quadratic equations

Quadratic EquationsLesson 5: CBSE Text book, 2008S Chand& Co2 Ram’s Workshops DefinitionsAny equation of the form p(x)=o ; where p(x) is a polynomial of degree 2 is a quadratic equation.i.e. ax2+bx+c=0 is a standard form of quadratic equation, where a≠0525:............5....5.....25:25)5(25)5(25)5(*)5(25)5(*)5(22±=−+==−=+=−−=++aswrittencommonlyisThisorbecanThereforeImportant: please bear in mind21*aaaaa==Note:The quadratic Equation should have the second degree polynomialSee Example 5.2 on page 1743 Ram’s Workshops What is a quadratic equationThe power of the variable x should be = 2 ( not less or more)The powers of the other variables should be less than 2 : but only as integersintegeran not but -2 than less thoughish power whic a has termsecond the...........;...0122aseqnquadraticanotisxx=−+ax2+bx+c=0 is a standard form of quadratic equation, where a≠0If k is the root of the quadratic eqn:Then x-k is a factor of the equation:Also substituting k in place of x shall be able to satisfy the condition of the equation4 Ram’s Workshops Some examples of Quadratic Equationsyourself 3-for above test theNowzero is :61046)2(5(-2):see usLet :satisfied isequation above x theof placein 2-substitute we..if...............3..2..0)3(*)2(:........06522whicheiEqnabovetheofRootstheareandxthenxxaswrittenbecanWhichxx+−=+−+−−==++=++5 Ram’s Workshops Understanding the Roots…(1)7)2(0612044222−=+=++=++xxxxx()realandsameTwoandareRootsx.........2....2........022−−=+()realandrootsdifferentandareRootsxx.........3....2........0)3(*2−−=++()numberrealnotisitnegativeforrootsquarenoisTherexx..........................;72.722−=+−=+6 Ram’s Workshops Understanding the Roots…(2)......:4444;1611;4;1:)3()3,2(..........:4...244;1446;12;1:)2()2,2(....:4...164;164;4;1:)1(02222222rootsrealNoacbacbcbarootsdistinctandrealtwoacbeiacbcbarootsequaltwoacbeiacbcbacbxax−−−−−−<==−−−−−−−===−−−−−−−−−>==−−−−−−===−−−−−−===−−−−−====++()7)2()3(01147)2()3(*)2()2(06122)1(04422222−=+=−−−−−−=++−=+++=−−−−−=+++=−−−−=++xxxxxxxxxxxWe are comparing the coefficients7 Ram’s Workshops E5.1/2(v)rootaisxThereforeandasxofvaluethengSubstitutixxEqnQuadraticfollowingtheofrootaisLet......12..02)2..(..22*2......(1224122211*222*2211*22)2(1)12(22)12(........0122................12xrout whethe find us 22222+====+−−++=+−−++=++−+=+−+=Now check the other condition given in the problem8 Ram’s Workshops The hypotenuse of a right angled triangle is 6 mtrsmore than twice the shortest side. The third side is two meters less than the hypotenuse04)3612(444)3612(4426)2(262222222222=+−−−+−+−++−=⎥⎦⎤⎢⎣⎡−+−=−=−=hhhhhhhhhhhhhhshlhslh = 2s+6----(1)l = h-2-----(2) )3(222−−−−−−−−−+=slh24&10..........26:.1026-26s242-26so..l26h hence0)l then 2h if i.e ( d.is..absur2..as..it. ..'...26....2;0)26(*)2(:052280361216160)3612()44(*4......4.........