Completing the square (video) | Khan Academy

To complete the square, first, you want to get the constant (c) on one side of the equation, and the variable(s ... Ifyou'reseeingthismessage,itmeanswe'rehavingtroubleloadingexternalresourcesonourwebsite. Ifyou'rebehindawebfilter,pleasemakesurethatthedomains*.kastatic.organd*.kasandbox.orgareunblocked. CoursesSearchDonateLoginSignupSearchforcourses,skills,andvideosMaincontentMathAlgebra1Quadraticfunctions&equationsCompletingthesquareintroCompletingthesquareintroCompletingthesquareThisisthecurrentlyselecteditem.Workedexample:Completingthesquare(intro)Practice:Completingthesquare(intro)Workedexample:RewritingexpressionsbycompletingthesquareWorkedexample:Rewriting&solvingequationsbycompletingthesquarePractice:Completingthesquare(intermediate)NextlessonMoreoncompletingthesquareCurrenttime:0:00Totalduration:14:060energypointsMath·Algebra1·Quadraticfunctions&equations·CompletingthesquareintroCompletingthe squareGoogleClassroomFacebookTwitterEmailCompletingthesquareintroCompletingthesquareThisisthecurrentlyselecteditem.Workedexample:Completingthesquare(intro)Practice:Completingthesquare(intro)Workedexample:RewritingexpressionsbycompletingthesquareWorkedexample:Rewriting&solvingequationsbycompletingthesquarePractice:Completingthesquare(intermediate)NextlessonMoreoncompletingthesquareVideotranscriptInthisvideo,I'mgoingtoshow youatechniquecalledcompletingthesquare.Andwhat'sneataboutthisis thatthiswillworkforanyquadraticequation,andit's actuallythebasisforthequadraticformula.Andinthenextvideoorthe videoafterthatI'llprovethequadraticformulausing completingthesquare.Butbeforewedothat,we needtounderstandevenwhatit'sallabout.Anditreallyjustbuildsoff ofwhatwedidinthelastvideo,wherewesolved quadraticsusingperfectsquares.Solet'ssayIhavethe quadraticequationxsquaredminus4xisequalto5.AndIputthisbigspace hereforareason.Inthelastvideo,wesaw thatthesecanbeprettystraightforwardtosolveif theleft-handsideisaperfectsquare.Yousee,completingthesquare isallaboutmakingthequadraticequationintoa perfectsquare,engineeringit,addingandsubtractingfrom bothsidessoitbecomesaperfectsquare.Sohowcanwedothat?Well,inorderforthis left-handsidetobeaperfectsquare,therehastobe somenumberhere.Therehastobesomenumberhere thatifIhavemynumbersquaredIgetthatnumber,and thenifIhavetwotimesmynumberIgetnegative4.Rememberthat,andI thinkit'llbecomeclearwithafewexamples.Iwantxsquaredminus4xplus somethingtobeequaltoxminusasquared.Wedon'tknowwhata isjustyet,butweknowacoupleofthings.WhenIsquarethings--sothis isgoingtobexsquaredminus2aplusasquared.Soifyoulookatthispattern righthere,thathastobe--sorry,xsquaredminus2ax-- thisrightherehastobe2ax.Andthisrightherewould havetobeasquared.Sothisnumber,aisgoingto behalfofnegative4,ahastobenegative2,right?Because2timesaisgoing tobenegative4.aisnegative2,andifais negative2,whatisasquared?Well,thenasquaredisgoing tobepositive4.Andthismightlookall complicatedtoyourightnow,butI'mshowingyou therationale.Youliterallyjustlookatthis coefficientrighthere,andyousay,OK,wellwhat's halfofthatcoefficient?Well,halfofthatcoefficient isnegative2.Sowecouldsayaisequalto negative2--sameideathere--andthenyousquareit.Yousquarea,you getpositive4.Soweaddpositive4here.Adda4.Now,fromtheveryfirst equationweeverdid,youshouldknowthatyoucannever dosomethingtojustonesideoftheequation.Youcan'tadd4tojustone sideoftheequation.Ifxsquaredminus4xwasequal to5,thenwhenIadd4it'snotgoingtobe equalto5anymore.It'sgoingtobeequal