Key to Algebra Book 10 Square Roots and Quadratic Equations

Key to

$ebra Sqaare Roots and Qundratic Equstisns

W Ana22 nne*12*

By fulie King and PeterRasmussen

Name

Class

TABLEOF CONTENTS ............1 ...................2 ............4 ..............5 ..............8 10 ........,......... .................12

Squares SquareRoots lrrational Numbers SolvingQuadratic Equations WithoutFactoring. The Pythagorean Theorem Formulas Involving SquareRoots S q u a r eR o o t so f M o n o m i a l s . . . . . , . . . . The ProductandQuotientRules Completing the Square T h eQ u a d r a t F i co r m u l a . . . . . . . . . . Usingthe Quadratic Formula to SolveProblems G r a p ho sfFunctions............ Functions Quadratic Written Work Practice Test........ EmmyNoether Vastchangesoccurredinalgebraduringthe1gthcentury.Theteenagers AbelandGaloisshowedthatgeneralequations of degreefiveandhigher cannotbe solved. Then Galoisintroducedgroupsto determinewhich equationscould be solved. He died in 1832. Althoughhis advances weren'tunderstood for another25 years,withinfiftyyearsgrouptheory wasbeingusedto defineall kindsof geometries andtransformations on them. just Aroundtheturnof thecenturyabstraction enteredmathematics, as it had in the art world. Not only were groupsstudied-so were structureswithsuchstrangenamesas ringsandfields. However,theredidn'tseemto be any drreadto tie thesestrands together.ThencameEmmyNoether(pronounced netter I 882-1935). Noetherwasn'tthe first womanto make a vital contributionto mathemalics.Therehad been Hypatia,the leadingmathematician in Greecearound400n.o.,andMariaAgnesi,whowroteoneof theearliest calculusbooks.A curvecalledthe 'witchof Agnesi'gotits namefroma mis-translation of an ltalianwordin her book. An importantmathemalician fromthe lastpartof the 19thcentury, (1850-1891 SofiaKovalevsky ), excelledin highschoolin Russiabutwas forbiddenfrom attendingclassesin collegebecausewomenwerenot permitted.So she arrangeda marriagein orderto studyin Germany. Kovalevskybecame the first woman to be awarded a doctoratein mathematics, thefirstto holda chairin mathematics at a maioruniversity, andthe firstto holda positionon the editorialboardof a majorscientific joumal.All of thisin 41 years! Butit wasEmmyNoetherwhorevolutionized algebrain ourcentury. LikeKovalEvsky, Noetherhadto fightmanybattlesbecauseof hersex. lt is ironicthatGermany,thecountrywhichpermittedKovalevsky to pursue advancedstudies,preventedNoetherfrom teachingin a university becauseshewas a woman. DavidHilbert,the leadingmathematician in the worldat thattime, pleaon Noether'sbehalf. He told the lacultyat madean impassioned GdttingenUniversity,"Gentlemen,I do not see hat he sex ol the candidateis an argumentagainstheradmission as a professor.Afterall, the facultyclubis nota bathhouse.'

Historicalnoteby DavidZitarElli

........26 ...........30 ...........32 .......33 ..................35 ..........36 fuoA*eaja+

lllustrafonby Jay Flom

buthe solved' Hilbertwasn'tableto winhera desiredprofessorship, problem lectures underhis of keepingherat Gottingenbyannouncing the namebut havingthemdeliveredby FrAuleinNoether. A few yearslat€r Noetherhad to flee Gennanybecauseof Nazi persecution.She becamea professorat the prestigiousBryn Mawr CollegenearPhiladelphia. Not to mathemaucs. Noethermademanyoutstanding contributions onlydid she becomethe leadingexpertin one areaof abstractalgebra tiedtogetherall (Noettrerian ringsarenamedfor her),butsheeffectively "abstract the differentstructuresof the subject.Today,coursescalled algebra'dealwithexacflythesametopics,andinthe algebra'or'modern '1920's. sameorder,thatshedevelopedin the On the coverof thisbookis a pofiraitof EmmyNoether.

IMPORTANTNOTICE:This book is sold as a studentworkbookand is notto be used as a duplicating master. No part of this book may be reproducedin any form without the prior written permission of the publisher. Copyrightinfringementis a violationof FederalLaw. Copyright@1992by KeyCurriculumProject,Inc.All rightsreserved. @Key to Fractions,Key to Decimals,Key to Percents,Key to Algebra,Key to Geometry,Key to Measurement,and Key to Metic Measurement arc registeredtrademarks of KeyCuriculumPress. Publishedby KeyCurriculumPress,I 15065thStreet,Emeryville, CA 94608 Printedin the UnitedStatesof America 21 20 19 08 07 06 05 lsBN 1-55953{10-3

Squares polynomials Squaring numbers, youhavedone andrational is something expressions "squaring" manytimesin algebra.Weusetheword to meanmultiplying a numberor expression by itself,because thisis whatwe doto findtheareasof squares. Findtheareaof eachshadedsquare.

i-i--i-i i

1

5

r

r

l

ffi-_-i_-l 2

5

1

Writean expression fortheareaof eachsquare.

