Tài liệu Sáng tạo và giải phương trình, hệ phương trình, bất phương trình-nguyễn tài chung

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AdST~ NGUYEN TAI CHUNG gmi # ? 9 PHO fI DG TRiHH HE PHIfOnC TRn iH BAT PHIDnG TR P H l J d N G PHAP XAY DL/NG B E CAC TDAN DANG TCAN, CAC PHLfdNG PHAP GIAI C A C O E T H I H O C S I N H G I C I GJUCC E I A , O L Y M P I C 3 0 / 4 T A I LIEU B D I DL/QNG H O C S I N H K H A GICI fx T A I L I E U O N L U Y E N T H I B A I HOC T A I LIEU T H A M K H A C C H D GIAO VIEN N H A X U A T e X N T r a N G H d P T H A N H P H D HO CHI M I N H LMnoidau SANG T A O V A GIAI P H U O N G T R I N H , HE PHLfdNG T R I N H , B A T PHaONG T R I N H HQC sinh hoc toan xong roi lam cac bai tap. Vay cac bai tap do 6 dau ma ra? Ai la nguai dau tien nghi ra cac bai tap do? Nghl nhu the nao? Ngay ca nhieu N G U Y I N TAI C H U N G Chiu trach nhiem xuS't ban giao vien cung chi biet suti tarn cac bai tap c6 trong sach giao khoa, sach tham 4 ; N G U Y E N THI THANH HlJdNG Bien tap : QUOC NHAN Si^abanin . : HOANG NHlTX Trinh bay : C6ng ty K H A N G V I E T Bia : C6ng ty K H A N G V I E T khao khac nhau, chua biet sang tac ra cac de bai tap. Mpt trong nhimg each do la tim nhirng hinh thiic khac nhau de dien ta ciing mpt npi dung roi lay mpt hinh thiic nao do phii hop vai trinh dp hpc sinh va yeu cau hp chiing minh tinh diing dan ciia no. ' • Nhu chiing ta da biet phuong trinh, h$ phuong trinh c6 rat nhieu dang va phuong phap giai khac nhau va rat thuong gap trong cac ky thi gioi toan ciing nhu cac ky thi tuyen sinh Dai hpc. Nguoi giao vien ngoai nam dupe cac dang phuong trinh va each giai chiing de huong dan hpc sinh can phai biet each xay NHA XUAT BAN TONG H0P TP. HO CHf MINH NHA SACH TONG HOP 62 Nguyen ThI Minh Khai, Q . l D T : 38225340 - 38296764 - 38247225 Fax: 84.8.38222726 Email: tonghop@nxbhcm.com.vn Website: www.nxbhcm.com.vn/ www.fiditour.com Tong phdt hanh dung nen cac de toan de lam tai li^u cho vi|c giang day. Tai lifu nay dua ra mpt so phuong phap sang tac, quy trinh xay dimg nen cac phuong trinh, he phuong trinh. Qua cac phuong phap sang tac nay ta ciing rut ra dupe cac phuong phap giai tu nhien cho cac dang phuong trinh, hf phuong trinh tuong ling. Cac quy trinh xay dyng de toan dupe tnnh bay thong qua nhiing vi du, cac bai toan dupe xay dung len dupe dat ngay sau cac vi du do. Da so cac bai toan dupe xay dung deu c6 loi giai hoac huong dan. Quan trpng hon niia la mpt so luu y sau loi giai se giiip chiing ta giai thich dupe "vi sao lai nghl ra loi giai nay". Nhu vay cuon sach nay se trinh bay song song hai van de: Phuong phap CONG T Y TNHH MTV DjCH V g VAN HOA KHANG V I E T ( ^ D i a chi- 71 Oinh Tien Hoang - P.Da Kao - Q.1 - T P . H C M Dien thoai:'08. 39115694 - 39105797 - 39111969 - 39111968 Fax: 08. 3911 0880 Email: khangvietbookstore ©yahoo.com.vn 1 Website: www.nhasachkhangviet.vn In ian thLT i, so lUdng 2.000 cuon, kho 1 6x24cm. Tai: C O N G T Y C O P H A N T H L / O N G M A I N H A T N A M Dia chi: 006 L6 F, KCN Tan Binh, P. Tay Thanh, Q. Tan Phu, Tp. Ho Chi Minh So DKKHXB: 1 55-1 3/CXB/45-24ArHTPHCM ngay 31/01/201 3. Quyet dinh xuat ban so: 296/QD-THTPHCM-2013 do NXB Tong Hop Thanh Pho H 6 Chi Minh cap ngay 19/03/2013 In xong va nop luU chieu Quy II nam 201 3 sang tac eae de toan va Cac phuong phap giai ciing nhu phan loai cac dang toan ve phuong trinh, hf phuong trinh. Diem moi la va khac bi?t ciia cuon sach nay la quy trinh sang tac mpt de toan moi (dupe trinh bay thong qua cac vi du) va each thiie chiing ta suy nghl, tim ra loi giai mpt bai toan (dupe trinh bay thong qua eae luu y, chii y, nhan xet ngay sau loi giai cac bai toan). Ngoai ra cuon sach nay con danh ra mpt ehuong (ehuong 5) de trinh bai cac bai toan phuong trinh, he phuong trinh, bat phuong trinh trong cac de thi Dai hpc trong nhiing nam gan day. Tot nhat, doe gia tu minh giai cac bai toan eo trong sach nay. Tuy nhien, de thay va lam chii eae ky xao tinh vi khac, cac bai toan deu dupe giai san (tham chi la nhieu each giai) voi nhiing miie dp chi tiet khac nhau. Npi dung sach da c6' gang tuan theo y chii dao xuyen suo't: Biet dupe loi giai ciia bai toan chi la yeu cau dau tien - ma hon the - lam the nao de giai dupe no, each ta xir ly no, nhiing suy lu^n nao to ra "c6 ly", cac ket lu^n, nhan xet va luu y tir bai toan dua ra... Hy vong cuon sach nay la tai li§u tham khao c6 ich cho cac em hpc sinh kha gioi, hoc sinh cac lop chuyen toan Trung hpc pho thong, cac em hpc sinh dang luypn thi Dai hpc, giao vien toan, sinh vien toan cua cac tmong DHSP, D H K H T N cung nhu la tai phyc v\ cho cac ky thi tuyen sinh D ^ i hpc, thi hpc sinh gioi toan THPT, thi Olympic 30/04. Cac ban hpc sinh, sinh vien, giao vien va nhirng nguoi quan tarn khac se c6 the a m tha'y thieu sot a cuon sach nay trong qua trinh su dung. Do vay, su gop y va chi trich tren tinh than khoa hpc va huang thien t u phia cac ban la dieu chiing toi luon mong dpi. H y vpng rang tren buoc duang tim toi , sang tao toan hpc, ban dpc se tim dupe nhiing y tuong tot hon, mai hon, nham bo sung cho cac y tuong sang tao va loi giai dupe trinh bay trong quyen sach nay. ChiMng 1. PhUcfng phdp sang tdc va giai phUcfng trinh, hf phUOng trinh, bd 1.1 PhiTdng phdp he so bat djnh 3 1.2 Phifdng phdp duTa ve h$ 5 1.3 Phufdng phap diTa phiTdng trinh ve phifdng trinh ham 15 1.4 Mot so phep dSt an phu cd ban khi giai h$ phufdng trinh 26 1.5 PhiTdng phdp cpng, phufdng phdp the 35 1.6 PhiTdng phdp dao an. Phifdng phap hiing so bien thien 54 1.7 PhiTdng phdp sijf dung dinh l i Lagrange 63 Nha sach Khang Viet xin trdn trgng gi&i thi?u tai Quy dgc gia va xin 1.8 Phu'dng phdp hinh hpc 69 idng nghe moi y kieh dong gop, de cuon sach ngdy cang hay hem, bo ich hon. 1.9 PhiTdng phap ba't dang thtfc 82 1.10 PhiTdng phap tham bien 95 Tac gia Thac sy: NGUYEN T A I CHUNG Thuxingici ve: Cty T N H H M p t Thanh Vien - Dich V u Van Hoa Khang Vi?t. 71, D i n h Tien Hoang, P. Dakao. Quan 1, TP. H C M ChUcfng 2. PhUcfng phdp da thiic va phUcfng trinh phdn thitc hOu ti. 11 Tel: (08) 39115694 - 39111969 - 39111968 - 39105797 - Fax: (08) 39110880 2.1 Cdc dong nha't thiJc bo sung 116 Hoac Email: 2.2 PhiTdng trinh bac ba 117 2.3 Phu'dng trinh bac bon 127 2.4 PhiTdng phdp sdng tdc cdc phiTdng trinh da thiJc bac cac 137 2.5 PhiTdng trinh phan thiJc hi?u ti 149 khangvietbookstore@yahoo.com.vn ChUcfng 3. PhUcfng trinh, bdtphUcfng trinh chiia can thiic 158 3.1 Phep the trong doi vdi phiTdng trinh 3/A(X) ± }JB{X) = 3^C(x) .... 