to5plus4.Weadded4ontheleft-handside becausewewantedthistobeaperfectsquare.Butifyouaddsomethingtothe left-handside,you'vegottoaddittotheright-hand side.Andnow,we'vegottenourselves toaproblemthat'sjustliketheproblemswe didinthelastvideo.Whatisthisleft-handside?Letmerewritethe wholething.Wehavexsquaredminus4x plus4isequalto9now.Allwedidisadd4toboth sidesoftheequation.Butweadded4onpurposeso thatthisleft-handsidebecomesaperfectsquare.Nowwhatisthis?WhatnumberwhenImultiplyit byitselfisequalto4andwhenIaddittoitselfI'm equaltonegative2?Well,wealreadyanswered thatquestion.It'snegative2.Sowegetxminus2times xminus2isequalto9.Orwecouldhaveskippedthis stepandwrittenxminus2squaredisequalto9.Andthenyoutakethesquare rootofbothsides,yougetxminus2isequalto plusorminus3.Add2tobothsides,yougetx isequalto2plusorminus3.Thattellsusthatxcouldbe equalto2plus3,whichis5.Orxcouldbeequalto2minus 3,whichisnegative1.Andwearedone.NowIwanttobeveryclear.Youcouldhavedonethiswithout completingthesquare.Wecould'vestartedoff withxsquaredminus4xisequalto5.Wecouldhavesubtracted5from bothsidesandgottenxsquaredminus4xminus 5isequalto0.Andyoucouldsay,hey,ifI haveanegative5timesapositive1,thentheirproduct isnegative5andtheirsumisnegative4.SoIcouldsaythisisx minus5timesxplus1isequalto0.Andthenwewouldsaythatxis equalto5orxisequaltonegative1.Andinthiscase,thisactually probablywouldhavebeenafasterwayto dotheproblem.Buttheneatthingaboutthe completingthesquareisitwillalwayswork.It'llalwaysworknomatterwhat thecoefficientsareornomatterhowcrazy theproblemis.Andletmeproveittoyou.Let'sdoonethattraditionally wouldhavebeenaprettypainfulproblemif wejusttriedtodoitbyfactoring,especiallyifwe diditusinggroupingorsomethinglikethat.Let'ssaywehad10xsquared minus30xminus8isequalto0.Now,rightfromtheget-go,you couldsay,heylook,wecouldmaybedivide bothsidesby2.Thatdoessimplify alittlebit.Let'sdividebothsidesby2.Soifyoudivideeverything by2,whatdoyouget?Weget5xsquaredminus15x minus4isequalto0.Butonceagain,nowwehavethis crazy5infrontofthiscoefficentandwewouldhaveto solveitbygroupingwhichisareasonablypainful process.Butwecannowgostraightto completingthesquare,andtodothatI'mnowgoingtodivide by5togeta1leadingcoefficienthere.Andyou'regoingtoseewhythis isdifferentthanwhatwe'vetraditionallydone.SoifIdividethiswholething by5,Icouldhavejustdividedby10fromtheget-go butIwantedtogotothisthestepfirstjusttoshow youthatthisreallydidn'tgiveusmuch.Let'sdivideeverythingby5.Soifyoudivideeverythingby 5,yougetxsquaredminus3xminus4/5isequalto0.So,youmightsay,hey,whydid weeverdothatfactoringbygrouping?Ifwecanjustalwaysdivideby thisleadingcoefficient,wecangetridofthat.Wecanalwaysturnthisintoa1 oranegative1ifwedividebytherightnumber.Butnotice,bydoingthatwe gotthiscrazy4/5here.Sothisissuperhardtodo justusingfactoring.You'dhavetosay,whattwo numberswhenItaketheproductisequalto negative4/5?It'safractionandwhenItake theirsum,isequaltonegative3?Thisisahardproblem withfactoring.Thisishardusingfactoring.So,thebestthingtodoisto usecompletingthesquare.Solet'sthinkalittlebit abouthowwecanturnthisintoaperfectsquare.WhatIliketodo--andyou'll seethisdonesomewaysandI'llshowyoubothwaysbecause you'llseeteachersdoitbothways--Iliketoget the4/5ontheotherside.Solet'sadd4/5toboth sidesofthisequation.Youdon'thavetodoitthis way,butIliketogetthe4/5outoftheway.Andthenwhatdoweget ifweadd4/5tobothsidesofthisequation?Theleft-handhandsideofthe equationjustbecomesxsquaredminus3x, no4/5there.I'mgoingtoleavealittle