Quadratic Equations.ppt

Solving Quadratic Equations AquadraticequationisanequationequivalenttooneoftheformWherea b andcarerealnumbersanda 0 Tosolveaquadraticequationwegetitintheformaboveandseeifitwillfactor Getformabovebysubtracting5xandadding6tobothsidestoget0onrightside 5x 6 5x 6 Factor UsetheNullFactorlawandseteachfactor 0andsolve Soifwehaveanequationinxandthehighestpoweris2 itisquadratic Inthisformwecouldhavethecasewhereb 0 Rememberstandardformforaquadraticequationis Whenthisisthecase wegetthex2aloneandthensquarerootbothsides Getx2alonebyadding6tobothsidesandthendividingbothsidesby2 6 6 2 2 Nowtakethesquarerootofbothsidesrememberingthatyoumustconsiderboththepositiveandnegativeroot Let scheck Nowtakethesquarerootofbothsidesrememberingthatyoumustconsiderboththepositiveandnegativeroot Whatifinstandardform c 0 Wecouldfactorbypullinganxoutofeachterm Factoroutthecommonx UsetheNullFactorlawandseteachfactor 0andsolve Ifyouputeitherofthesevaluesinforxintheoriginalequationyoucanseeitmakesatruestatement Whatarewegoingtodoifwehavenon zerovaluesfora bandcbutcan tfactorthelefthandside Thiswillnotfactorsowewillcompletethesquareandapplythesquarerootmethod Firstgettheconstanttermontheothersidebysubtracting3frombothsides Wearenowgoingtoaddanumbertotheleftsidesoitwillfactorintoaperfectsquare Thismeansthatitwillfactorintotwoidenticalfactors Ifweaddanumbertoonesideoftheequation weneedtoaddittotheothertokeeptheequationtrue Let sadd9 Rightnowwe llseethatitworksandthenwe lllookathowtofindit 9 9 Nowfactorthelefthandside twoidenticalfactors Thiscanbewrittenas Nowwe llgetridofthesquarebysquarerootingbothsides Rememberyouneedboththepositiveandnegativeroot Subtract3frombothsidestogetxalone Thesearetheanswersinexactform Wecanputtheminacalculatortogettwoapproximateanswers Okay sothisworkstosolvetheequationbuthowdidweknowtoadd9tobothsides 9 9 Wewantedthelefthandsidetofactorintotwoidenticalfactors WhenyouFOIL theoutertermsandtheinnertermsneedtobeidenticalandneedtoaddupto6x 3x 3x 6x Thelasttermintheoriginaltrinomialwillthenbethemiddleterm scoefficientdividedby2andsquaredsincelasttermtimeslasttermwillbe 3 3 or32 Sotocompletethesquare thenumbertoaddtobothsidesis themiddleterm scoefficientdividedby2andsquared Let ssolveanotheronebycompletingthesquare Tocompletethesquarewewantthecoefficientofthex2termtobe1 Divideeverythingby2 2 2 2 2 Sinceitdoesn tfactorgettheconstantontheothersidereadytocompletethesquare Sowhatdoweaddtobothsides 16 16 Factorthelefthandside Squarerootbothsides remember Add4tobothsidestogetxalone themiddleterm scoefficientdividedby2andsquared Bycompletingthesquareonageneralquadraticequationinstandardformwecomeupwithwhatiscalledthequadraticformula Rememberthesong Thisformulacanbeusedtosolveanyquadraticequationwhetheritfactorsornot Ifitfactors itisgenerallyeasiertofactor butthisformulawouldgiveyouthesolutionsaswell Wesolvedthisbycompletingthesquarebutlet ssolveitusingthequadraticformula 1 1 1 6 6 3 Don tmakeamistakewithorderofoperations Let sdothepowerandthemultiplyingfirst There sa2incommoninthetermsofthenumerator Thesearethesolutionswegotwhenwecompletedthesquareonthisproblem NOTE Whenusingthisformulaifyou vesimplifiedundertheradicalandendupwithanegative therearenorealsolutions Therearecomplex imaginary solutions butthatwillbedealtwithinyear12Calculus SUMMARYOFSOLVINGQUADRATICEQUATIONS Gettheequationinstandardform Ifthereisnomiddleterm b 0 thengetthex2aloneandsquarerootbothsides ifyougetanegativeunderthesquareroottherearenorealsolutions Ifthereisnoconstantterm c 0 thenfactoroutthecommonxandusethenullfactorlawtosolve seteachfactor 0 Ifa bandcarenon zero seeifyoucanfactorandusethenullfactorlawtosolve Ifitdoesn tfactororishardtofactor usethequadraticformulatosolve ifyougetanegativeunderthesquareroottherearenorealsolutions Ifwehaveaquadraticequationandareconsideringsolutionsfromtherealnumbersystem usingthequadraticformula oneofthreethingscanhappen 3 The stuff underthesquarerootcanbenegativeandwe dgetnorealsolutions The stuff underthesquarerootiscalledthediscriminant This discriminates ortellsuswhattypeofsolutionswe llhave 1 The stuff underthesquarerootcanbepositiveandwe dgettwounequalrealsolutions 2 The stuff underthesquarerootcanbezeroandwe dgetonesolution calledarepeatedord