*+3

A=

6y z

2x-1

A=

Tofindtheareaof anysquareallwe needto knowis thelengrth of oneside. lt is also possible to findthelengrth of a sideif we knowthearea.Tryto figureoutthelenghsof the sidesof eachsquarebelow.

A=576 A= 144

Wecalltheareathesquareofthelenghof a side.Wesaythatthelenghof a sideis the squareroot of thearea.Inthisbookwewillstudysquareroots.Knowing aboutsquare rootsenablesus to solvesomequadratic equations we couldnotsolveby factoring.

@19Qby KsyCurbulumPreos,Inc. Do not dupllcalewithoutpormlssbn.

SquareRoots Hereis a definition of a squarerootof a number. o is a squarerootof b if and onlv if a2 = b.

Weusethesymbol{- to showa squarerootof a number.f signor a radlcalsign.

J T = 3 b e c a u s3 e" = 1

,m

is calleda squareroot

= 1.4 because = 1.16 (l.r+)'

Writeeachsquarerootas an integer, a fraction or a decimal.

/BT=

n

I -

m--

Jt6

/im

-l 36

,lT= JA=

ln

lw

, -

,l I

=

IE =

,tfr = ,.mmT

=

/im =

lE

I - =

/81 Usea calculator witha f, neyto helpyoufindthesesquareroots.Enterthe numberfirst. Thenpressthe{ keyto seethesquareroot.

.lm

2

,tm

,IM= ,/M=

lw

Jm=

JM=

JW6 =

,lw6 =

./ffib=

,/mF6=

lw6q

= O1S2 by KeyGorlcubm Pross,lnc. Do nol duplbatewithoutpermlsslon,

Everypositivenumberhasa squareroot. Infact,it hastwosquareroots- onepositive andonenegative.3 and-3 arebothsquarerootsof 9 because

3"=1

a n d ( - 3 ) "= I

we usethesymbol is meantOVlI]91To avoidconfusion Howwillwe knowwhichnumber -{- for the negative squarerootandsave{- torthe positivesquareroot. squarerootof o." fi means"thepositive - {o teans "thenegative squarerootof o." Findeachsquareroot.

lm=-8

-reom=

JA -JE JT:

-.m 1m

-lr

--

J49

-J+oo=

-'4e0O =

.m=

-tT l

-

/81

=

tT

{ 2500

IT { 900

Negative numbers do nothavesquarerootsin ournumbersystem.Youcanseewhyif you be positive tryto finda squarerootof -9. lt'simpossible! A squarerootof -9 couldn't because thesquareof a positive numberis positive.lt couldn't be negative eitherbecause exist. thesquareof a negative number is alsopositive.Certainly it isn't0. {-9 justdoesn't It hasnomeaning.Sometimes we sayit is undefined. Crossouttheexpressions whichareundefined.

- t00

.m

./g

.,m

-,1T6

-,/m

,m

,lw

@19Cby KeyCurrlculum Pross,lnc. Oonotduplicate withoulpormissbn.

.o025

-.mi -2.2s 2.25 3

lrrationalNumbers squareroots. do nothaverational like7 and1000,whicharenotperfect squares, Numbers a line,butwecannotfindan integer, Theirsquarerootscanbe located ona number whosesquareis exactly equalto either fraction or a repeating or terminating decimal 7 or 1000.Wesaytheirsquarerootsareirrational. eachof these.Thensquarethenumberyouget. Usea calculator to compute

n= JEN givesyouis closebutnotexactlyequalto the As youcansee,thenumber thecalculator number foran irrational squareroot. lt is an approximation.Thedecimal expression goesonforeverwithoutrepeating. fora actually Whenwewantto usethe number calculation thousandth, or measurement we roundit offto the nearest tenth,hundredth, "is etc. Weusethesymbol= instead equalto." of = to mean approximately cc 2.6

(tothenearest tenth)

:'l, 2.65 (tothenearest hundredth) p 2.6+ 6 (tothenearest thousandth) Usea calculator to approximate eachsquarerootto the nearestthousandth.

Jj

n,

n N m H lfr N,

m*

,/B o

Jm?ffi^' JmN lmx

J$x

m/fffffff N

JMF

-{Wp

,/86

A,

,re3

Conveileachfraction itssquarerootto to a decimal.Thenusea calculator to approximate the nearesthundredth.

tt, N

n

Ji

=

le 4

@1992by Key Curiculum Pross, Inc. Do not duplicate without p€rmission.

SolvingQuadraticEquationsWithoutFactoring howto solvean equation likex2= 16 byfactoring? Doyouremember x" - 16 = 0 Firstwesubtract 16to geta 0 ontheright. Thenwefactorthepolynomiat ontheleft.

