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Nguyn Phi Hng - V Thnh Vni hc Khoa hc Hu**************Phng php t n phtrong gii phng trnh v tA. Li ni uQua bi vit ny chng ti mun gii thiu cho cc bn mt s k nng t n ph trong giiphng trnh v t. Nh chng ta bit c nhiu trng hp gii mt phng trnh v t m tabin i tng ng s ra mt phng trnh phc tp , c th l bc qu cao ...C l phngphp hu hiu nht gii quyt vn ny chnh l t n ph chuyn v mt phng trnhn gin v d gii quyt hn .C 3 bc c bn trong phng php ny :-t n ph v gn lun iu kin cho n ph-a phng trnh ban u v phng trnh c bin l n phTin hnh gii quyt phng trnh va to ra ny . i chiu vi iu kin chn n ph thchhp.- Gii phng trnh cho bi n ph va tm c v kt lun nghim* Nhn xt :- Ci mu cht ca phng php ny chnh l bc u tin . L do l n quyt nh n tonb li gii hay, d , ngn hay di ca bi ton .-C 4 phng php t n ph m chng ti mun nu ra trong bi vit ny l :+ PP Lng gic ho+ PP dng n ph khng trit + PP dng n ph a v dng tch+ PP dng n ph a v h

2B. Ni dung phng phpI. Phng php lng gic ho :1. Nu |x|ath ta c th ttaxsin,t 2;2hoc;0,costtaxV d 1 : Gii phng trnh: )121(1122xxxLi gii : K :|1|xt 2;2,sinttxPhng trnh cho tr thnh : 2cos23sin22sinsin2cos2)cos21(sincos1tttttttt 346)12(2123sin02cos0)123sin2(2cosktktttttKt h p vi iu kin ca t suy ra :t6Vy phng trnh c 1 nghim : 216sinxV d 2 : Gii phng trnh: 3132)1()1(112332xxxxLi gii : K :1||xKhi VP > 0 .Nu 0)1()1(:0;133xxxNu 0)1()1(:1;033xxx.ttxcos, vi 2;0tta c : tttttttsin2sin211cos62sin22sin2cos2cos2sin6233 61cos0sin21cos6tttVy nghim ca phng trnh l 61xV d 3 : Gii phng trnh: xxxxxx212121212121Li gii : K : 21||xt;0,cos2ttxphng trnh cho tr thnh : 0cos02sinsinsin4sin122cot2tan22cos2sin232ttttttantttVy phng trnh c nghim duy nht0x

3V d 4 (THTT): Gii phng trnh: 233xxx(1)Hng dn :Nu2x: phng trnh khng xc nh .Ch vi2xta c : 24323xxxxxxxVy gii phng trnh (1) ta ch cn xt vi2;2xt;0,cos2ttxkhi phng trnh cho tr thnh : 2cos3cos tt2. Nuax||th ta c th t : 0,2;2,sintttaxhoc 2;;0,costttaxV d 5 : Gii phng trnh: 111122xxLi gii : K :1||xt 2;2,sin1ttxPhng trnh cho tr thnh : 0sin1coscoscotcos1cot1sin122tttanttantt kttt12212sin0coskt hp vi iu kin ca t suy ra 12tVy phng trnh c 1 nghim : 13212sin1xTng qut: Gii phng trnh axax1122V d 6 : Gii phng trnh: 2932xxxLi gii : K :3||xt 2,;0,cos3tttx, phng trnh cho tr thnh : 234cos3412sin2sin22sin122sin1cos12xtttttt(tho mn)Tng qut: Gii phng trnh: baxaxx22viba,l cc hng s cho trc3. t 2;2,tanttx a v phng trnh lng gic n gin hn :V d 7 : Gii phng trnh: 0333323xxx(1)

4Li gii :Do 31xkhng l nghim ca phng trnh nn (1) 331323xxx(2)t 2;2,tanttx, Khi (2) tr thnh : 3933tankttSuy ra (1) c 3 nghim : 97tan;94tan;9tanxxxV d 8 : Gii phng trnh: 22222121211xxxxxxLi gii : K :1;0xxt 4;0,2;2,tantttx, phng trnh cho tr thnh : 012cos2cos.sin202cos.sin21sin211cos14sin22sin1cos1tttttttttt 262221sin1sin0sin0sin2sin1sin0sin2sin21sin2222ktkttttttttttKt hp vi iu kin suy ra : t6Vy phng trnh c 1 nghim : 316tanx4. Mc nh iu kin :ax||. Sau khi tm c s nghim chnh l s nghim ti a ca phngtrnh v kt lun :V d 9 : Gii phng trnh: xx2163Li gii :Phng trnh cho tng ng vi :1683xx(1)t;0,costtx, Lc (1) tr thnh : Zkktt329213cosSuy ra (1) c tp nghim : 97cos;95cos;9cosSVy nghim ca phng trnh cho c tp nghim chnh l SII. Phng php dng n ph khng trit * Ni dung phng php :a phng trnh cho v phng trnh bc hai vi n l n ph hay l n ca phng trnh cho :a phng trnh v dng sau : xxPxfxQxf..khi :t 0,ttxf. Phng trnh vit thnh :0.2xPxQttn y chng ta gii t theo x. Cui cng l gii quyt phng trnh txfsau khi n gin hav kt lunV d 10 : Gii phng trnh 169244222xxx(1)Li gii : K :2||x