(.222aresidestheandhypoANSbetcanhorhhhorhhhhhhhhsidesbothonwithgmultiplyinorLCMtaking==========−−=−+=−++−=+−−−9 Ram’s Workshops E5.3/212..21..2..210)2)(12(0)2(*1)2(202420252:..........Re1;1025222222=====−−=−−−=+−−=+−===+−xorxOraandaaaaaaaaaaaeqnabovethewritingaxaxxxUnderstand these steps:We have to the split the middle term (5a) to enable the common elements for factorizationTaking the common factor:a-210 Ram’s Workshops E5.3/302.....;....202......020)2)(2()2()2(2224:exp)(24222222222222222222axbxaxandbxOraxbxaxbaxxbaxbxaxandingbaxbax===−=−=−−=−−−=+−−++−Two roots11 Ram’s Workshops Please remember the following222221211**21xxxxxxxx++=++=⎭⎬⎫⎩⎨⎧+132)3()2()32)(32(22−=−=−=+−etcxxxxxxxxbaba....***43527===+12 Ram’s Workshops Find the two numbers whose sum is 27 and the product 18214......13...............0)14(*)13(....0)13(14)13(01821413:..........018227......27182.................27182)1.(..........182)2......(182)1.....(27222andarenumberstheeixxxxxxxxionfactorizatforsplittingOrxxorxxxbysidesbothgMultiplyinxxinyforngsubstitutixyxyyx=−−=−−−=+−−=+−=+=+===+13 & 14 are knownas the roots of theQuadratic Equationx2-27x+182=013 Ram’s Workshops Roots of Quadratic Equation: ProofaacbboraacbbxoraacbabxoraacbabxsidesbothonrootSquaretakingthenacbifaacbabxaacbabxacababxasabovethewritecanwesquareperfectasitmaketoInorderacxabxabydividecbxaxequationquadratictheConsider24...24.......242...242:...............;04..;..442024222:...........:........................0....;...0:........222222222222222−−−−+−=−±−=⎟⎟⎠⎞⎜⎜⎝⎛−±=⎟⎠⎞⎜⎝⎛+≠−−=⎟⎠⎞⎜⎝⎛+=−−⎟⎠⎞⎜⎝⎛+=+⎟⎠⎞⎜⎝⎛−⎟⎠⎞⎜⎝⎛+=++=++Only if b2≥4ac then only the quadratic equation has real roots14 Ram’s Workshops E5.4/9bamandtakingmbabbmamaababmmbmambmaababmabbammabbammmLet======−−−=−−++=++=++=+=.......abmbma b..andma0bm)-b)(a-(ma:factorcommon the0)()(0)1(1b-xa-x 222222222baxorbaxbaxbabababaxbabaxbbxaaxabbxaxmmngSubstitutibamand+==−−=+−−+−−−=−−−−=−=−−====00))()(())(()(0)()(;b-xa-x.........abm2222Similarly x=0 can be shown15 Ram’s Workshops E5.4/2001322562 012*2*22*2*2:eqngiven in the 2*222*224;216:under as write-recan webase theas 2 of powers thesing012*164*162x4424142)2(22412=+−==+−====−−−=+−+++++aaaLetngSubstitutiUgivenxxxxxxxxxANSxHencethereforeaaoraiswhichx..4...::22:2161611160)116(.....4412−=======−=−−−16 Ram’s Workshops E5.5/17rootsalandEqualTwohaseqntheacbasacbcbatwothecomparingcbxaxxx....Re............;..4...44149*49*444412149;21;49...0049214922222=========++=++143.....14349*22104..224:....22−−=−=−−=−+−andacbasabaacbbareRootsThe17 Ram’s Workshops Summarytiscriminan.........404 if :roots realdistinct twoHas04 if : roots real equal twohasIt 04 if :roots real no hasIt :properties following thehaseqn qudratic above the:24...24.......2420:........22222222DasKnownisDacbacbacbacbSoaacbboraacbbxoraacbabxcbxaxequationquadratictheIn=−>−=−<−−−−−+−=−±−==++18 Ram’s Workshops E5.7/141......32..0)1)(23(0)1(2)1(3022330234...........084120)2(*1*41604:...........2;4:1024x0.......:.........02)14(22222222222−>>>+−>+−+>−−+>−+>−+>+−−>−+−====+−++=++=+−++korkorkkkkkkkkkkorcommonfactbydividingkkkkkacbrootsrealofconditiontheForkkckbakkkxcbxaxofformtheinequationquadraticabovetherewrtingkxkx19 Ram’s Workshops