(t - 4) (x * 4 ) = O

FinallyweusetheZeroProductRute. r-4=0

or l+4

= 0

r=4 ort=-4 wecannotfactorx2- 15. Thismethod wouldnotworkfortheequation x2= 15 because of squareroottellsusthatr But 12= 15 is veryeasyto solveanywaybecause thedefinition mustbethesquarerootof 15. Sincetherearetwosquarerootsof 15,we canwrite

x=-ffi-

x=*E

We willwriteirrational solutionswiththe radicalsignexceptwhenwe wanta decimal approximation. Solveeachequationwithoutfactoring.

x 2= l O 5

xu=38

nt=5

x2= t{00

a z= 2 . 6

x " -1 =O

x 2 -2 0 = O

X=rms-or t=-@ x2 = 283

. . _ + r t ^2+ + ?

x2-q2 =O

r.2=42

x = ,l{7. or r =-tlfi 3x" -- 15

@199by KeyCurr'rculum Prsss,Inc. Do notduplicate withoutp€rmi$bn.

2x" = IOO

-5x' = -85

5

Solveeachequation withoutfactoring.Approximate thesolution to the nearesthundredth if it is nota rational number.

3x"= 2l

lOx" = 5t+O

l2x"= 5

2(2=7

x = {7 or x=-JT * 2.65 or x*'2.65

^

7x' = 56

3x' = 20

6

2 x ' - 2 0 =0

1s

5x'

7x' = 7.7

5x'-3=O

320-8x'=O

56-8x'=0

3x'-?+2=0

fOx'

25 = {x"

tlx"

7 =O

Ol92 by KeyCunlculum Press,lnc. Do nd dupllcatewilhoutp€rmiseion.

if it is nota Solveeachequation.Approximate the solutionto the nearesthundredth rationalnumber.

( x - 5 ) " =to ? ( - 5 = / i d o r x - 5 =-1m ,\ = 5 + fid of ? (= 5 - . m x P ' g $ + 3 , 1 bo r ? ( ^ ,5 - 3 . 1 6 X x 8.16

o? t( p l.8t+

k * 3)' = 50

k - 1 ) ' =?

(x+ q'= ?O

@199by KeyCurriculum Pro6s,Inc. Do notduplicate wilhoutpermlsslon.

(x- lI = 12

k + 2)'= 36

k-qf = ?O

k + 2)" = 8l

7

The PythagoreanTheorem with right Right triangles(triangles angles)havea veryspecialproperty. Theareasof squaresbuilton the two shortersidesadd up to the areaof the squarebuilton the longestside. The longestsidein everyrighttriangleis the onlyonewhichis nota sideof the right angle. lt is calledthe hypotenuse. In symbols,we write

ii i i i ii :i t ;

ll a, b andc aresidesof a righttriangle then andc is the hypotenuse, a2+b2=c2.

Thisis calledthe Pythagorean Pythagoras.You Theoremafterthe Greekmathematician, cancheckthatit worksforthe righttriangleabove.(16+ 9 = 25) The Pythagorean Theoremenablesus to findthe lengthof the thirdsideof a righttriangle whenwe knowthe lengthsof the othertwosides.All we haveto do is usethe formula. Findthe lengthof thethirdsideof eachtriangle.lf the lengthis nota rationalnumber, roundit off to the nearesthundredth. o.12

b=s'

-a't t

q'o5

bslz 5t + l1t = ga 25+l$rlr Ct 159 = c1

35

c e 13or Fsfil /\./?---

613

.N 8

-do.rn't 'rot\o"

W

Pross,Inc. @1S2by KeyCurriculum Do notduplbalewithoutp€rmission.

Makea sketchfor eachproblem.Labelthe lengthsthatare known.Thenusethe Theoremto solvethe problem. Pythagorean A baseballdiamondis a square90 feeton The bottomof a 25-footladderis placed eachside. Howlongis a throwfromhome 7 feetfroma wall. Howfar up thewallwill plateto secondbase(tothe nearestfoot)? the ladderreach? I I

I cl I I I

home

A pathleadsacrossthe parkfromone corner.Thepark cornerto theopposite is 150meterswideand200meterslong. Howfarwouldyouwalkif youtookthe pathinstead of walkingaround? Howmanymeterswouldyousave?

o19e by KeyCurdculum Pr6s. Inc. Do na dupliate withoutpemissbn.

by nailing Samwantsto bracea bookcase a stripof woodfromthe lowerleftcornerto is theupperrightcorner.Thebookcase 1 meterwideand2 metershigh.How longshouldthebracebe (tothe nearest of a meter)? hundredth

FormulasInvolvingSquareRoots of Manyusefulformulas contain squareroots.Theformulat = .25{A givesthenumber if therewerenoair (r)it wouldtakean objectto falld feetafterbeingdropped seconds to fall100feetwe resistance. lf wewantto knowhowlongit wouldtakeforsomething 100ford. couldsubstitute

t = . 2 5 . ' r c 0 = . 2 5 ( l O ) = 2 .2.5 5

se"onJs

to the nearesttenth. Usethisformulato answereachquestion.Approximate Howlongdoesit takea wrench,dropped froma 3O-footroof,to reachthe ground?

how lf therewereno air resistance, longwouldit takean objectto fallone mile(5280feet)?