5t 242xtLc :(1) xxxxxxxx8421648169216421642422222Phngtrnh tr thnh :0816422xxttGii phng trnh trn vi n t , ta tm c : 42;221xtxtDo2||xnn02tkhng tha iu kin0tVi t2xth : 324480242222xxxxxx( tha mn iu kin2||x)V d 11 :Gii phng trnh 361122xxxLi gii : K :1xt 01xt,phng trnh cho tr thnh : xtttxt66036122* Vi txt66, ta c :66tx(v nghim v :0;0VPVT)* Vi txt66, ta c :tx)6(6Do6xkhng l nghim ca phng trnh nn : xxxt66166Bnh phng hai v v rt gn ta c :3x(tha mn)Tng qut: Gii phng trnh: 222baxbaxxV d 12 : Gii phng trnh: 12831112322xxxxLi gii :t 1122txPhng trnh cho vit thnh :033838313132222xxtxtxtxtxtT ta tm c t3xhocxt31Gii ra c :0x* Nhn xt : Ci kho lo trong vic t n ph c th hin r trong phng php ny v c thl v d trn . bi trn nu ch dng li vi vic chn n ph th khng d gii quyt trn vn n .Vn tip theo chnh l vic kheo lo bin i phn cn li lm bin mt h s t do , vic giquyt t theo x c thc hin d dng hn .V d 13 : Gii phng trnh: 3420073420082xxxxLi gii : K : 43xt 034txphng trnh cho tr thnh :02007200822txtxGii ra :txhoc x2008t(loi)*txta c :310342xxxxVy3,1 xxl cc nghim ca phng trnh cho .V d 14 : Gii phng trnh: 12211433xxxx

6Li gii : K :1xt 13xt,Phng trnh cho tr thnh01214214121222xtxttxxtPhng trnh trn kh n gin !!!!!!!III. Phng php dng n ph a v dng tch1. Dng mt n phV d 15 : Gii phng trnh: 49232xx(1)Li gii : K : 23xt 023txphng trnh (1) tr thnh : 2013001349233322tttttttt(2) gii oc bng cch p dng phng php I :t;0,cos2ttx a v dng : 213costTng qut: Gii phng trnh: 22aaxxvial hng s cho trc .V d 16 :Gii phng trnh: 16223323xxxxLi gii : K :2xVit li (1) di dng : 20222333xxxxt 02xt, Khi (2) tr thnh : 2222020232323xxxxtxtxtxtxtxtx 3222084002022xxxxxxxxVy phng trnh cho c 2 nghim : 322,2xxV d 17 : Gii phng trnh : 015xxLi gii : K :6;1x(1)t 01xt(2) , phng trnh cho tr thnh : 552tt(3)054020102224ttttttti chiu vi hai iu kin (1) v (2) thay vo v gii ra : 21711xV d 18 : Gii phng trnh: 2112006xxxLi gii : K :1;0x(1)t 101txt, Khi : 2221,1txtx,phng trnh cho tr thnh :010031212007111120061222222222ttttttttttV10tnn010032ttDo phng trnh tng ng vi :101ttDo vy0x(tha (1))

72. Dng 2 n ph .V d 19 : Gii phng trnh: 391215422xxxxxLi gii :t 12;15422xxbxxa01392222babababaxba 65560312923931010xxxxaxbaxbabaVy tp nghim ca pt l 6556;0;31SV d 20 : Gii phng trnh: 8323232xxx(1)Li gii : K :212xx(*)t 2,422xvxxuta c :2322xxvuLc (1) tr thnh :vuvuvuuvvu20223222(Do02vu)Tm x ta gii : 133046224222xxxxxx(Tha (*))Vy (1) c 2 nghim : 1332,1xV d 21 : Gii phng trnh: 1520914522xxxxxLi gii : K :5xChuyn v ri bnh phng hai v phng trnh mi ,ta c: 045454354215410524951222xxxxxxxxxxxxx(2)t 0,,4,542vuxvxxu,th :(2)0562540953203205322222xxxxvuvuvuvuuvvuGii ra ta c 2 nghim tha mn : 8;261521xxV d 22 : Gii phng trnh: 424343421111xxxxxxxxLi gii : K :10xt : 10014444vuvuxvxuT phng trnh ta c :1001232322vuvuvuvuvuvuuvvuvu( Do0vu)t ta gii ra c cc nghim : 21;1;0xxx3. Dng 3 n ph .V d 23 : Gii phng trnh: 21881732323xxxxx