158 3.2 PhiTdng trinh (ax + b)" = pJ^a'x + b' + qx + r 160 3.3 PhiTdng trinh [ f ( x ) ] " + b ( x ) = a(x)!i/a(x).f ( x ) - b ( x ) 168 3.4 PhiTdng trinh d i n g cap d6'i vdi ^ P ( x ) v^ ^ Q ( x ) 174 3.5 Phu'dng trinh doi xiJng d6'i vdi ^ P ( x ) vd ^ Q ( x ) 179 3.6 Mpt so hiTdng sdng tac phiTdng trinh v6 ti 184 ChiMng 4. //# phUcmg trinh, h? bat phUOng trinh 231 4.1 He phiTdng trinh doi xuTng 231 4.2 He c6 yeu to d^ng cap 253 4.3 H$ bac hai tdng qu^t 266 4.4 Phi/dng phdp dilng tinh ddn dieu cua ham so' 271 4.5 He lap ba an (hodn vi vong quanh) 277 4.6 SuT dung can bac n cua so phuTc de sang tac va giai he phiTdng trinh ... 4.7 Phi/dng phap bien doi ding thiJc 314 4.8 MotsohekhongmaumiTc 317 307 ChUctng 5. Cdc bai todn phUcmg trinh, h^ phUcfng trinh, bat phUcfng trinh trong dethidt^ihQc Chi:fc?ng 1 Phi:fdng phap sang tac va giai phifcfng trinh, he phi:fcfng trinh, bat phi:^dng trinh 328 5.1 Phtfdng trinh, bat phi/dng trinh chiJa can 328 5.2 He phiTdng trinh dai so 332 5.3 PhiTdng trinh liTdng gidc 337 5.4 Phi/dng trinh, bat phi/dng trmh c6 chlJa cdc so n!,Pn, A^, C\5 5.5 PhiTdng trinh, ba't phiTdng trmh mu 5.6 PhuTdng trinh, ba't phifdng trinh logarit 373 5.7 H? mu va logarit 387 5.8 Phtfdng phap dilng dap ham 392 368 Trong chitcJng nay ta se trinh bay nipt so phUdiig phap cO ban va mot so phUdng phap dac biet di giai va sang tac phitdug tiinh, lie phUdng trinh, bat phUdng trinh. Co mot so vi du, bai toan c6 sii dung den kien thi'tc cua plntdug trinh da thi'tc bac ba, ban doc c6 the xcni bai i)liitdiig tiiiih bac ba d chUdng 2 (chi can c6 kign thv'tc ve lUdng giac la co the hieu bai phUdng trinh bac ba) trudc khi xem cac bai toan, vi du nay. - . , • 1.1 PhifcTng phap he so bat dinh PhUdng ])hap he so bat djnh la chia khoa giup ta i)han tich, tim dudc l a i giai cho nhieu locii phUdiig trinh. Chung ta se Ian lUdt tini hieu phUdng phap nay thong qua cac bai toan va cac km y ngay sau do. Bai toan 1. Giai phiCdng trinh 2^^ - l l x + 21 - 3^4.x- - 4 = 0. Giai. Tap xac dinh D = E. Plntdug trinh da cho tu'diig ditdng vdi •^ikf •'A/nh 6" ^ ( 4 x - 4 ) 2 - I ( 4 x - 4 ) + 1 2 - 3 x / 4 : r : ^ = 0. • ^, (1) Dat t = ^4x - 4, thay vao (1) ta dUdc f - Ut^ - 2-it + 96 = 0, hay (f - 2)2(t'' + 4i^ + 12/2 ^ ^ 24) = 0. (2) Neu t < 0 thi f' - Ut^ - 24f. + 9G > 0, neu t > 0 t hi + 4r* + 12*2 + 18( + 24 > 0. 3 wid ;:,v»fb im V >. D o do (2) <=> i = 2 => X = 3. L U L U y . De c6 (1) ta can t i i i i a, (3,7 sao cho B a i t o a n 3 . Gidi phuang trinh 2x2 - l l x + 21 = IGrtx^ + (4/^ - 32rv);r + (16^ - 4 / ^ + 7) f 16a = 2 ^ { 4/3-32a = -11 I 1 6 a - 4 / 3 + 7 = 21 , , fl ^{a;(i\-i) = 7 Dap so. X = - - — - . J \ B a i t o a n 4. dung phuong phap he so bat dinh cho t a 15i giai bai toan m o t each rat tijt nhien va ro rang. Gidi phuang trinh 4 + 2\/l - x = - 3 x D a p so. PhUdng t r i n h c6 tap nghiem S — I 0; — ; — 25 B a i t o a n 2. Giai phiCcfng trinh - x + 3 = 2\/r^-/m^+3\/r^. (i) Dat u = ^/^+x,V 2v^r^ + \/l + = Vl - X {u >0,v>0), - + 1) = 0 ^ 3\/l- x2 = 0. 10x2 + 3 x + 1 (2) Thay vac (*) : • Vdi w - 1 ^ 0 • ^/3^ [ « Z I . tah s v , ;: i; j lOx^ + 3 x + 1 = ( 6 x + l ) V x 2 + 3. = i(6x (*) + 1)2 + (x^ + 3) - - = — + ^2 _ 9 4 t a duoc {u^ - 2uv) + [u - 2v) - {uv - 2^^) = 0 + 2^2 - 2u + It - 3 u v = 0 ^ ( • a - 2v){u .T - B a i t o a n 5. Gidi phuang trinh 2 + SVxTT + V l - x^. G i a i . Dat u = 6 x + 1, ?; = \/.x2 + 3. Ta c6 G i a i . Tap xac d i n h D = [ - 1 ; 1]. PhUdng t r i n h (1) viet lai n h u sau : (1 + x ) + 2(1 - x) - 5s/TTx. V3 X - l l x + 21 = a(4x - 4)^ +/3(4a; - 4) + 7 4 \ / l - x = x + 6 - 3 \ / l - x^ + 2D = 4 4 4 5, + 1,2 _ 5 = ^ i K : ^ (u - 2t;)2 = 9 <^ u - 2u = ± 3 . 3, 1ta+ C6Ox - 2v/x2 + 3 = 3 < ^ 3 x - l = \ / x 2 + 3 v 3x - 1 > 0 ^ , x2 + 3 = ( 3 x - l ) 2 ^x^l. -'^'^i'l'S • Vdi u - 2t; = - 3 , t a c6 L u ^ i y. D l CO ( 2 ) , t a t i m a, f3 sao cho - x + 3 = a ( l + x) + / 3 ( l - x ) o { ^ ; ^ ^ 3 ^ ^ { g = i Vay p h i M n g t r i n h c6 tap nghiem 5 = < 1; ——^ Doi v 6 i bai toan t o n g quat : Giai phvfdng t r i n h p{x) = as/1 Ta bieu dien p{x) - X + bVl + X + cVl - x^, X (u > 0, i; > 0 ) . K h i do dirdc phitdng t r i n h doi v d i u, v c6 thg phan t i c h ditdc. V i d u 1. , Lvfu y. Phudng phap he so b a t dinh de giai he phudng t r i n h se ditdc de cap trong phan phan t i c h t i m Idi giai cac bai toan c i i a bai 1.5 : PhUdng phap cong, phiTdng phap the (d trang 35). theo 1 - x , 1 + x va dat u = - / I + X, i; = \ / l - I. 1.2 Phifcfng phap difa ve he. Ta sc. sang tdc mM phUdng trinh duclc gidi hhng phiMng phdp he Dg giai phUdng t r i n h b a n g each dua ve he phUdng t r i n h t a thutdng dat an so bat dinh nhu sau : Ta c6 {a-b+ l ) ( 2 a - 6 + 3) = 0 ^ 20^ + 6^ - 3a6 + 5 a - 46 + 3 = 0. Tii day lay a — ^Jl + x vd b = \/l - x ta diidc 2x + 2 + 1 - X - 3 \ / l - x2 + 5VI+X Rut gon ta duac bai toan sau. p h u , p h e p dat an p h u n a y c i m g vdi phUdng t r i n h trong gia thiet c h o t a m p t h$ phitdng t r i n h . Sau day t a se t r i n h bay phuldng p h a p s a n g t a c (thong q u a cac V I d u ) , phUdng p h a p giai (thong q u a Idi giai c a c b a i toan va q u a n trong h d n - 4s/l-x + 3 = 0. n i i a la cac hru y s a u Idi giai). Cac p h u d n g p h a p s a n g t a c c i i n g nhiT phifdng p h a p giai cac phUdng t r i n h b a n g each dUa ve he con dildc de c a p r a t n h i i u 6 s a u b a i n a y ( c h a n g h a n b a i 3.2 d t r a n g 4 5 160). Lay (1) tru' (2) tlico ve ta du'dc V i d u 1. Xet I y ~ 2 ^ 3^^ ^ x = 2 - 3 (2 - 3x•'^)^ Ta c6 hdi todn sau. 2(y - x ) B a i t o a n 6. Gidi phUdng trlnh Giai. D a t , = 2 - 3 x ^ . 