bitofspace.Andthat'sgoingto beequalto4/5.Now,justlikethelastproblem, wewanttoturnthisleft-handsideintotheperfect squareofabinomial.Howdowedothat?Well,wesay,well,whatnumber times2isequaltonegative3?Sosomenumbertimes 2isnegative3.Orweessentiallyjusttake negative3anddivideitby2,whichisnegative3/2.Andthenwesquare negative3/2.Sointheexample,we'll sayaisnegative3/2.Andifwesquarenegative 3/2,whatdoweget?Wegetpositive9/4.Ijusttookhalfofthis coefficient,squaredit,gotpositive9/4.Thewholepurposeofdoingthat istoturnthisleft-handsideintoaperfectsquare.Now,anythingyoudotooneside oftheequation,you'vegottodototheotherside.Soweaddeda9/4here,let's adda9/4overthere.Andwhatdoesour equationbecome?Wegetxsquaredminus3xplus 9/4isequalto--let'sseeifwecangetacommon denominator.So,4/5isthesame thingas16/20.Justmultiplythenumerator anddenominatorby4.Plusover20.9/4isthesamething ifyoumultiplythenumeratorby5as45/20.Andsowhatis16plus45?Yousee,thisiskindofgetting kindofhairy,butthat'sthefun,Iguess,ofcompletingthesquaresometimes.16plus45.Seethat's55,61.Sothisisequalto61/20.Soletmejustrewriteit.xsquaredminus3xplus 9/4isequalto61/20.Crazynumber.Nowthis,atleaston thelefthandside,isaperfectsquare.Thisisthesamethingas xminus3/2squared.Anditwasbydesign.Negative3/2timesnegative 3/2ispositive9/4.Negative3/2plusnegative3/2 isequaltonegative3.Sothissquaredis equalto61/20.Wecantakethesquarerootof bothsidesandwegetxminus3/2isequaltothepositive orthenegativesquarerootof61/20.Andnow,wecanadd3/2toboth sidesofthisequationandyougetxisequalto positive3/2plusorminusthesquarerootof61/20.Andthisisacrazynumberand it'shopefullyobviousyouwouldnothavebeenableto--at leastIwouldnothavebeenableto--gettothisnumber justbyfactoring.Andifyouwanttheiractual values,youcangetyourcalculatorout.Andthenletmeclear allofthis.And3/2--let'sdotheplus versionfirst.Sowewanttodo3dividedby2plusthe secondsquareroot.Wewanttopickthatlittle yellowsquareroot.Sothesquarerootof61divided by20,whichis3.24.Thiscrazy3.2464,I'll justwrite3.246.Sothisisapproximatelyequal to3.246,andthatwasjustthepositiveversion.Let'sdothesubtraction version.Sowecanactuallyputour entry--ifyoudosecondandthenentry,thatwewantthat littleyellowentry,that'swhyIpressedthe secondbutton.SoIpressenter,itputsin whatwejustput,wecanjustchangethepositiveorthe additiontoasubtractionandyougetnegative0.246.Soyougetnegative0.246.Andyoucanactuallyverify thatthesesatisfyouroriginalequation.Ouroriginalequation wasuphere.Letmejustverify foroneofthem.Sothesecondansweronyour graphingcalculatoristhelastansweryouuse.Soifyouuseavariableanswer, that'sthisnumberrighthere.SoifIhavemyanswersquared-- I'musinganswerrepresentsnegative0.24.Answersquaredminus3times answerminus4/5--4dividedby5--itequals--.Andthisjustalittle bitofexplanation.Thisdoesn'tstoretheentire number,itgoesuptosomelevelofprecision.Itstoressomenumber ofdigits.Sowhenitcalculateditusing thisstorednumberrighthere,itgot1times10to thenegative14.Sothatis0.0000.Sothat's13zeroes andthena1.Adecimal,then13 zeroesanda1.Sothisisprettymuch0.Oractually,ifyougotthe exactanswerrighthere,ifyouwentthroughaninfinite levelofprecisionhere,ormaybeifyoukeptitinthis radicalform,youwouldgetthatitisindeedequalto0.Sohopefullyyoufoundthat helpful,thiswholenotionofcompletingthesquare.Nowwe'regoingtoextendit totheactualquadraticformulathatwecanuse,we canessentiallyjustplugthingsintotosolveany quadraticequation.Workedexample:Completingthesquare(intro)UpNextWorkedexample:Completingthesquare(intro)