Quadratic Equations

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1、SolvingQuadraticEquationsAquadraticequationisanequationequivalenttooneoftheformWherea,b,andcarerealnumbersanda0Tosolveaquadraticequationwegetitintheformaboveandseeifitwillfactor.Getformabovebysubtracting5xandadding6tobothsidestoget0onrightside.-5x+6-5x+6Factor.UsetheNullFactorlawandseteachfa

2、ctor=0andsolve.Soifwehaveanequationinxandthehighestpoweris2,itisquadratic.Inthisformwecouldhavethecasewhereb=0.Rememberstandardformforaquadraticequationis:Whenthisisthecase,wegetthex2aloneandthensquarerootbothsides.Getx2alonebyadding6tobothsidesandthendividingbothsidesby2+6+622Nowtakethesquare

3、rootofbothsidesrememberingthatyoumustconsiderboththepositiveandnegativeroot.Let'scheck:Nowtakethesquarerootofbothsidesrememberingthatyoumustconsiderboththepositiveandnegativeroot.Whatifinstandardform,c=0?Wecouldfactorbypullinganxoutofeachterm.FactoroutthecommonxUsetheNullFactorlawandseteachf

4、actor=0andsolve.Ifyouputeitherofthesevaluesinforxintheoriginalequationyoucanseeitmakesatruestatement.Whatarewegoingtodoifwehavenon-zerovaluesfora,bandcbutcan'tfactorthelefthandside?Thiswillnotfactorsowewillcompletethesquareandapplythesquarerootmethod.Firstgettheconstanttermontheothersidebysubt

5、racting3frombothsides.Wearenowgoingtoaddanumbertotheleftsidesoitwillfactorintoaperfectsquare.Thismeansthatitwillfactorintotwoidenticalfactors.Ifweaddanumbertoonesideoftheequation,weneedtoaddittotheothertokeeptheequationtrue.Let'sadd9.Rightnowwe'llseethatitworksandthenwe'lllookathowtofindit.99N

6、owfactorthelefthandside.twoidenticalfactorsThiscanbewrittenas:Nowwe'llgetridofthesquarebysquarerootingbothsides.Rememberyouneedboththepositiveandnegativeroot!Subtract3frombothsidestogetxalone.Thesearetheanswersinexactform.Wecanputtheminacalculatortogettwoapproximateanswers.Okay---sothisworkst

7、osolvetheequationbuthowdidweknowtoadd9tobothsides?99Wewantedthelefthandsidetofactorintotwoidenticalfactors.WhenyouFOIL,theoutertermsandtheinnertermsneedtobeidenticalandneedtoaddupto6x.+3x+3x6xThelasttermintheoriginaltrinomialwillthenbet