1 ^ CO 5(.T2 - y2) ^ x + 3 (2 - 3x^)^ = 2. ^[^ZlZ^. he Vdi y = X , thay vao (1) ta dUdc 5x2 _2x y = X 5x + 2 ^ -\=i) 1 ± \/6 ^ x = — . thay vao (1) ta du'dc • Vdi y = (1) t n r (2) t a fhrac y - X = 0 2 = - 5 ( x + y) >;V;!; o .• y = X x - y = 3{x'^ - y^) ^ X - y = Q 3(x + y) = l ^ y = x G |~^' Vc'ri y = x, thay vao (1) ta diWc Sx^ + x - 2 = 0 Vdi y = 1 - 3x 3 1 - 3x —^—• , thay vao (2) ta dudc 1 - 3x = 2 - 3x^ <^ 9x2 - 3x - 5 = 0 = -1, 2 X = 1-V21 - , 3 X = — — , 6 1 + V21 () . Lxiu y . T i r Idi giai t r c n ta thay iftng neu kliai Irien (2 - 3x'^)'^ t l i i sc dira phitdng t r i n h da cho vc phiWng t r i n h da thiitc bac bon, sau do Ijien doi thanh ^ = 5x2-l.=.25x2 + ay + 6 = 5.r2 - 1 8x - 5 («y + bf = - 4 72 1 ± v/O - 1 ± ^ { + b + i = hx'\.T + 4 - 5^2 = 5a2y2 + C fl _ I _5_ _ , i{)aby. ^+1 r , _ . Dg he tren la he doi xftng loai TT t h i < g ~ 5„2 ~ 4 _ 5/^2 => | Z 9 Vfw 10a/; = 0 ' I « - ta C O phep dat 2y — 5x'^ - 1. f\ . -jj.,., B a i t o a n 8. GidirphtMng trinh 5{5x~ - 17)2 - 343x - 833 = 0. Y tvtdng. D a t ay + b^5x^- (x + l ) ( 3 x - 2)(9x2 - 3x - 5) = 0. Vay ncu k h i sang tac de toan, t a c6 y lam cho plnMng t r i n h khong v6 nghiem h i i u t i t h i phildng phap khai trien dua ve phUdng t r i n l i bac cao, sau do phan tich dua ve phu:dng t r i n h tich se gap nhieu klio khan. l ( , x - l = 0 ^ x = - ^ = ^ ^ 25 5 5 L u t i y. Phep dat 2y = 5 x 2 - 1 chrdc t i n i ra n h u sau: Ta dat n:y-\-b = 5x2 _ ^ vdi a, h t h n sau. K h i do t i i u dUdc he 1 ± v/21 X = l PhUdng t r i n h da cho c6 bon nghic'm X PhUdiig t r i n h da cho co bon nghiem X - 17 (a ^ 0). Klii do . jay + b = 5.7:2 _ ^7 \ 5 ( a y + 6)2 - 343x - 833 - 0. (*) Tir (*) ta , , ,, • ' ' CO 5(ay)2 + lOa^y + ^2 - 343x - 833 = 0 ^ x = 5(ay)2 + 10a6y + ^2 - 833 V i d u 2. Xet mot phucing trinh bac hai c6 cd hai nghiem Id so v6 ti 5x'^ - 2x - 1 = 0 ^ 2x = 5x2 _ 5x2- 1\ 2x = 5 B a i t o a n 7. Gidi phUdng trinh 8x - 5 (5x2 _ - 1. Ta CO bdi todn sau. _ _^ G i a i . Dat 2y = 5x2 _ ^ YAn do 2y = 5 x 2 - 1 . ^ r 2y = 5 x 2 - 1 (1) 8x - 5.42/2 = - 4 ^ \x = 5y2 - 1. (2) 6 „ * ^ 5a-^y2 + i0ft2.;;.y-|.^2^j_y33^ Suy ra ax + b = + b. (**) Ta hy vong c6 ax + h = by^ - 17, ket hdp vdi (**) suy ra Hi v, . 2 5 a ' ^ y 2 + l ( ) o 2 . 6 y + 6 2 . a - 833a , o/y-l7= 46 •M! I <^343.5y2 5831 = 5a''.y2 + l()a2.6y + 62,„ f 343 = a'* Dong nhat he so ta ditcJc I al^h = 0 t--833a+ 3 4 3 6 = - 5 8 3 1 Idi giai sau. 7 - g 3 3 „ _|. 3435. r - 7 1 / = (1 "^''^^ ' ''"^ COS 7y = 5x2 _ 17 = 5x2 _ 17 1^5,y2 - 343a; - 833 = 0 J7y 7x = 5 y 2 _ 1 7 . (1) (2) x = y 5x + 5y = - 7 . 7±\/389 x = 10 . Lay (1) t r i t (2) t a c6 7{y - x) = 5(x + y ) ( x - y) ^ * Neu X = •(/, thay vao (1) : Sx^ - 7x - 17 = 0 * N i u 5x + 5j/ = - 7 , ket hdp (1) t a c6 llTT 1 llTT llTT = 4 cos-* 18 - 3 cos 18 ' 6 137r 137r 137r COS — — = 4 cos'' 18 — 3 cos 6 18 • llTT 137r TT la tat ca cac nghiem ciia phuong r = cos ——, X = C O S 18' • 18 ' 18 t r i n h (4) va cung la t a t ca cac nghiem cua phUdng t r i n h da cho. L t f t i y. Phep dat 6y = 8x3 _ ^ dUdc t u n ra n h u sau : Ta dat G i a i . D a t 7y = 5x^ + 17, t a c6 h? phitdng t r i n h ay + 6 = 8x3 - V3 ^^^^ ^jj^ ^ Ket hdp v6i phudng t r i n h da cho c6 he Ket luan: Phudng t r i n h c6 tap nghiem 5 = V i d u 3. Ta CO Ax^ - 3x = ~ <^ Qx = 8x^ fSx^ { >1296x + 216v/3 = 8 (Sx^ - ^/fj Ta CO hai todn ' 7 ± \/389 - 3 5 50 10 - V^. - v/3\ ay + 6 = 8x3 - v/3 162x + 27V3 = a3y3 + 3a26y2 + Safety + ^3 ± 5 v ^ l J • 73 8 Can chgn a va 6 sao cho : 162 o? 27\/5 - 63 3a26-3a62 = 0 Vay t a c6 phep dat 6y = 8x3 _ ^ Vdy xet V i d u 4. Xet tarn thiic bdc hai luon nhdn gid tri duang : x'^ + 2. Khi do -V3 ^ 162x + 2 7 ^ = (Sx^ - Vs^ / . x^ + 2) dx = — + 2x + C. ^ 3 x3 Chi cdn chon C = 0 ta diicfc mot da thiic bdc ba dSng bien la h{x) = — + 2x. sau. B a i t o a n 9. Gidi phUOng tnnh 162x + 27\/3 = (8x^ - s/zf Ta CO /i(3) = 15. Vdy ta thu duoc mot ham so da thiic bdc ba dong bien g{x) x3 vd thod man g{2>) = Q la g{x) — — + 2x - lb. Ta se tim mot da thiic bdc ba . G i a i . Dat 6y = 8x^ - \/3. Ta c6 h ^ r 6?y = 6y - 8x3 - v/3 162x + 27\/3 = 216?y3 dong bien k{x) 8x3 _ ^ 3 \x = 8?y3 - v/3 (2) o sao cho k{x) = x <^ g{x) — 0, muSn vdy ta xet ^ + a x - 15 = X 0 nen 8 (x^ + xy + y^) + 6 > 0. Do do t i t (3) t a dUdc x = x3 ^ A;(x) = y tuang dudny U(H .• — + 3x - 15 = y <^ x^ + 9x - 45 = 3y. TH phiiang Thay vao (1) t a dUdc tnnh cuoi ndy thay x bdi y ta thu duac he doi xilng loai hai 6 i = 8x3 - \/3 <^ 4x3 - 3x = ^ a x3 + 9x - 45 = 3y y3 + 9y - 45 = 3x. 4^3 _ 3^, = cos ^ . Til: he tren, sU dung phep the ta thu duoc phuong a Sii di^ng cong thiic cosa = 4cos3 - - 3cos - , t a c6 x3 + 9x - 45 cos-=4cos 1 8 - 3 C O S - , 8 I + 9 trinh /x3 + 9 x - 4 5 \ 9 - 45 = 3x 0fiOJ i.' <^ ( x ' ' + 9x - 4 5 ) ^ + 81 {x^ + 9x - 45) = 1215 + Vay ta thu dUdc hai todn sau. Bai Gidi t o a n 10. phuong 81x. G i a i . D i c u kiCm , hdp v d i phUdng t r i n h d a cho, t a c6 he | ; V trmh (x^ + 9 x - 4 5 ) V 8 1 D a t y = l o g n ( l O x + 1), Ivhi d o I P = 1 0 x + 1. K e t > JJy ^ L a y (1) t n r (2) tlieo ve t a d i t d c (x^ + 9 x - 4 5 ) = 1215 + 81X. (1) I F - ir^ = ^ ^ | "» " • - - w., • f-: , A. l O y - l O x <^ ir^ + • lOx = I P + lOy, (3) ..ji G i a i . T a p xac d i n h M. D a t :^i^:,^.^7j • • . - • / x^ + 9 x - 45 = 3y (2) \ + 9 7 / - 45 = 3x. (3) J X e t h a m so / ( / ) = 1 1 ' + 10/.. T a c6 f'{f.) + 9 x - 45 = 3y. K e t h d p v 6 i (1) t a c6 he ^ = 1 1 ' h i 11 + 1 0 > (1) t a d i f d c I L ' ' = 1 0 x + 1 <^ I F - (4) X e t h;\ so f / ( x ) = I F - lOx - l O x - 1 =0. .• 1 tren khoang ( ; +00 V Lay (2) t i i f (3) t h e o v e , t a dUdc x^ - g\x) + 9 x ^ 9)/ = 3?