TheWashington Monument is 555feettall. lf youholdoutyourhandanddropa penny,abouthowlongdoesit taketo hit In howmanysecondswouldan object droppedfromthe top reachthe ground? the floor?

Theformulas = 5.45fi can be usedto findthe speedat whichan objectdroppedfroma heightof d feetwillhitthe ground.Thespeed(s)is in milesperhour. Howfastwillan objectdroppedfromthe Monument top of the Washington be goingwhenit reachesthe ground?

10

withwhatspeed Withno air resistance, wouldan objectdroppedfromone mileup hitthe ground?

Press,Inc. @1992by KeyCurriculum Do notduolicatswithoutoermission

on whereyouare. Ontheoceanor onflat Howfaryoucanseeon a cleardaydepends landthe distance(d)to the horizonin milescanbefoundby usingtheformula Thevariableh standsfortheheightof youreyeabovethe land(infeet). d,=@. Usetheformulato answereachquestion. Measure or estimate the heightof your owneyeto the nearesttenthof a foot. Howfar outcanyouseefroma beach?

Howfarawaywouldthe horizonbe if you werestandingontop of an 80-foottower? (Remember to addtheheightof youreye to thetowe/sheight.)

Howfar is the horizonfromthetop of a 2000-footmountain?

Howfar couldyou see froma planeflying threemilesup?

Whentheareaof a circleis known,thediameter(d)canbefoundby usingtheformula

d=2\m ptzzacovercabout80 A smafl-size squareinches.Whatis itsdiameter (tothe nearesttenthof an inch)?

ole bt K.t Crrrlculrrn PrB.' lrrc. Dond duCbabrilhoulpflmb.bn.

ol apizza Whatwouldbethediameter twiceas large(twiceas muchto eat)?

11

SquareRootsof Monomials Tofindthesquarerootof a number wecanthinkof thenumber astheareaof a squareand thesquarerootasthelengthof a side.

Wt = 26 Whatwouldbe meantby ,lxz c Thinkof 12as the areaof a square.

Thelengthof a sidemustber, so

17 = x We can findthe squarerootof any expression thatwe can writeas a square. We will assumethatall of ourvariablesstandfor positivenumbers. Findeachsquarerootby writingthe expression as a square.

J""=

=xt

,l'^"=

,lm= ( 5 n l00x'

=5n

,TT Bla+

r"y* a6b. 12

@1992 by KsyCuniculum Prsss,Inc Do notduplicalewithoutp€rmisEion.

The Productand QuotientRules Maybeyou sawthata simplewayto findthe squarerootof a monomiallikex2yatsto find the squarerootof eachfactorseparately.We can do thiswheneverwe havethe square rootof a productbecauseof the Product Rule.

{ob = fi./b Ruleto findeachsquareroot. UsetheProduct

JxT = vG*b; =

1 4 4n '

4'""t = rtO0c' 9m{'3bn"

By writingthe ProductRulethe otherwayaroundwe seethatwe can use it to multiply squarerootsas wellas to findthem.

..6fi = {ab leaveit as if youcan. Otherwise or a polynomial Multiply.Writeyouransweras an integer a radicalexpression.

lt/s =.F

{ T J n = J i f , = 5E.m =

Jr'J^' =

/3" J3o =

JTvq =

Jtvth,=

fiJ* =

,tb,[n=

J;lq =

Jtr,[6=

@199by KsycurrbulumPrsss,Inc. Do notduplicatswithoutp€rmission.

-

t-

13

Everyquotient canbewrittenas a product, soyoumighthaveguessed thatthereis alsoa QuotientRulefor radicals.

^t; {a ,^[a \a=Gano*= Usethe QuoJient Ruleto findeachsquareroot. I-

@-3; o--I-

lT6"r-T . lz s . . 5 =

, -

lr

@cl

=

tt6

t -

I

\^-v\/

lqq

=

11

E

t-

E /Et -

14

m

ltE

-

, lz e q

tzl

nt} + T o

=

-

Z

ffi lffi

IT = Jtoo

n -l

l

I

ttr[

ltoo ,

=

=

Usethe QuotientRuleto divide.

/E = -15

@

=lT=3

= 1 7 ' t _ r -- 1 '64 { f r r - J ? 3 ,TT J6

ln

-m

@

,m

F

hs E 14

.tT

,TT ,l-qq @192 by Key Curiculum Press, Inc. Do not duplicats wilhout p€rmission.

