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
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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.
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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
,tm
,IM=
,/M=
lw
Jm=
JM=
JW6 =
,lw6 =
./ffib=
,/mF6=
lw6q
2
=
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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
.o025
.,m
-,1T6
-,/m
,m
,lw
-.mi
-2.2s
2.25
3
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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
A,
,/86
,re3
Conveileachfraction
itssquarerootto
to a decimal.Thenusea calculator
to approximate
the nearesthundredth.
tt, N
n
=
Ji
le
4
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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
2x" = IOO
-5x' = -85
5
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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
2 x ' - 2 0 =0
1s
5x'
7x' = 7.7
5x'-3=O
320-8x'=O
56-8x'=0
6
3x'-?+2=0
fOx'
25 = {x"
tlx"
7 =O
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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
(x- lI = 12
o? t( p l.8t+
k * 3)' = 50
k + 2)'= 36
k-qf = ?O
k - 1 ) ' =?
k + 2)" = 8l
(x+ q'= ?O
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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
-do.rn't 'rot\o"
W
8
Pross,Inc.
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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?
by nailing
Samwantsto bracea bookcase
a stripof woodfromthe lowerleftcornerto
is
theupperrightcorner.Thebookcase
1 meterwideand2 metershigh.How
longshouldthebracebe (tothe nearest
of a meter)?
hundredth
o19e by KeyCurdculum
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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?
withwhatspeed
Withno air resistance,
wouldan objectdroppedfromone mileup
hitthe ground?
10
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@1992by KeyCurriculum
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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)?
ol apizza
Whatwouldbethediameter
twiceas large(twiceas muchto eat)?
11
ole bt K.t Crrrlculrrn
PrB.' lrrc.
Dond duCbabrilhoulpflmb.bn.
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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
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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=
-
t-
13
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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
.tT
,TT
,l-qq
14
@192 by Key Curiculum Press, Inc.
Do not duplicats wilhout p€rmission.
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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 *
,E-mp
,166+m
-.fi-n N
J6rez
{m + Is.3 N
15
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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=
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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
,t2 +
+
JN
JN
J54
Jn + ,@o
/iq4
,@
hz+.'&
./m + Jtr
J5
,E
J@
.m
reO6+ ./To
17
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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.
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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
19
@1992by KsyCurriculum
Prsss,Inc.
Do notduplicats
withoutpsmission.
www.pdfgrip.com
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
.3
G
zfi
I
il2
J5€
:
=
1f3 €
JZ
m
tr
{E
&
.6
?E
{z
20
Ol9
byK€yCunid*nnPrsss,hc,
Do nol d.p{catg wtlhod poflr*rdon.
www.pdfgrip.com
Usethe QuotientRule. Writethe answerin simplestradicalform.
3l
32
o19e by KeyCurrlculum
Pr$s, lnc.
Oo nol duplh.atewlihorrlpermlsslon,
www.pdfgrip.com
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
12x
rt" + 4x
Add:
Add:
l8x
x z- l 6 x
Add:
:72 24x
Bx
Add:
30x
x2-2x
Add:
22
Add:
Press,Inc.
@192by KsyCuniculum
p€rmission.
rvithout
Do notduplicato
www.pdfgrip.com
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
21
6x
Bx
l0x
-5
-15
23
O19Q by KeyCu?rlcuhrm
Pr6s, Inc.
Oo nol dupllcalesrlthoutpermissbn.
www.pdfgrip.com