/ - 3 x ^ . ^ ( x - y)(x^ + xy + - x = Vi 6. Ta se su dung trinh = 0, y ( 0 ) = 0 n e n x = 0 v a x = 1 l a t a t ca phuang phdp doi xvtng loai hai. Xuat the ta diMc phuang c6 cac lap di phdt sang tit \^^Z tdc phuang tit. he •^^ ^^'^'"'(1 P^^'^P ^^^4^30 trmh Ax = ^ 3 0 + | v / x + 30. Til phuong trinh nay ta lai, trinh thu dtWc he doi xtCng loai liai do x^ + 9 x - 45 = ay a'Uj^ + Slay = 1215 + S i x ^ ; | xj^ + 9 x — 45 = ay_ \ + Slay - 1215 = 8 1 x . , . , «3 81« 1215 81 „ „ Vay can chon a thoa m a n dieu kien — = — = — 7 - = — => a = J . U o cto ••' • 1 9 45 a d a t x^ + 6 x ~ 45 - 3 y , t a so t h u ducJc m o t ho d o i x i ' m g l o a i h a i . V i d u 5. Chon mot phMng phiCOng trinh trmh nay trmh chi c6 hai nghiem / d 0 v d 1 IdlV = ^ - 10 quay 4x = l o g , , ( l O r + 1) logiiuux+ij. Ta CO bdi todn Bai ra I F = l O l o g n ( 1 0 x + 1) + 1 ^ - v / a M ^ 4x= -v/^r+30. ^/30+ .Zi the ta thu diMc phuang trmh = \ sau. t o a n 12 ( D e n g h i O l y m p i c 3 0 / 0 4 / 2 0 1 0 ) . Gidi phuang I F = 2 1 o g i i ( 1 0 x + 1)^ + 1. Ta c6 bdi sau. t o a n 11. Gidi phuang trinh IV = 2 1 o g i i ( l O x + 1)^ + 1. 10 nst,- lOx+1. ta thiel lap mot he doi xvCng loai hai, sau do lai ^ I y = l o g n ( l O x + 1) ^ \ l F = 1 0 y + l 4M = ^ / 3 0 + Tir he niiy, ticp tuc s'li dung phcp nhu sau : f l l ' - = lOy + 1 \ i r y = 10x + l Bai du phuang x^ + 9 x - 45 = ay ( v d i a t i n i s a u ) . todn 1 ^ - j ^ ; + 0 0 j , s u y r a d o t h i cvia n g h i e m c i i a ( 4 ) . N g h i e m c i i a p h U d n g t r i n h d a cho l a x = 0 v a x = 1. 3. LuTu y . P h c p d a t x^ + 9 x - 45 = 3y dittfc t u n r a u h u s a u : T a d a t Suy J h a m g v a t r u e h o a n h co v d i n h a u k h o n g q u a h a i d i e m c h u n g , s u y r a (4) P h i l d n g t i i i i h d a c h o c6 n g h i e m d u y n h a t x = 3. Tii . T a c6 10 = l F ( l n l l ) 2 > 0. / k h o n g q u a 2 n g h i e m . M a g{l) x^ + 9 x - 45 = 3 x <=> ( x - 3) (x^ + 3 x + 15) = 0 ^ ve phuang 10, g'\x) V a y h a m so g c6 d o t h i l u o n 16m t r e n k h o a n g T h a y vao (2) t a d i W c Khi = ll-^nll - + 1 2 ( x - ?y) = 0 + 12) = 0 <^ X = y. 0, V/. G K . V a y h a m so / d o n g b i e n t r e n E . M a (3) c h i n h l a / ( x ) = / ( y ) n e n x = y. T h a y vao 4x = 30 + - W 30 + - ^ 3 0 + - \/^T30. 11 trinh 4 G i a i . Do x la n g h i f n i t h i x > 0. Dat u = 30 + --^x + 30, t i t phitdng t n n h da cho t a c6 ho 4u = J 3 0 + -y/xT3Q (1) 4x = A / 3 0 + - v / w + SO. Gia s\t X > u. (2) Dat . = \V^FT30, t i t (2) ta c6 he | ^J I (3) V. Nhit vay dang nay la j)hn'dng t r i n h vo t i , infi san k h i dat an phu dita ve he, r o i dimg phep the dan t d i phudng t r i n h da thitc, do do k h i sang tac de toan t a phai dac biet chii y cac chi so can. Chang han d v i d n 7 t h i m = n = 4 nen t a yen tam rang se dan tdi phitdng t r i n h da thite bac 4 co i t nhat m o t nghiem dep. B a i t o a n 14. Gtdi phiMng trinh + 30 > ^ 3 0 + - \ / u + 30 = 4x =^ u > x =^ x = ?i. Vay t i t he (1) t a c6 a; = u va 4x = ^ 3 0 + - y x T S O . Gia sii x> - JJx) + "\/b + / ( x ) = c, t a co each giai : Dat u = 'ija - fix), v = '^h + f(x), dan den he { ^ H ^+,n"s'^['^ V i d u 8. Vd'i. x = - 2 thi 2 0. K h i do 5 I"2 — Z t'0, 7 ^ox + Sv'^ = 5(3x - 2) + 3(6 - 5x) = 8. . M a t khac t a lai co 2u + 3r - 8 = 0. Vay t a co he K h i do 4v = Vx + 30 > VtTTSO = 4a: 4u > 4a: =^ V > a; =^ u = X . , f T> 0 Vay r = .x va 4.T = ^ | J g p ^ ^ ^, ^ 30 1 + 71921 t r i n h da cho co nghiem d n y nhat x = 32 1 + \/l921 — . PhUdng {^t +'fv= 8^ =^ + 3 ( ^ ^ ) ' = 8 ^ 15..^ + 4^2 - 32z. + 40 = 0 Phu'dng t r i n h nay c6 nghiem d u y nhat u = - 2 nen v'Sx - 2 = - 2 B a i t o a n 1 5 . Giai phiMng X- = -2. trinh V i d u 7. Vdi X = 8 thi ^/x-\-8-\- \ J x - l — 3, ia c6 bai todn {ch&c chan co mot nghiem 1 + \ / l - x 2 [ V ( l + :r)-* - ^ ( 1 - x)'A^ =2+ dep x = 8) sau. B a i t o a n 1 3 . Giai phUdng trinh y/x + 8 + \/x - 7 = 3. G i a i . Dieu kien - 1 < x < 1. D a t ^l + x = a, \ / r ^ G i a i . Dieu kien x > 7. D a t u = ^x + 8 > 0 va u = v ' x - 7 > 0. T a c6 he u + r = 3 {V = — u ^ i 0/2--,,2)(„2 [u,(;>0 U.,V>{) { 2) u4-t;'*-15 u = 3- u < 3 u2 + (3 - uf 3 ^ \ 4 u ^ - 18u2 + 36u - 32 = 0 T i t do t a t h u dudc 1 = 2 ^ ro < ^ \ <=>{^ u< 3 = 2 - + f=p ^ x = 8 (thoa m a n dieu kien). Vay phitdng t r i n h da cho co nghiem d u y nhat x = 8. 12 s/lT^=-^{a vdi a > 0, 6 > 0. S "" ' (2) , + b) [do a,b>0). V2 Ket-hop (2) t a co ft; = 3 - u = K h i do a' + l? = 2. T a co he sau ( \ . . \l + ab{a-^ -b^) = 2 + ab. (1) =^ {a + bf = 2+ 2ab^ ,,2)^15 0 < i< < 3 ^ ' ( 2 u - 3 ) ( 2 u 2 _ 6 u + 9) = 5 = 5 ^ ro < 1/- < + ffif, t uM.. yjl - xK ' . ,| 1 1 ' ( -7= (a + b){a - b){a^ + b^ + ab) = 2 + ah => ^ ( a ^ ~ h'-) = I. v2 v2 T i t do t a c6 he | ~ ^2 ! l 2 ^ ' Cong hai phitdng t r i n h ve theo ve t a co 2 a 2 - - = 2 + y 2 ^ a 2 = l + 4 = ^ l + a ; = l + ^ ^ x = 4=V2 s/2 V2 , Vay phitdng t r i n h co nghiem d u y nhat x = — . 13 vj / j B a i t o a n 16. Gidi phuang irinh 1 -x + \/\/2 = n + J' = v2 G i a i . D i n i kien 0 < x < \/2 - 1. Dfit \/\/2 - 1 - x = u va ^ 0 < u < \/s/2-l va - 1. N h i t vay t a c6 u = —^ ( .u:^ + v^ = v/2 - 1 1 - < 8 - vfei 8+ h"^^ 18 he 8 Vay w, f la nghiem ciia V 8 + 1 Ttr phudng t r i i i h thi'i: hai, ta co - vfei 18 •, yi94 18_ 3' 18 nen nghiem duy nhat ciia phudng t r i n h la + 7-4 = v / 2 - 1 . V ^ = v. K h i do 71 + 7) = - !i i = 0 (1) = 0. (2) Do (2) v6 nghiem / -2 + ^ 2 ( 7 1 9 4 - 6 ) + ^ ^ / 1 2v \ v/2 v/2 + i ; ' = \/2 - 1 + 1.3 1.3.1 Phifcfng phap difa phifdng trinh ve phifdng t r i n h ham Phu'dng phap giai. Dita vao ket ciua : Neu ham so y = f{x) - 3 1 ± ,72 B a i t o a n 17. G'jdv phifdng trmh G i a i . Dieii kien | 2 « > 0, v < - . v / l - -x^ = Q r 1 - x^ = 1 - « 4 Do do 1(1 \2 2u..v . Ta CO he I [(i7,+ r ) 2 - 2 n . r 2 3 W+ U= - 2u2.,;2 ^ 1 9 14 - t a CO the sang tac va giai dUdc nhien phitdng t r i n h hay va kho, thudng gap trong cac k}' t h i hoc sinh gioi. D6 van dung dildc phitdng phap nay, t a thirdng bien ddi phiWng t i i n h da cho thanh phitdng t r i n h ham f {{x) = tpix). De giai dUdc cac bai toan bang phitdng phap nay t h i nhftng kien thifc ve ham so nlut dao ham, xet sit bien thien