Doyouthinktheremightbesimilarrulesforaddingandsubtracting squareroots? youthinktheserules Whatwouldtheylooklike?Writeyourideashere,andcheckwhether aretrueor nottrue. Addition Rule:

trueI

nottrueI

Subtraction Rule:

trueI

nottruef,

youusedin yourrulesto seewhether Substitute somenumbers forthevariables therules seemforworkforallnumbers.

Wecaneasilyfindapproximations forsumsanddifferences of squarerootsby using tne{ keyon a calculator. youmustusethe memoryto storeone Onsomecalculators squarerootwhileyoucompute theother.lt is easiest to findthesecondsquarerootfirst. Usea calculator to findan approximation hundredth. to thenearest

lB

{6x

.E + {j*

/i6 + Jyz =

lTs*n-* J5+b m * /63 +m * O19P by KeyCur&ulumPr6s, Inc. Do notduplicalowtthoutpemissbn.

,E-mp ,166+m -.fi-n N J6rez {m + Is.3 N 15

As you probablydiscovered , ^{o* b is notalwaysequalto rio * {b sn6 {o - 6 may notequal{o - {b. We can'talwayscombinesquarerootsby additionor subtraction. Sometimes we cando it by usingthe ProductandQuotientRulesto simplifythe square roots. Simplifying the squarerootof a wholenumbermeansfindingan equivalent expression with the smallestpossiblenumberunderthe radicalsign. Thisis calledwritingthe numberin simplest radical form. To writea squarerootin simplestradicalformwe firstfindthe largestpossiblesquare factorof the numberunderthe radicalsign. Thenwe usethe ProductRule. Lookat thisexample.

Rewriteeachsquarerootin simplestradicalform.

ffi =n,E =38

lf a numberis largeit is sometimes helpfulto finditsprimefactors.

I

T

J@.=

m=

JiG =

Jm=

lmE=

Jm=

JT6=

16

@1992 by KsyCurriculum Press,Inc. Do notduplicate withoulpsndssion.

as like Nowwe havea wglto addsomesquareroots- thosethatcanbeexpressed becausethey canbewrittenas 2{3 and5{3. terms.fi2 and{75 canbecombined Property them. to combine Theseareliketerms,as2x and5r are. Weusethe Distributive

,ln

+

2,15+ = 7,13 Simplifyandcombineliketerms.

{8

+

,/tr

ffiJn

.m - /40

JN

.N

,// + JE *,/n

@19eby KsyCurftulumP€ss, Inc. Do not duplicatewilhoutpermlssbn.

,t2 +

+

JN

JN

J54

Jn + ,@o

/iq4

,@

hz+.'&

./m + Jtr

J5

,E

J@

.m

reO6+ ./To

17

Rewriteeachsquarerootin simplestradicalform. Combineliketermsif possible.

2"ln + JE

q,lfr Jry

3,/28* Jm

g+Jry+JT

zJzoo- .'6--0

sJn + 7Jn

2 A + + J q + 8 { 6 q,/n-31@ +rm

.,e + 3.,6 ,|EJZ + sJqJZ 517 + 3'2,12

5,lZ + 6JZ

nJZ

,R,*m

JM-JW

J*

+./G

01902by KsyCurriculum Press,Inc. withoutpermission. Do nol duplicate

An expression radicalformif (1)no numberundera thatcontains a radicalis in simplest radicalsignhasa factorthatis a perfectsquare,(2)no fractionis undera radicalsign, and(3)no radicalis in a denominator. is in simplestradicalform,because the numberunderthe radicalsigndoesn't

zG havea perfectsquarefactor,no fractionis underthe radicalsign,andno radical 5 is in thedenominator.

E5

radicalform,because the numberunderthe radicalhas4 as a is nofin simplest factor,and4 is a perfectsquare.

IT is nofin simplestradicalform,becausea fractionis underthe radicalsign. .JEi 2

s?tris nofin simplestradicalform,becausea radicalsignis in thedenominator. whicharein slmplestradicalform. Circlethe numbers

3

to

E

T

&

€

L

2

Ev'

n

E

E

Z

T{r

J6 3

anddenominator. Circlethe resultif Usethe QuotientRule. Thensimplifythe numerator it is in simplestradicalform.

lz 25

3

t

t+

27

9

t:rt

I

z

I IL

25

2+

+8

+1

5

@1992by KsyCurriculum Prsss,Inc. Do notduplicats withoutpsmission.

19

Thereis a simpletrickwe can use to changea fractionwitha radicalin the denominator intoonethat is in simplestradicalform. We multiplyby 1. Lookat this example.

7

J7

7JZ _ i l T {q l7{2

7JZ 2

Doyouseewhymultiplying./Z by itselfgetsridof the squarerootsign? Rewriteeach fractionin simplest radicalform.

