Quadratic Equations

SolvingEquationsAquadraticequationisanequationequivalenttooneoftheformWherea,b,andcarerealnumbersanda002cbxaxTosolveaquadraticequationwegetitintheformaboveandseeifitwillfactor.652xxGetformabovebysubtracting5xandadding6tobothsidestoget0onrightside.-5x+6-5x+60652xxFactor.023xxUsetheNullFactorlawandseteachfactor=0andsolve.02or03xx3x2xSoifwehaveanequationinxandthehighestpoweris2,itisquadratic.Inthisformwecouldhavethecasewhereb=0.02cbxaxRememberstandardformforaquadraticequationis:02cax002cxaxWhenthisisthecase,wegetthex2aloneandthensquarerootbothsides.0622xGetx2alonebyadding6tobothsidesandthendividingbothsidesby2+6+6622x2232xNowtakethesquarerootofbothsidesrememberingthatyoumustconsiderboththepositiveandnegativeroot.3xLet'scheck:0632206322066066Nowtakethesquarerootofbothsidesrememberingthatyoumustconsiderboththepositiveandnegativeroot.02cbxaxWhatifinstandardform,c=0?002bxaxWecouldfactorbypullinganxoutofeachterm.0322xxFactoroutthecommonx032xxUsetheNullFactorlawandseteachfactor=0andsolve.032or0xx23or0xxIfyouputeitherofthesevaluesinforxintheoriginalequationyoucanseeitmakesatruestatement.02cbxaxWhatarewegoingtodoifwehavenon-zerovaluesfora,bandcbutcan'tfactorthelefthandside?0362xxThiswillnotfactorsowewillcompletethesquareandapplythesquarerootmethod.Firstgettheconstanttermontheothersidebysubtracting3frombothsides.362xx___3___62xxWearenowgoingtoaddanumbertotheleftsidesoitwillfactorintoaperfectsquare.Thismeansthatitwillfactorintotwoidenticalfactors.Ifweaddanumbertoonesideoftheequation,weneedtoaddittotheothertokeeptheequationtrue.Let'sadd9.Rightnowwe'llseethatitworksandthenwe'lllookathowtofindit.996962xx6962xxNowfactorthelefthandside.633xxtwoidenticalfactors632xThiscanbewrittenas:Nowwe'llgetridofthesquarebysquarerootingbothsides.632xRememberyouneedboththepositiveandnegativeroot!63xSubtract3frombothsidestogetxalone.63xThesearetheanswersinexactform.Wecanputtheminacalculatortogettwoapproximateanswers.55.063x45.563xOkay---sothisworkstosolvetheequationbuthowdidweknowtoadd9tobothsides?___3___62xx99633xxWewantedthelefthandsidetofactorintotwoidenticalfactors.WhenyouFOIL,theoutertermsandtheinnertermsneedtobeidenticalandneedtoaddupto6x.+3x+3x6xThelasttermintheoriginaltrinomialwillthenbethemiddleterm'scoefficientdividedby2andsquaredsincelasttermtimeslasttermwillbe(3)(3)or32.Sotocompletethesquare,thenumbertoaddtobothsidesis…themiddleterm'scoefficientdividedby2andsquaredLet'ssolveanotheronebycompletingthesquare.021622xxTocompletethesquarewewantthecoefficientofthex2termtobe1.Divideeverythingby20182xx2222Sinceitdoesn'tfactorgettheconstantontheothersidereadytocompletethesquare.___1___82xxSowhatdoweaddtobothsides?161616Factorthelefthandside154442xxxSquarerootbothsides(remember)1542x154x154xAdd4tobothsidestogetxalone228themiddleterm'scoefficientdividedby2andsquaredBycompletingthesquareonageneralquadraticequationinstandardformwecomeupwithwhatiscalledthequadraticformula.(Rememberthesong!!)aacbbx242Thisformulacanbeusedtosolveanyquadraticequationwhetheritfactorsornot.Ifitfactors,itisgenerallyeasiertofactor---butthisformulawouldgiveyouthesolutionsaswell.Wesolvedthisbycompletingthesquarebutlet'ssolveitusingthequadraticformulaaacbbx2421(1)(1)66(3)212366Don'tmakeamistakewithorderofoperations!Let'sdothepowerandthemultiplyingfirst.02cbxax0362xx212366x224662642426262632There'sa2incommoninthetermsofthenumerator63Thesearethesolutionswegotwhenwecompletedthesquareonthisproblem.NOTE:Whenusingthisformulaifyou'vesimplifiedundertheradicalandendupwithanegative,therearenorealsolutions.(Therearecomplex(imaginary)solutions,butthatwillbedealtwithinyear12Calculus).SUMMARYOFSOLVINGQUADRATICEQUATIONS•Gettheequationinstandardform:02cbxax•Ifthereisnomiddleterm(b=0)thengetthex2aloneandsquarerootbothsides(ifyougetanegativeunderthesquareroottherearenorealsolutions).•Ifthereisnoconstantterm(c=0)thenfactoroutthecommonxandusethenullfactorlawtosolve(seteachfactor=0).•Ifa,bandcarenon-zero,seeifyoucanfactorandusethenullfactorlawtosolve.•Ifitdoesn'tfactororishardtofactor,usethequadraticformulatosolve(ifyougetanegativeunderthesquareroottherearenorealsolutions).aacbbxcbxax24022Ifwehaveaquadraticequationandareconsideringsolutionsfromtherealnumbersystem,usingthequadraticformula,oneofthreethingscanhappen.3.Thestuffunderthesquarerootcanbenegativeandwe'dgetnorealsolutions.Thestuffunderthesquarerootiscalledthediscriminant.Thisdiscriminatesortellsuswhattypeofsolutionswe'llhave.1.Thestuffunderthesquarerootcanbepositiveandwe'dgettwounequalrealsolutions04if2acb2.Thestuffunderthesquarerootcanbezeroandwe'dgetonesolution(calledarepeatedordoublerootbecauseitwouldfactorintotwoequalfactors,eachgivingusthesamesolution).04if2acb04if2acbTheDiscriminantacb42=81+40=121•Thediscriminantispositiveandaperfectsquare.•Tworeal,rationalroots.•Solvingbysubstitution•Equationswithradic