va k l nang doan nghiem la cite k i ciuan trong, c6 nhitng bai doan dUdc dap so la da hoan t h a n h den hdn 90% Idi giai. Phitdng phap nay ditdc si't dung nhien, chang han d muc 3.6.3 d trang 191. M o t so tritdng hdp dac biet thitdng gap : • Neu / la ham ddn dieu tren khoang (a; 6) t h i plutdng t r i n h / ( x ) = k {k la hang so) CO khong qua 1 nghiem tren khoang {n; h). B a i t o a n 18 ( H S G Q u a n g N i n h 2 0 1 1 ) . Gidi phieang tnnh u.v 2u^.i)-^ vdi a; = 7^ • Neu f yk g \h hai ham ddn dieu ngitdc chieu tren khoang (a; b) t h i phitdng t r i n h / ( x ) = g{x) c6 khong qua 1 nghiem tren khoang (o;6). • Neu t a thay cum t i t " / la ham ddn dieu tren khoang (a; 6)" bdi cum t i t " / la ham ddn dieu tren m5i khoang (a; 6), {c]d)" t h i hai ket qua d tren se khong dung, ti'tc la plutdng t r i n h co thc^ sc c6 nhien hdn mot nghiem. Ban doc hay xein bai toan 20 d trang 16. - («2+t.^)^-2»^.7>2^1 - - - ddn dieu tron khoang (a; b) va (a; b) t h i /(^) = /(y) - ^' <^ 0 < x < 1. Dat u = sji: va v; = ^ - U + V= I x,ye 1 =0 81 v^5x 15 =7 + 1 v / ^ ^ = 0. (1) Vay p h U d n g t r i n h da cho c6 t a p n g h i e m l a 5 = {3, - logs 2}- G i a i . D i c u k i c u ^ - > T- K l i i d o o (1) ^ (5.T ^ 6)2 - - f{5x-6) • ,; ^ , , hiiu y. X e t h a m so /(x) = 5 ^ . 8 " ^ , Vx ^ 0. K h i do / ( 3 ) = 500 v a ,. ' 1 ^ = 5 ^ 8 ^ . In 5 + ^ . 5 ^ 8 ^ . In 8 > 0, Vx ^ 0. fix) v d i f{t) = = f{x), (2) t^- ' " Suy r a h a m s6 / d o n g b i e n t r e n m o i k h o a n g (-CXD; 0 ) , (0; + 0 0 ) . T u y n h i e n n l u k e t l u a n 3 l a n g h i e m d u y n h a t c i i a p h U d n g t r i n h t h i se m i c p h a i sai l a m . T a c o f'{f) = 2t + _ 1 , > 0,V/, > 1. Vay / doug bieu t r c u (1; +oo), t h i p h U d n g t r i n h / ( x ) = k {k \h h a n g so) c6 k h o n g q u a 1 n g h i e m t r e n {a; b)". 2v/rn:(t-i) tit (2) CO 5x - 6 = X Vay t a c a n n h d c h i n h x a c k e t q u a " N e u / l a h a m d d n d i e u t r e n k h o a n g (a; b) b) T u d i i g t i t c a u a ) . X = 1,5. Phifdng t r i n h c6 nghiem duy nhat x = 1,5. B a i t o a n 19 ( H S G L a m D o n g , n a m h o c 2 0 1 0 - 2 0 1 1 ) . Gidi phuang trinh G i a i . Dieu kien x > 1. Dg thay x = 1 khong l a nghigm ciia phirong t r i n h nen B a i t o a n 21 ( C h o n d o i t u y e n N i n h B i n h n a m h o c 2 0 1 0 - 2 0 1 1 ) . phuang trinh Xet ham so / ( t ) = s/T+G + = - + + \ / x - 1 = 7. (2x^ - , + 2t + = 7 nen phitdng t r i n h d a cho c6 nghiem duy nhat la x = 2. -.fu.:': h) 3 ^ 8 ^ X ( l + ilog52) 2^ ^ X . X logs 2 [(x^ + 2x) - (2x^ - hix) = k[g{x) = 3 , log3 X sao cho + 2)] =^k = l. ^ fix)]. , =0 2x- 1 , = 3 x 2 - 8 x + 5. (x - 1)^ G i a i . Dieu kien 0,5 < x 7^ 1. K h i do (1) titdng ditdng „ X = - logs 2. logs (2x - 1) - log3 (x - 1)2 = - (2x - 1) + 3 (x - 1)2 + 1. trinh (1) =• (2) ^ l o g g (2x - 1) + (2x - 1) = log3 (3(x - 1)2) + 3 (x - 1)^ 4»/(2x-l) = /(3(x-l)2), 16 -2 1. Phitdng t r i n h da cho c6 hai nghiem x = - 2 , x = 1. = 500 ^ 5 ^ 2 ^ = 5^2^ ^ 5 - ^ 2 ^ - 2 ^ 1 ^ 5 ^ ^ - ^ ^ logs ( 5 ^ - ^ 2 " ? ' ) = logs 1 ^ logs 5 " " ' + log5 2 ^ = 0 l + -logs2 = 0 X = X = B a i t o a n 22 ( H S G T h a i B i n h n a m h o c 2 0 1 0 - 2 0 1 1 ) . Gidi phUdng = 1 x-3 = 0 = 3 * h i 3 + 1 > 0 , G E . Vay + 2 = x^ + 2x <^ x^ - 3x + 2 = 0 <^ ,^ —log52 = 0^(x-3) , , ( 3 ) t h i t a dung plntdng phap he so bat (Hnh u h u tren de difa vc Giai. + h)\ Con phitdng t r i n h t6ng quat a-^(^) - a^^^) = h{x) dUdc giai tUdng t u . T h u d n g = 36. a) Dieu k i c u x 7^ 0. K h i do (2) + 2) = / (x^ + 2x) , v d i /(<) = 3* + i . - (x^ - 3x + 2)=k 5 ^ 8 ^ = 500 ; ^ x - 3 X L i r t i y . Phep phan tich (2) difdc t i m r a n h u sau : T a can t i m ' B a i t o a n 2 0 . Gidi cdc phiCcfng trinh 5 - 8 ^ (1) tiino H a m so / d o n g b i e n t r e n E v i f'{t) > 0 , V i > 1. Do do ham so n a y d o n g bien. Suy ra (*) c6 khong qua mot nghiem, mat khac a) Gidi ' ' ^ 3 2 x 3 - x + 2 ^ (2x^ - X + 2) = 3 ^ ' + 2 . ^ (^3 ^ 2x) (*) + yTH:,Vi > 1. K h i do 1 I s/^Te (3) ^ 2x^ - f{2) + x^ - 3x + 2 = 0. 3^'+^^ 3 2 x ^ - . + 2 _ 3 x 3 + 2 x ^ _ (23,3 _ ^ + 2) + (x^ + 2x) + x^ = 7 - v / ^ ^ ^ f'(t) 32^'-^+2 - G i a i . PhUdng t r i n h (1) viet lai ta chi xet x > 1. T a c6 ^/^T6 '' vdi fit) - log3 t + t. THLT VIEN TiNHBINHTHUAN r-\^ A 1 AQ , , A A (3) Vi /'(/:) = — ^ + 1 > 0 , /.In 3 \2 ^ Dong nhat he so vdi ve trai cua (1) ta dildc > 0 neii / (long bicii ticii (0; +oo), tit (3) c6 2x - 1 = 3(a; - 1)' <^ 3x' - 8x + 4 <^ x e | 2 , ? | o 2 -12u = -36 6w2 _ 1 = 53 -u^-u + 5 = -25 (thoa di^u kien). Tap nghiem ciia phudng trinh (1) la 5 = | 2 , Lifti y. Pliep phan t i d i (2) dUcJc tini ra nhit sau : Ta can tini n, ft, 7 sao cho 3x-2 - 8x + 5 = a (2x - 1) + ( x - - 1)2 + 7 (3x - 5)2 + 3x " 5 = 9x2 _ 28^; + 21 = sjx - 1. 3 2 Giai. Dicu kien x > 1. Neu fix) X - 1 + v/x - 1 ': : ^ / ( 3 x - 5) = / ( ^ F ^ ) , v6i J{t) = e + t phap he so bat dinh nhir tren dfi dua ve mot trong cac tnrdng hdp <^3x - 5 = \/x - 1 ^do ham / dong bien tren ( - - ; + 0 0 ) j + a'=g(x) + k, hix) = fix) - a'gix) Bai toan 23. Giai phucing trinh + k, /i(x) = k[gix) - Sx^ - SGx^ + 53x - 25 = \/3x - 5. fix)]. (1) Giai. Ta CO (1) ^ 8x^ - 36x2 + 54x - 27 + 2x - 3 = 3x - 5 + ^ + X - ^ ^ 1 ^ fii - 3x) = fi^/^i^) X G {2; ^ - ^ ^ } . r 4 - 3x > 0 1 (4-3xf = x - l L i f u y• Bai toan nay con c6 each giai khac, dirdc de cap 6 bai toan 8 d trang 163. • Do ve trai c6 bac 3 con v6 phai co bac - nen ta can difa 2 ve ve bieu thiic o dang fit) = mt'^ +nt. De y rang hang tit v'3x - 5 6 ve phai c6 bac thg,p nhat nen no tiidng ling vdi nt trong / ( i ) , vay n= 1. Lifu y r^ng 8x^ = 8(x^) = (2x)^ nen d day ta phai xet 2 trUdng hdp, m = 8 hoSc m = 1. Ngu m = 1 thi fit) = t^ + t. Do do can dita (1) ve dang (2x - M)3 + (2x - u) = 3x - 5 + ^3x - 5 <^8x^ + x2(-12u) + x(6i/2 -l)-u^ 18 -u + 5= \/3x - 5. = 2 (^thoa a: > (vdi fit) =t^ + t) 4 - 3x = i / x - 1 ^do f { t ) dong bien tren ( - ^ ; +c)o)^ • ^ (2x - 3)^ = 3x - 5 <^8x^ - 36x2 + 51x - 22 = 0 <^ X (1) <^ (4 - 3x)2 + 4 - 3 a ; = x - l + \/x - 1 (2) Vi fit) = 3/2 + 1 > 0, Vi e R nen / dong bien tren R, vay tir (2) ta c6 2x - 3 = r 3x-5>0 5p = ^ 1 (3x - 3 1 Neu l < x < - = > 4 - 3 x > - - thi N/3X^ (2x - 3)^ + 2x - 3 = 3 x . - 5 + s / 3 ^ ^ ^ / ( 2 x - 3 ) = / ( s y 3 ^ ^ ) , vdi/(«) = ^ (1) 1 3x - 5 > - - . Ta co Dang tong quat log„ —T-T- = h(x) dUdc giai titdng tu. Ta thitcing dung phudng h{x) = -fix) *' s Vay trifcJng hdp rn = 1 da cho ket qua, do do khong can xet m = 8. • Nhiing budc phan ti'ch tren nhhi tuy dai nhung khi da quen roi tlii ta c6 t h i tinh rat nhanh. Tuy nhien, trong mot so bai toan, ham f { t ) khong dong bien tren R nhitng ta co th^ chi can xct ddn dicu tren mien xac dinh D. Bai toan 24. Giai phuang trinh ( ft = 3 ( a=-l => ^ 2Q - 2/3 = - 8 ^\ S [ -a + ft + 'y = 5 I 7 = 1- : u = 3. 