3 {=t

3Jr = 3{F F

7

{5{r {2s

{z

I

4 {r_ qE {7a- {+

:

,13

---

z

re, = zJiE 2 I

3

.''r3 I

20

.3

G

zfi

I il2

J5€ : = 1f3 €

JZ

m

tr {E

& .6

?E

{z

Ol9 byK€yCunid*nnPrsss,hc, Do nol d.p{catg wtlhod poflr*rdon.

Usethe QuotientRule. Writethe answerin simplestradicalform.

3l 32

o19e by KeyCurrlculum Pr$s, lnc. Oo nol duplh.atewlihorrlpermlsslon,

Gompletingthe Square Eartierwe solvedsomequadraticequationsby findingthe squarerootof eachside. We cansolveany quadraticequationwhichhasrealnumbersolutionsthe sameway. equationwhichhasthe squareof a binomial We just haveto be ableto lind an equivalent on onesideanda numberon the other. whenwe squarethe binomialr + 5. Lookat whathappens x

+

5

x2

5x

5r

25

(x*5)' = x" + 5x + 5x + 25 = ^2*lOx+25

+ 5

t

l

-

2times5

\ r

l

5qucred

of r is always2a andthe constant Whenwe squareanybinomial,x,+ a,, the coefficient workto findout whatto addto a termis alwayso2. Knowingthis,we can do somedetective binomial to makeit a square. oo oo

l{ is 2 timcs7. 7t rs {9, so I shouldcdd 49.

xz + ltlt + to makeit a square. Decide whatmustbeaddedto eachexpression

x' + ZOx

6x Add:

l0x

Add:

Add:

l8x

x z- l 6 x Add:

:72 24x

Bx Add:

30x

x2-2x Add:

22

12x

rt" + 4x

Add: Press,Inc. @192by KsyCuniculum p€rmission. rvithout Do notduplicato

intothe To completethe squaremeansto adda numberwhichmakesan expression the bycompleting equation squareof a binomial.Hereis howwe cansolvea quadratic

square. Add ? to

, o o o o o ox z + 6 { r =

7*' t6

Xz + 6x + 1 = ( l * 3 ) " = l6 th" sguare. complete

x +

r'=4-, o,' x

x =

l o t

+ 3-3= -q-3

x =-7

Herearesomeforyouto try.

2t

l0x

O19Q by KeyCu?rlcuhrm Pr6s, Inc. Oo nol dupllcalesrlthoutpermissbn.

21

6x

Bx

-5

-15

23

the square.Writethe solutionsin simplestradicalform. Solveby completing

x'-4x=lO

x"

x 2 - 4 1 1+ t l = l O +

(t - 2)'= 14 N - 2*'=/i4t1, I -

4x = 6

tL\tfr

r = Z + f l + o rx = 2 - . 1 i 4 ; x ' +2 x = 6

xz - lOx

-l xz+ lTx =

^ z- f 4 x = - 4 7

xt+6x = 3

24

@1s2 by KsyCunbulumPrees,Inc. withoutpsrmis8bn. Do notduplicate

in simplestradicalform. the square.Writethesolutions Sotveby completing

r' + 4r - llttt= O+ll

N " - 2 x -5 = O

?(t+tlt = Il 7 6 24+r + 4 = l l + 4

(r * 2f = t5 ?(+|''=-in! or x*2'2=i/i5 N = - 2 * 1 1 f foi r * = ' 1 - " t r x"*8x*3=O

2 x " + 4 x =2 B 2 ^z+ U

3x'*l8x=48

2

-

lr+

5x'+40x=-55

@1992 by KeyCuriculumPross,Inc, Do notdupllcate wlthoutp€rmissbn.

xz- l0r * 18= O

2 x . - l / x * 1 4= O

25

The QuadraticFormula and equation) Bystartingwith as2+ bx + c = 0 (whichcanstandforanyquadratic forthesolutions. completing thesquare,we cangetformulas -b - .[bL

-b +.,[b2-4ac x= 2a

x=T

4a c

Thetwoformulas thesameexceptforthesignin themiddle.Weusually areexactly combine themintoonebywriting+ ('plusor minus')in frontof thesquarerootsign. -bt

b2- 4ac 2a

forthe last Thisis calledtheQuadraticFormula.Let'suseit to findthesolutions equation on page25. +l+ =O 2x"-lZx

in Firstwe rewrite theequation (ox2*bx+c=0). standardform

@^'+ @r t I a

b

* @ = O t c

rhen-:]::T:'iJ[""13hff'r = -:rtz\xrffii 2(2) Thenwesimplify theresult.

^ = tZ , Jffi

+ a =12rEd,

+ x =lLt\JZ

t 3

t

'(

t(

^=flrftJZ r

l

^ = 3 x.l7

26

olg! by l(6t Or||lculrm Prs!, Inc. Oo not dtpllcde dthouf perni$bn.