E5.8/35.....30)5)(3(0)3(5)3(0155301520164120)4(*1*4)1(04:...........1;1:40........:........01)1()4(2222222orkkkkkkkkkkkkkkkkacbrootsequalofconditiontheForckbkacbxaxwithequationquadraticthecomapringxkxk−==−+=+−+=−−+=−−=−−++=+−+=−=+=+==++=++++20 Ram’s Workshops E5.8/587403545.35.3*45.3*5.3*45.305.35.30)(5.30)(.............5.3;035100151050015)5(2)5(*2:get wexofplacein 5-substutingroot then a5..is-..222222======++=++=++==+−=−−=−−+−kkkxxkxxkxxpinpofvaluethengSubstitutippppAs21 Ram’s Workshops ProblemTwo Water Taps together fill a tank in 9 3/8 hrs. The tap of the large diameter takes 10hrs less that the small diameter one to fill the tank separately. Find the time in which each tap can separately fill the tank22 Ram’s Workshops Solution --------(page1)2(75811 :2) ( )2(11183911839 fill they will together 839 tan 1L1 : fill willhour they onein Together tank theof S1 fills PipeSmaller hour the oneIn tank theof L1 fills pipeLarger hour oneIn -(1)-----10S tank thefill tohrs S takepipesmaller The tank thefill tohrs L takepipeLarger let the−−−−−=⎥⎦⎤⎢⎣⎡+−−−−−=⎥⎦⎤⎢⎣⎡+⎥⎦⎤⎢⎣⎡++=−SLgSimplyfyinSLThereforeTankSLhrsinktheofportionSportionportionL23 Ram’s Workshops ----(page2)15hrsLand25S839 than less bet can'it as 415 bet can' 415830Sor 250)25(30)25(8075030200807502308:8087507575))(10(8)10(7575*: Re7581101(2)in ngSubstituti10L (1) from 222======−−−=+−−=+−−=−+−=−+=⎥⎦⎤⎢⎣⎡+−−=HenceSSSSSSSSSSOrSSSSSSSSLCMwitharrangingSSSBut24 Ram’s Workshops E5.13/140432122046836122468)3612(468)6()..1...(..)3(6..6)2(...Re)2(2444)1(46822222222=−−=−+−=+−+=−+−−−−−−−−==−−−−−−−−=−−−−−−−−=+xxxxxxxxxaboveinSubstitutexyoryxeqnwritingyxyx12.................18472................,......'......460124601246012436001243456144122*2)432(*2*4)12()12(24:.....22===−+=±=±=+±=−−−±−−−±−issquareanotherofsideothertherootpositveThenegativetheignorehencenegativeinbetcanlengthsTheoraacbbarerootsThe25 Ram’s Workshops Find the value of k in the following eqn)1(0322−−−−−−−=++kxx6224....243*2*4....4:..................22±=====korkeiacbmeansrootsequalhaveshouldeqnThe26 Ram’s Workshops E5.11/556:fraction the..min....521021112111....21112120111*230*1*41130;1;1030:03006161303060300)1(6130)1(30)1(*)(*30;...3061112222222SofigureustheignoringorxcbaxxOrxxxxxxxxxxxxxLCMThexxxx==+−−−±−=+±−=−−±−=−====−+=+−−=−−+++=+−+++=+++27 Ram’s Workshops E5.15/41)61x1(*4:.....)61x1( docan they .....61x1docan day they oneIn days 6-xakesAbhishek tworkfinish the todays x takeanubhav ==−+−+−whichxdaysfourinHenceperdayxTogetherworkofpiecexandLetAnswertheisimpossibleissolutionfirsttheasorxacbacbcbacbxaxwithComparingxxrxxxxgSimplifyin......12..............12224......224210141*296196)14(049624*1*44;..19624,14;1....02414:)6(4)6(4:2222===±=−±−−=>−====−==++=+−−=+−28 Ram’s Workshops E5.16/857624Answer xbe toassumed are birds ofnumber the..2428672:........2842025228176400252281128966350425214*22016*14*4)252()252(222===±=±=+±−−−±−−TheAsvaluepositivetheonlytaking0201625214036*567*3614:Re3636*5636*794936567494.............22222222222=−−=−−=++++==+⎭⎬⎫⎩⎨⎧+++xxxxwritingxxxxxLCMxxxxxxbebirdstotalofnumbertheLetThanks for your attention