4 " - 3 ,25±\/l3. 18 «>x = 25 - \ / l 3 18 Vay (1) CO tap nghiem S'= {2; ^ ^ - j ^ ^ } . Lxiu y. Ta xay dvtng ham fit) = mt^ + nt. Dg y rftng ha,ng t i i v/x - 1 d v6 phai CO bac thap nhat nen n = 1. V i 9x2 _ 9 (^.2^ ^ l.(3x)2 nen ta phai xet 2 tritdng hdp m = 9, m = 1. • Ngu m = 9 t h i / ( / ) = 9/2 + / . Can dua (1) ve dang 9(x - 'u)2 + X - u = 9(x - 1) + v/x - 1 -»9x2 + x ( - 1 8 u - 8 ) + t i 2 - u + 9 = v ^ x - 1 . 19 Dong nhat he so t a dUdc: <=>/(3x - 3) = / ( ^ / 9 ( : : 3 x 2 + 21.7?T5)) vdi i{t) / - 1 8 w - 8 = -28 ^ j i i = — ^ 3 x - 3 = > y 9 ( - 3 x 2 + 21x + 5) o f . . s = + 27t. 3x^ - 6x2 _ ^^ - 8 = 0. Day l a phUdng t r i n h da thUc b a c 3, dUdc do cap d b a i 2.2.1 (d t r a n g 117). • Neu 771 = 1 t h i fit) L i i u y. N h a n 9 cho 2 ve c i i a (1) t a dUdc = t'^ + t. Ta can diia (1) ve dang (3x - uY + 3x <^9x^ + x{-6u Dong nhat he so t a duoc | — u = {x — 1) + \/ X — 27X''' - 54x2 1 - u + 1 = y/x - 1. + 2) + ^^^1'^^'^'^" = 5. Den day c6 le bai - 30x + 25 + 3x - 5 = ^/(3x-5) = /(\/^^), L u u y rang f{t) X ^27x^ - 1 + \/x - 1 v d i / ( ( ) = f2 + i . + x 2 ( - 2 7 7 / + 27) ^f^^ "2*^J " "f^ la - + 3 i . Viec nhan trinh (1) Hu'dtng d a n . N h a n 9 vao hai ve ciia (1) t a d\tdc ( 3 x - 3)'^ + 27(3x - 3) = 9 ( - 3 x 2 + 21x + 5) + 2 7 ^ 9 ( - 3 x 2 + 21x + 5) 20 -2777 - 9 <^ 77 = 3. 7x^-45 = ' -153 chi ddn gian la k h i i mau so. 1627/2 + 27 ^ 3 = 7i = 4. 3x^ - 6x2 _ 3^. _ 17 = 3 y 9 ( - 3 x 2 + 21x + 5). 45) wi-iart.R ciing c6 the Si'ssi;: 30.7:2011 ^ 4^^,^2010 ^ 30y4022 ^ 4y2012 3 1 K i e m t r a lai : T a c6 x < - <;=^ 4 - 3 x > do do chon u = 4. Den day bai z z toan mdi thuc sU dudc giai quyet. N h u vay t a can l i n h , h o a t t r o n g viec xay dung ham so, nhat l a doi vdi ham bac chan. Ta cuiig c6 the giai bai toan tren bang each dat ^x-\ 3?/ - 5 de dUa ve he doi xi'tag loai 2. B a i t o a n 25. Giai phuong 2777 - B a i t o a n 26 ( D e n g h i O l y m p i c 3 0 / 0 4 / 2 0 1 1 ) . Giai he phxcang + 77, + 1 = V x - 1. ^ - 3x'' = Q(3.X)^ (trudng hdp nay t h a t ra hiem gap). N h u vay f{t) ( u - 3x)2 + u - 3 x = x - l + \/x - 1 D6ng nh^t he s6 : { (2) Co the ban doc se thac mac t a i sao lai nhan 9 ma khdng phai la so khac. T h a t r a dieu nay da dUdc de cap den roi. K h i xay dUng ham f{t) = mt^ + 3t, ta thudng nghi t d i 3x'^ = 3(x^) nen cho m = 3 ma quen r i n g ngoai ra con c6 3 Con 1 < x < - t h i sao ? L a i de y rang ham so + x ( ~ 6 7 / - 4) + + (-u^ f -2717 + 27 = - 5 4 D5ng nhat he so t a dirdc : { 9u'^ - 108 = - 2 7 I bac 2 cung c6 cai hay cua no, do l a ( - / ) 2 — t r e n , dua vao he so bac cao nhat l a 9, t a chi m d i xet t = 3 x - u nen bay gid t a se xet i = u - 3 x . Can dUa (1) ve dang ^Six^ + x ( 9 7 7 2 ~ 108) = 2 7 ^ 9 ( - 3 x 2 + 21x + 5). (2) 1 1 ( - o o ; - - ) , hon niTa \/x - 1 > 0 > - - . N h u vay t a chi c6 1 3 khi 3 x - 5 > - - < ( ^ x > - . ' (3x - uY + 27(3x - 77) = 9 ( - 3 x 2 + 21x + 5) + 27 V 9 ( - 3 x 2 + 21x + 5) = t^ + t chi dong bien tren ( - ^ ; +oo) v a nghich bien tren (2) 4^ 3 x - 5 = 153 = 2 7 ^ 9 ( - 3 x 2 + 21 + 5 ) . Do bieu thiJc chita can c6 he so la 27, hang t i i bac cao nhat la 27x^ = (3x)^ nen t a se difa hai ve ciia (2) ve dang f{t) = t^ + 27t. Ta ])ien doi (2) thanh : toan d a ditdc giai quyet, nhung that r a "chong gai" con ci phia trudc. (1) <^ 273. , (8x^-^3) ^ , ,,, f;,? trinh „ (2) G i a i . T h a y 7/ = 0 vao he thay khong thoii man, vay chi xet y 7^ 0. T a c6 ™2011 „ ( 1 ) ^ 3 0 . ^ + 4 . - = 30^2011 + 47/. (3) y y Xet ham so f{t) = 30<20ii ^ 4^^^^ ^ y/^^^ ^ 3 0 . 2 0 i u 2 0 i o + 4 > Q nen ham / dong bien tren M . Do do t i f (3) t a c6 ' x \, , X f ( - ) = f i y ) ^ = y^x \y/ y o = y\ Vay (!) <^ x = 7/2. T h a y vao (2) t a dUdc 162x + 2 7 V 3 = (Sx^ - ^ / 3 ) ^ 21 • Mki ... ( • (4) Then l)ni loan 9, cJ trang 8, cac nghiem cua (4) la cos-;^. cos^-i^. c o s i ^ lo 18 18 117r 137r TT Do cos —— < 0, ('OS - — < 0 nen ta chi nhan nghiem x = cos — . Cac nghiem 18 18 18 cua he phuring trinh da cho la ^ , Xet ham so dac trung f{t) = t'^ + t, t eR. Ta co / ' ( i ) = Si^ + 1 > 0, vay ham so dong bien tren R nen (*) <=i> a + 1 = Ta co he sau : Sii dung phep the ta co ^3 _ ; _ 1)2 = l<=^ 6'^ - 362 + 66 - 4 = 0 o ;; . (6 - ^^(,^2 _ 26 + 4 ) = 0 ^ 6 = 1. Tit do X = - 1 . Vay phUdng trinh co nghiem duy nhat x = - 1 . Bai toan 27 (De nghi Olympic 30/04/2011). Gim phuang trinh I' 1.3.2 V i du 1. Xudt phdt tic mot phuong Hu'dng dan. Xet, /(,;:) = Vx'-^ + 3.7-2 + g^. _ 13 _ ^^332^. Ta c6 6^. + 0 3.7:2 + 2\/jr'^ + 3a;2 + 9.7: - 4 Vay ham / rlong bion trcn 13 + 1 v/3 - 2x , mat khac / / 4 \ \/543 - \/27 Uj ~ cd each gidi rat cd ban, do la : 0(7--^)=0(6x-5) ^ 7 - ^ - ' + 6 log7 7^-^ - (6x - 5) + 6 logy (6x - 5) nen - la 3 9 trinh -'..