Solveeachequationby writingit in standardformandthenusingthe QuadraticFormula.

N2+2x-3=O

3x.+lQx*5=O

2 x "- 7 t * 6 = O

2x' + 1x - 5 = 0

5x^+2x-3=0

^^z-5X-2=Q

x'* 2x* l=O

6x" 8r*l=0

x z- 3 x + 2 = 0

(D"'*@x+@= O

@19eby KeyCurriculum Press,lnc. 0o notdupli:alowithoutpermi66ion.

27

Writeeachequationin standardformandsolveby usingthe QuadraticFormula.

x z+ 5 x = 0

x z+ 2 x = 0

x"-6r=O

(Dr'* @n* @=o c b e

2x"- Bx=0

? ( 1- l f = O

51" = t{0x

-3x" = f5x

2x' - 7x = 0

*@=O @* 0t'* a b s

28

01SIz by KsyCunbulumPree, Inc. Do nd dupficdswnhoutpemlsslon.

youranswers to the Formula.Approximate Solveeachequation by usingtheQuadratic nearesthundredth. , x z+ 8 x + 6 = 0

x2+6x*4=O

(Drtt@t*@=o c

b

lr

c -

€trl5'j-t+.f.6

T

r- -8 l, I con Bse rny

cclculotorhcre,

r=-Etam

T

so f won't

o

simphfy.

t=@"o,

l=-8--ffi

t nr -o.Er+

7r* -7.15

z

x'-2x'30=0

o19e by KoyCurrbubmPre6s,Inc, Do not drJplbal€withoutp€m*ssbn.

z

2 x , - l 0 x + l =l O

29

Usingthe QuadraticFormulato SolveProblems to the nearesthundredth. Findthe lengthsof the sidesof eachrectangle 3x-1

r+3 A=80

A= 20

Chcck: A s 7.S7(to.S7)

r(x + 3) = 60 x 2 + 3 x = 8 0 A:: 8o.ol q,z+ 3x +-60 = O

1=-3sm

z r=ffi--3!Jffi T

T

t, * 7.57 or

,'[^=;l

x+7

r+10

30

r-1

A=500

Press,Inc. @1s2 by KeyCufriculum withoutpsrmission. oo notduplicate

Findthe lengthsof the sidesof eachrighttriangle.

"ffi

@ x+1

r z + ( r * l ) t = ttj

xt + xt+ 2x * l-2 1i-*e Z x z + 2 x + - { 8 =o

@19eby KsyGurriculum Pre66,Inc. Oonol duplicale wilhoutp€rmissbn.

31

Graphs of Functions pairupnumbers. Functions Foreverynumber r thatwechoose, thefunction f gives numbercalledflr). Graphing usanother is easy.Wejustnamethe horizontal a function pairsof numbers axisr, thevertical in thefunction. axisflr) andgraphtheordered Makea tablefor eachfunctionanddrawa graphof thefunction. The SquaringFunction

{rxl = Nz ----t---i----i

j

-2

:L

o

I 2 3

frxt

---t.--.+---.1----t----l---

The AbsoluteValue Function t f ( * ) = tl*l

f,nt

iji*ii-i-i-8 i t :..t....i....i....i....i..:.

i i i i -.-1.--|. "-t.-.t---f

i

i --t--

/

i - * i i - i , - : - i - -t+

"r-'r--i'-'i-'i'-l--i-l--

-i-t--i-i-l'--l--i-i-

i i i i i i i i -t--i--i-'i-i"-t-'i-i'-

- * ii i i t i r :

-2

o 2 4

E

A linearfunctionis a function whosegraphis a straight linewhichis notvertical.Therule fora linearfunction lookslikea linearequation: we fl*) = rrlx+ 6. Tographa linearfunction canjustgraphthelinearequation ! = mx + b. Grapheachlinearfunctionwithoutmakinga table.(Usetheslopeandthey-intercept.)

0192 by KeyCuniculum Press,Inc. Do notdupli:atewilhoutp€rmission.

QuadraticFunctions equation: Therulefora quadraticfunctionlookslikea quadratic fl*) = o*2+ bx + c. Thegraphof a quadratic function is a curvecalleda parabola.Wecangrapha quadratic points. function by plotting function.Thendrawitsgraph. Complete thetableforeachquadratic

f ( x )= x " ' 2 x - 3

ftxl= x'-4x

I I

2

I I

5

frr) =-2x"+ l0

f { * 1= x " * B x + 7

-8 -7 -6 -5

-4 -3 -2 ts--

@19@by KeyCurriculum Pross,Inc. Do nol duplicatowhhoulpennisGbn.