8Li gii :t 3232318,8,17xxcxxbxata c :2818817182223333xxxxxcbacbacbaT (1) v (2) ta c :033333accbbacbacbaNn :accbbaaccbba0t d dng tm ra 4 nghim ca phng trnh :9;1;0;1SV d 24 : Gii phng trnh: 034925133333xxxx(1)Li gii :t 33392,5,13xcxbxa,ta c:34333xcbakhi t (1) ta c :03333accbbacbacbaGii nh v d 23 suy ra c 3 nghim ca phng trnh : 58;4;3xxxIV. Phng php dng n ph a v h1.Dng n ph a v h n gin gii bng php th hoc rt gn theo v .a.Dng mt n ph .V d 25 : Gii phng trnh: 552xxLi gii : K :5xt 0,5txtTa c :52tx 221122111505015055522222222xxtxtxtxtxtxtxtxxttxtxxttxTng qut: Gii phng trnh: aaxx2b.Dng 2 n ph .* Ni Dung : cxfbxfanm* Cch gii :t : nmxfbvxfau,Nh vy ta c h :bavucvunmV d 26 : Gii phng trnh: 5405744xx(1)Li gii : K :5740xt 4440,,57xvxuKhi :(1)05281025972259752222244uvuvvuvuuvvuvuvuvu

92332654465vuvuuvvuuvuvvu(Do h445uvvuv nghim)n y ch vic thay vo tm nghim ca phng trnh ban u .V d 27 : Gii phng trnh: 442112xxLi gii : K : 120xt : vxux412vi 4120120vu(*)Nh vy ta c h : )1(12212112214244424vvvuvuvuGii (1) :(1) 023241021102112,142,1422422vvvvvvVy2,1vtha (*) chnh l 2 nghim ca phng trnh cho .V d 28 : Gii phng trnh: 221147xxxLi gii :t : (*)147111471471104444yyyzyyxzyzyxzxyGii phng trnh (*),ta c: 169043004342xxyyyy2. Dng n ph a v h i xngDng 1 : Gii phng trnh: nnbaxabxCch gii: t nbaxtta c h :axbtatbxnnVic gii h ny tr nn d dngV d 29 : Gii phng trnh: 331221xxLi gii :t : 312xtta c h :0221221212122333333txtxtxtxxttxtxxttx 251104011202211012222222233xxtxxtxxxtxtxtxxxtx

10Vy tp nghim ca phng trnh l : 251;1SDng 2 : Gii phng trnh: xaaxCch gii : t xat,phng trnh cho tng ng vi xattaxV d 30 : Gii phng trnh: xx20072007Li gii : K :0xt : xt2007(1), PTLy (3) tr (2) ta c : txxtxtxttx01(1) 480292803002007xxx(Do0x)Dng 3 : Chn n ph t vic lm ngc :V d 31 : Gii phng trnh: 12222xxxLi gii : K : 21xt bayx12Chn a, b h :122222xbaybayxx1, y21x(*) l h i xng .Ly1,1bata c h :012212212222222yxyxxxyyyxxGii h trn ta c : 22yxi chiu vi iu kin ca h (*) ta c nghim duy nht ca phng trnh l : 22xDng 4 :Ni dung phng php :Cho phng trnh : xedxcbaxnnvi cc h s tha mn :EbceacdCch gii :t nbaxedyV d 32 : Gii phng trnh: 7728942xxLi gii : K : 49xPT 4721728942xx- Kim tra : 47,0,21,1,7,289,71edcba(tho mn)t : yyxxyyxyyxy7721494777289441289421222(1)

11Mt khc : xxy77212(2)T (1) v (2) ta c h : xxyyyx7721772122y l h i xng loi II bit cch gii .V d 33 : Gii phng trnh: 3,3362xxxxLi gii :PT 3632xx- Kim tra :6,0,3,1,1,3,1edcbat : 3633963322yyxxyyxy(1)Mt khc :3632xxy(2)T (1) v (2) ta c h :36336322xxyyyxn y kh d dngV d 34 : Gii phng trnh: 255336853233xxxxLi gii :PT 232532272.9.33.4.325333233xxxxxxxx- Kim tra :2,1,3,2,1,5,3edcba(tho mn)t : 332553368532754368533223233yxyyyxyyyxy(1)Mt khc :32255336823yxxx(2)T (1) v (2) ta c h :3225533683325533682323yxxxyxyyyGii h trn tht n gin !!!!!!!!!Hu , ngy 15 thng 4 nm 2007