^ - r x - i = 6x - 5. Xet mot ham. so dOn dicu (j){t) = t + 6 logy t. Khi do 04 > 0, V x e Phu'dng phap sang tac bai toan mdi. ^r~^ + 6 (x - 1) = (6x - 5) + 6logy (6x - 5) ^7^-^ = 1 + 2 logy (6x - 5)^ . i : '-ih ; / •[ ', , , , , , , nghiem duy nhat cua phitdng trinh da cho. x < ^, Ta duoc bai toan sau. Bai toan 28 (De thi hoc sinh gioi cac trtrdng Chuyen khu vu'c Duyen Hai va Dong Bang B a c B o nam 2010). Gidi phiMng trinh Bai toan 30. Gidi phuang trinh 7'^''^ = 1 + 2 logy (6x - 5 ) ^ (1) Giai. Dieu kien .x > | . Ta co 6 2.x'* - x2 + v/2x^ - 3 x + 1 = 3 x + 1 + v/x2 + 2 (1) <^ 7^-^ - 6 logy (6x - 5) = - 6 (x - 1) + (6x - 5 ) . Hu'dng dSn. Tap xac dinh D = M. Bien d5i phildng trinh ve • (2) 7-^-^ + 6(x - 1) = (6x - 5) + 61ogy (6x - 5) 2x'* - 3x + \/2x'-^ - 3 x + 1 = x2 + 1 + \/x2 + 2.. <^ 0 (7^-1) = (;6 (6x - 5), vcii (/>(«) = « + 6 logy t. Xet ham ,s6 / ( / ) = t + 0, ^t > 0. Vay (4) 7^-1 = 6x - 5 <(=> 7"^-' - 6x + 5 = 0. Cach 1. De thay r i n g x = 1, x = 2 thoa ( 4 ) . Xet ham f{x) = T'^ - 6x + 5 tren R. Ta co / ' ( x ) = 7 ^ - i l n 7 - 6; / " ( x ) = 7^-^ (In7)^ > 0, Vx G R. Vay ham / CO do thi luon luon loin nen cat true Ox tai khong qua hai diem, suy ra (4) CO khong qua 2 nghiem. Vay x = 1, a; = 2 la tat ca cac nghiem cua (4). Phitdng trinh (1) co tap nghiem la 5 = {1,2}. f, ^ Bai toan 29. Gidz phuang trinh (x + 5)\/x + 1 + 1 = \/3x + 4. Giai. Dicu kiOn x > -1. Dtlt a = Ham so 0 dong bien tren (0; + 0 0 ) vi (i>'{t) = 1 + (3) _^ Cach 2. Ta co / ' ( x ) = 0 " ^ Cong ve fheo ve ta co 7^-' = — x = xo = 1 + logy (6. logy e). V i v6i moi x G R thi / " ( x ) > 0 nen suy ra / ' la ham dong bien tren R va a'* + 30.2 _^ 2 = 1/ + ( a + I ) ' * + (a + 1) = 22 + b. (*) / ' ( x ) < 0, Vx G ( - 0 0 ; xo) ; / ' ( x ) > 0, Vx € 23 (XQ; + 0 0 ) . , Vay ham / nghich bien tren {-OO;XQ) va dong bien tren ( x o ; + o o ) , do do (4) C O khong qua 2 nghiem. Vay x = 1, a; = 2 la l a t ca cac nghiem ciia (4). PhUdng t r i n h (1) c6 tap nghiem la S = { 1 , 2 } . L i f u y. Phep phan tich (2) diidc t i m ra nhil sau : Can chon a, /3, 7 sao cho l=a(x-l)+/3(6x-5)=.{ - + 6/?j0^ " ' a-''^^) + klog^gix) = ^(x) - / c / ( V ) , (doi vdi (*)) f /^V^ = 3' + t. Tic phuang 1\ l-x = 2x^-2x-l. - 2x 1\ 2x (v/3) ^ - 2x ( - B a i t o a n 31. Gidi phudng trinh = 2x2 _ 2x - 1. Hu'dng d a n . T i f d n g t i f bai toan 21 d trang 17. PhiTdng t r i n h c6 hai nghiem l + V^ - — , l-\/3 X = — - — . V i d u 3. Xet ham so nghich bien tren khodng (0; + 0 0 ) la f{t) Tii phuang trinh ham j (\{x \ logi (\{x-\f = x'^ ~ ISx - 3 1 . + in 9x2 + (ix + 3126 ^,^3^^^ . '• ' vdi / ( i ) = - 'Ux / = l o g i t — t. = f {2x + \) ta cd - if] = l o g i (2x + 1) - (2x + 1) ^ 3 + l o g ! ((x - 1)-) - l o g i (2x + 1) = (\{x 2 2 \ 24 trmh , ^ • • '' "' 6^56^30 = 30 g«l.>lifii| + 3125 < 1 nen ham c6 / ' ( < ) > 0, suy ra ham / dong bien tren E. -i • B a i t o a n 34 ( D e n g h i cho k i t h i h o c s i n h gioi c a c tru'dng C h u y e n k h u v y c D u y e n H a i v a D o n g B a n g B a c B o n a m 2 0 1 0 ) . Gidi phudng trinh Ta CO bai toan sau. — 8 l o g i -~—-j- ••• trmh 2x2-2x-l f^V~" 2x(v/3)' = 1 ' T a c 6 / ' ( i ) = 3i2 + i + - ^ J ^ i _ , ma , X ^ x^ - 2x = 1 + ^ / ^ /(x) = /(A/S^TT), = / ( ^ ~ 1)' ^" ^-'^ or-i Ta diCcfc bdi toan sau. i Htfofng d a n . PhiJdng trinh viet lai ( d 6 i v d i (**)), + kf(x), V i d u 2. Xet ham .so ddiig hiev. tren M Id f{t) 1 = x2 - 18x - 31. 2 ^ ^ - - ' B a i t o a n 33 ( D e n g h i O l y m p i c 3 0 / 0 4 / 2 0 1 1 ) . Gidi phuang Dudc giai tudng t i f n h i l tren bang phitdng phap he so bat d i n h , phan tich ham f 1)2 (v6i 0 < a < 1, A: > o) = hix) h{x)=!jix) - Hii'ding d a n . T u d n g t i t nhxi bai toan 22 d trang 17. Phitdng t r i n h c6 hai nghiem x = 9 - 2^22,x = 9 + 2\/22. /^y. i^...;^. .y) Q ..V,. = h{:r.) (vdi a > 1, A; > 0 - klog,jj{x) X B a i t o a n 32. Gidi phuang trinh ^ { ? = T.' Cac phiTdng t r i n h long quat ' '' ^ 8 logi 5 - lf \ (2x + 1) / (6^' - 3^) (19-^- - 5^) (10^ - r) + (15^ - 8^) (9^ - 4^') (5^ - 2^) = 2 3 F . (1) G i a i . Ta c6 cac nhan xet sau : N h a n xet 1. Vdi a > 6 > c > 1 t h i > 6^ neu x > 0 va N h a n xet 2. Vdi a > 6 > 0 cho tritdc t h i ham so / ( x ) = dong bien va lien tuc tren tap D = fO; + 0 0 ) do / ' ( x ) = a"" In a - 6^ in 6 > 0, Vx > 0. < 6^ neu x < 0. - 6^ xac djnh '' ' N h a n xet 3. T i c h hai ham so dong bien, nhan gia t r i ditdng tren tap D la ham dong bien, tong hai ham dong bien tren D la ham dong bien tren D. Ta se ap dung ba nhan xet tren de giai bai toan nay. Ngu X < 0 t h i 's.ji. - •< 'i t , (6^ - 3^) (19x - 5^) (10^ - 7^) + (15^ - 8^) (9^ - 4^) (5^ - 2^) < 0, 25 trong k h i 231'^ > 0, auy r a phitdng t r i n h khong c6 nghi^m khong ditdng. V6i X > 0, chia hai vg phitdng t r i n h cho 2 3 F = (3.7.11)^ dudc B a i t o a n 3 6 . Gidi he phitang trinh | ^ J ^2 _ ^ _ ^^^^ Hu'o'ng d i n . D a t u =^ x + y, v = x - y- K h i do u + iv ) Goi y l a hani so d ve t r a i ciia (2). T i t nhan xet 2 va nhan xet 3, suy ra y dong bien t r e n D = (0; +oo) va 9 _ 4\5 _ 2\ 5(1) = ( 2 - 1 ) ( { ^ _ 14 3 +-V = 2x, uv = x2 - y2, ^2 + 1-2 = 2(x2 + y^). (2) 7 7) [3 Thay vao he ta ditdc | 4v'^ - t;2 {v - 1) (4i;2 _^ 4,^ _^ ^2) = 0 11 7 ^ 1 1 ' 7 3 ~ • Co r a t nhieu each dat an phu k h i giai he phitdng t r i n h . D a t an p h i i nhit the nao con t u y thuoc vao tifng he phitdng t r i n h cu the. B a i nay se neu r a mot so phcp dat an p h u cd ban, thit6ng gap. N a m ditdc cac phep dat nay t a se CO dinh hudng t o t hdn k h i giai he phitdng t r i n h . 4 G i a i . Dieu kien x 7^ 0 va y M- + V X = K h i do uv = + X 9 u-v 9 = 2x (x" + 10x2y2 + 5y4) u^-v'' = 2y (3x2 ^ ^2) u'-v' = 8xy (x2 + y^) =»7/7; 2x + 2x2 _2y2 = 7 2 (.r2 + y2) ^ 5 26 — 2 W. - y = " I 2 2x 1 2 {u - -t)) z(, + w 2 (^2 - 7/2)' u+ 01; [uvY = - 5 {u^ - v^) = u - {uvfv. (3) 1 + =0 V {u^ - ( / ) = 1 - uS;" <=> u'^v{l + v'') = 1 + If' ^u'^v = 1. Khi 1 + = x2 - 2/2^^2 ^. ^2 ^ 2{x'^ + xf). T h a y vao he t a ditdc he doi x i i n g loai 1 doi v d i u va v. { ""2"^^ t .,. , Neu u = 0 t h i y = - . T , the vao phitdng t r i n h thit hai ciia he thay khong thoa man. Vay xet u 0. T i t (*) t a c6 Hvfdng d i n . D a t u = x + y, v = x - y. K h i do u + v = 2x,uv 1 0. D a t u = x + y, v = x - y. K h i do uv (^2 + t'2) ^ - v'^ = 2y (5.7;'* + 5x2y2 ^ ^^4^ B a i t o a n 3 5 . Giai he phiCOng trinh 3 trinh Thay vao he (*) t a du'dc = 2y + -.2 = 2 (x2 + y2) - y 4 1 4y u + v = 2x =0 = 1. (*) P h e p d a t « = x + y, v — x - y. 1.4.1 V V B a i t o a n 37 ( D l n g h i O l y m p i c 3 0 / 0 4 / 2 0 1 1 ) . Gidi he phxMng M o t so phep dat a n p h u cd b a n k h i giai he phufdng t r i n h . 