33

Didyoufindit hardto pickr-valueswhichgaveyoua goodpictureof thelastgraphon page33? Thiswouldhavebeeneasierif youhadknownwherethe graphcrossedthe r-axis- in otherwords,wherefl*\ --0. Wecanfindther-valuesat thesepointsbysolving theequation x2+ 8r + 7 =0. ffx)- x"*8x+7 0 = xt+$x+7 0 = k*7)(t+l) x+7=O or x+l=O x n-7 or x =-l ther-axisat (-7,0)and(-1,0).Wewouldgeta goodpicture Thismeansthegraphcrosses in ourtable. numbers between-7 and-1 astheotherr-values ofthegraphbychoosing function crosses ther-axis. Findthepointswherethegraphof eachquadratic Approximate ther-valuesif youneedto. Thenfindat leastthreeotherpointsonthecurve anddrawthegraph.

ffxt = ^Az-l6x * 60

f ( x ) =x 2 - 6 x + 4

34

o1$2 by Kdy Cunlculum Pf6q Inc Do nol duplbale wlthout pemissbn.

Written Work Dotheseproblems on somecleanpaper.Labeleachpageof yourworkwith yourname,yourclass,thedateandthebooknumber.Alsonumbereachproblem. Keepthiswrittenworkinsideyourbook,andturnit in withyourbookwhenyouarefinished. Please doa neatjob. 1. Explain why1.414285= f . |

2. Copyeachexpression whichstandsfor a rationalnumberandtellwhatrational number it equals.

./25 ./20 .80.tr .6.tr I V

G -G

-fi

./0

./T

fi

fi

fid

I

gV

EG

G

f

i

q

.Itoo ./m00 .f0000

3. Simplifyeachexpressron.

{50

L

{5

li

I

'

1;

\f

21112fi

.Et

.Ee,

'/u *./600

4. Approximate to the nearesthundredth.

^t4z

3{5

4-fi0

G*€

5. Solveeachequation.lf the solutionsare not rationalnumbers,givethemin simplest radicalform. x 2= 2 6

(x-2lz =19

3*2- 1 = 11

x2+2x-8=0

x2+8x-2=0

2x2-5r+3=0

6. Solvethe equationx2- S3= 0 in threedifferentways. 7. Findthe lengthsof the sidesof eachtriangleor rectangleto the nearesthundredth. 2r 7

252

.r+5 l

6

0

l

11

8. Tryto solvex2- 4x +7 = 0 by usingtheQuadratic Formula.Whathappens? -1, 9. Graphthequadratic function fli) = 12- 4x+ 7,using 0, 1,2,3,4and5 asthe .r-values in yourtable.Whatdoesthe resulthaveto dowiththe resultin Problem 8? O19(Pby KeyCurrbulumPr6s, Inc. Oo nol dupli:atowithoutp€rmission.

35

rI PracticeTest Findeachsquarerootif possible.lf thesquarerootis nota rational givean number, approximation to thenearest tenth.lf it is undefined, crossit out.

JB fi00 ffi

-./a 'rlt+ fr

{16

Multiply or divide.Writeyouransweras an integer, numberor in simplest radical a rational form.

J7J3 =

,rffi

:

,lnJn=

J30

-l@

3JTJ5=

"/5

Simplify eachexpression.

Jffi=

IT

"l 1 IT =

.tf^ =

lE

,tfi, =

ffi+Je=

q,=

ZJM Jffi =

Usetheformulat = .25{d to answereachquestion. Howmanyseconds willit takea rockkicked I Howmanyseconds willit takea rivetthat

off a 100-footcliffto hitthe waterbelow?

36

fallsfrpma plane5000feetup to reachthe ground?

OlSXlzby K€t Curlculrm Prsss,Inc. Do nd duplhalswlthoutpemiesbn.

tenthof a unit. of thesidesof eachfigureto thenearest Findthelengths

A=30

Sotveeachequation.lf the solutionis nota rationalnumber,writeit in simplestform.

5x' = 35

k - 3 ) ' =l ?

^^,2 - lQx * 18 = Q

Graphthe quadraticfunction.

fr"t = xz-?(- 6

ole by KeyCufrhubm Pr6s, Inc. 0o not duplkatewithoutpermissbn.

x z+ B r = O

Bmk t: Operotionson Intqgers : Bosk h Vartobles, Terms and Expressions Book 3: Eguctions Book 4: Polynomials Sook 5: Rotfonal Alurnbers Book 6: Malttp$tng and Dtvtdtng Rationql Expressions Sook 7: Addingand Subtrasttng Rotional Expressions Book 8: Graphs Book 9: Systemsof F4uatlons Book rO: $qucre Rmf6 cnd Q$cdratic Equations Ansurcrssnd N"otesfor Books l-4 Answercand Notes for Books 5-7 fuiswers and Notes for Bmks 8-10

Kry to Fractlonso Keyto Decimalso Key to Percentso Keyto {3€omctqf KsYto Measrrenrenf; Kry to Metrtc ll{easuremenf

CURRICULUMPRESS KEY ^ \ w Education Inno.tators in Mathematics

rsBN 1-55955-010-3

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