping .22Chapter 2: Toeing the Straight Line: Linear Equations . . . . . . . . . . . . .23Linear Equations: Handling the First Degree .23Tackling basic linear equations .24Clearing out fractions .25Isolating different unknowns .26

viiiAlgebra II For DummiesLinear Inequalities: Algebraic Relationship Therapy .28Solving basic inequalities .28Introducing interval notation.29Compounding inequality issues .30Absolute Value: Keeping Everything in Line.32Solving absolute-value equations.32Seeing through absolute-value inequality .34Chapter 3: Cracking Quadratic Equations . . . . . . . . . . . . . . . . . . . . . . . .37Solving Simple Quadratics with the Square Root Rule.38Finding simple square-root solutions .38Dealing with radical square-root solutions .38Dismantling Quadratic Equations into Factors .39Factoring binomials .39Factoring trinomials.41Factoring by grouping.42Resorting to the Quadratic Formula.43Finding rational solutions .44Straightening out irrational solutions.44Formulating huge quadratic results.45Completing the Square: Warming Up for Conics.46Squaring up to solve a quadratic equation .46Completing the square twice over .48Getting Promoted to High-Powered Quadratics (without the Raise) .49Handling the sum or difference of cubes .50Tackling quadratic-like trinomials.51Solving Quadratic Inequalities .52Keeping it strictly quadratic .53Signing up for fractions .54Increasing the number of factors .55Chapter 4: Rooting Out the Rational, Radical, and Negative . . . . . . . .57Acting Rationally with Fraction-Filled Equations.57Solving rational equations by tuning in your LCD .58Solving rational equations with proportions.62Ridding Yourself of a Radical.65Squaring both sides of a radical equation .65Calming two radicals.67Changing Negative Attitudes about Exponents.68Flipping negative exponents out of the picture.69Factoring out negatives to solve equations .70Fooling Around with Fractional Exponents .73Combining terms with fractional exponents .73Factoring fractional exponents.73Solving equations by working with fractional exponents .74