1.4 3 + 3^;2 = 4v^ 4v^ + 8t'2 - 12u = 0 {4V^ + 8v -12) =0<^v Vay (1) ^ g{x) = y{\) o x = 1. Suy ra x = 1 la nghiem d u y nhat ciia (1).'"'I 4r^ - ,-2 T i t (1) suy ra u thav vao (2) dUdc : 3) 7_b3_ ^f+^;^ I ^ = 0 t a CO v; = - 1 , suy ra u = v^S. Vay X = s/5 - 1 y= 27 V5 + 1 Khi 7/.^?; = 1 t a c6 1' ^ ~ ^ _ 2/- 1 = u-v= = —r, thay vao {v.ny = - 5 t a ditdc x+1 1 5 V-5 , X = —-—z ) y = 2 ' " (x + l ) ( y + l ) •i> ( S I ofiv "/,fii! X + y he phuang x - y | ^2 3-x 3y2^_""i^ Ta t h u dUdc he Hu'dng d a n . I C a c h 1. D a t D a t u = x + y, v = x - y. K h i do \+ <^u^ + - UV = 1 ^ = .,j3 _ ,^3 _^ 2ur2 - 2u^u <^ 2v^ + 2u^v - 2uv^ = 0 r ^' = 0 <^t; (t;^ - u r + u^) = 0 2 _|. " j ' ^ 4. ^ = 0 1.4.2 Phep dat u = x+ l u ^ 1 r u= 0 1-4.3 Phep dat u ^ x + 22:y - 2 r + x+1 y + 1 (x+l)(y+l) 28 = '.\> g i i b - J i i - i 2 - y • Jiftwrfs ,ijtjt.3,i VB 1 - 2w - 2 2 - v+l V - 1 ^-v u + 3' V+1 kb .QJTi u + •() ~ 2u + 4 uv — \ — V ^ / tx2 = 2z; ^ I = 3. v = ij + - . X K h i do u + u= X \. uv + 1 7' + 3 2^2 + 4u - 2uv - 4v = Au + 4(; - 2u'^ - 2uv I uv^ -v + 3uv - 3 = 3uz; + 3 - uv'^ - v ^ y + 1 K h i do - u - V _ 4 - 2u ^ / 4u2 - 8u = 0 ^\2ut;2 = 6 { " - 0 C a c h 2. D u a ve phildng t r i n h d i n g cap bac ba doi vdi x va y. 3 V = - — r - K h i do t; + 1 X + y uv — \ u + v' I + xy uv -\-1 3u - 3 1 u+\ 4 - 2 ^ 1 - 2 y " - 1 2it + 4 ' 2 - y 3u +1 1 -3x trinh 1 - xy u — V x-y I - xy '* ' < 1 + xy Thay \ao he t a dvtdc { JJ3 | J^3 ^ 35 x,,ff,ij 11 ^r:^g''\-^i-iu..W.a 3^ + y ^ 1 - 2y Hu'dng d a n . Dtit x = ^^—4u+1 Hifdfng d a n . D a t u = x + y, v = x - y. K h i do B a i t o a n 3 9 . Giai he phUOng trinh | ^3 :t | ^ 2 ^ ?! 17^ 5. V i . Ui*o.. trinh ,. , , (x;y) = B a i t o a n 4 0 . Giai he phuong B a i toan 38. (x + l ) ( y + l ) ^2/ - (x + y) + 1 ^ 2 (x + y) (x+1) (y+1) (x + 1) ( y + 1 ) " - ^ _ X- y , 1 — uz) (x; y) = 2xy + 2 \ - u v ^ \ -. Cac nghiem ciia h§ 1^ ' ^ f " 2 (x + 1) ( y + 1 ) x y - ( x + y) + l (x+1)(y+1) _ x y - (x + y) + 1 '^""^ Do do y + 1 x - l y - 1 uv — x + l ' y + 1 2 (x - y) + 1- + " + w = (x + y) \' y 2, v^ = r/ + \ 2 vk 1 1+ — ) u'-* + 29 = (x^ + y' + 4 n _ y x'^ + 1 V x' y'^ + 1 ' uv = xy -\ xy xy {x + y) B a i t o a n 4 1 . Giai he phudng trinh • xy + 1 xy / \ 1 4.4 1A 1+ — = 4 x2 + h xy B a i t o a n 4 2 . Giaijie phmng | Irinh ^ ^2) K l i i do u + t) = (x + y) f l + — = 4. Hifcing d a n . D a t u = x + i , u = y + ^, t a t l m dudc he { ^ uv = x y H = 4 . 4 1^-11 = ( x - y ) H V —) 1-2 B a i t o a n 46. Gidi he phiMng irinh 0. Ho titdng ditrmg vrJi f xyy 1 \ 1+ — =4 \ V xyy xy + — = 2 / ^xy (x + y) Hifctng d a n . D a t u = x + - i , v = y + - , t a t h u dudc he | " , + 2 ' r y X • u t — 4t. 1 1 \ = 208. B a i t o a n 4 7 . G?d? he phuong B a i t o a n 4 3 . Gidi he phMng trinh \ .2 _^ 2^ V 4 xyJ Hu-dng d a n . Dat u = x + ^, i ; = y + ^, t a t h u dUdc he | 4("2"^_j_^]2y f = 16. ^^ i trinh t a t l m diOTc he | "2"t|.\,2 ^ 2 1 2 . y (x2 + 1) = 2x (y2 + l ) / B a i t o a n 48. Gidi he. phudng Uvtdng xy V xy _^ 3.2^2^ ^ 208x2y-. Hvfdng d a n . De thay (x; y) = (0; 0) la nghieni cua h§. T i e p tlieo xet xy Dat u = X + ^ , v^y^-K P h e p d a t a — x + i , v = y + -• y • X •! •2\ 1\ 45. 25 trinh d a n . Dieu kieu x y ^ 0. Dat u = x + - , 7; = y + ^, t a t l m ditcfc he ^ y Hifdng d a n . Dat u = x + J , ^ = y + ^ , t a t h u dudc he | + i;2 = 20. { = 1 B a i t o a n 4 4 . Gidi he phmng trinh B a i t o a n 4 5 . G i d i he phUdng trinh / 1 X + - V H i f d n g d a n . He phimng t i i i i h tucing diMiig 2 (x+ x) 30 1-4.5 ( y + ..V/ xy (2x + y - 6) + y + 2x = 0 ^^2 _|_ y2^ ^2 _)_ ^ Hifdng d i n . D a t u = x + - , v = y + - , t a t l m dUdc he y X r 2x'^y + y^x + 2y 4- x = Qxy I J_ + 4_ E = 4. I xy X y = 4 = 6. + J J ^ j I ^35 M o t s o p h e p d a t §n p h u k h a c . _ g 2u + w = 6 u2 + t;2 ^ 8. \ Cac phep dat fiu p h u r a t da daug va phong phii. Ta can kliai thac cac dac fiipm rieng, cac t i n h chat dilc biet cua tftng he phUdng t r i n h de dita ra phep <5§.t phii hdp. f. 31 Bai toan 50. Gidi x = 4- y/W va y = 3 + / l O he phUdng trinh Giai. Bien d6i he da rho, ta tlni ditrJc x^ do. ta + x/ - xy{x + y) = 3 + xy{x + y) = 15. CO x-'^ + =9 >^ < xy{M '+y) = 6 • y = 21 2 X = . y. = 3 =2 .X = y = l- he CO nghiem (x; y) = (1; 2), (x; y) = (2; 1). Bai toan 51 (HSG Hai Phong, bang A, nam hoc 2010-2011). Gidt /x + - + ^ x + y - 3 = 3 he •phuanc) trinh sau V 2x + y + -1 = y Giai. Dieu kieii y 7^ 0, x + - > 0, x + y > 3. Dat 1 a= ^ x + i , "J'^^t He da oho viet lai la { • V6i a = 2 va 6 = 1, ta c6 xJr- y X+ 6= v^x + y - 3, a , 6 > 0. a = 2 va 6 = 1 a = 1 va 6 = 2. 5 '= 2 va Jx + y - i = l < ! = > x + 1- = 4 v a x + y = 4 V f x 2 - 8 x + 15 = 0 X = 3 va y — I = 4 <^ x ^ 4 <^ 4 - X X = 5 va ?y = - 1 . I y=4-x [ y= 4 Vdi a = 1 va 6 = 2, ta c6 - + v'lO va y = 3 - ^10. ' V' ''•'' Thii lai, thay tat ca deu thoa man. He phirdng trinh da cho c6 4 nghiem la (3,1). ( 5 , - 1 ) , (4 - v ^ , 3 + \/ro) , (4 + v ^ , 3 -/To) . LuM y- Dang he ijhitong tiinh giai bang each (hit an phu nay thittJng gap tj nhieii ky thi, tit DH-CD den thi HSG cap tinh va khu vuc. Bai toan 52 (HSG tinh Ha Tinh, nam hoc 2010-2011). Gidt he 3 . 2y = 1 + x2 + y2 _ 1 x' + i/V 2x = 4 X = X X• + - = 1 va Jx + y - 3 = 2^' v * / »» • .^.7^; ""'^'^ ^^ V i d u 1. Xuat phdt tit mot bien doi tiMng dudng do ta chon / f + y = 3 I z' + y = b k h i (y; z) = (4; - 1 ) : I CO fti:^^i*i,3> 1)' '^"y '•'^ (^; 2/) = (1; 0). G i a i . Xet x = 0 => y = 0. Vay (0;0) la mot nghiem ciia he. Xet x ^ 0, chiii hai ve cua (1) cho x, hai ve ciia (2) cho x^, ta dudc X 4COS(Q ^ - 2 cos(3a - 45") = ^ 3 ^ = x - y, dieu kien u > 2. Thay vao (1), ta ditoc { a + 6 =^3"" f ^ f ^ I'^i •; 8sin(rv - 45"). sin(a + 15") cos(a - 15") = \/3 ..ill G i a i . Dieu kien x + y 7^ 0. He viet lai £ /Of v 3(x2+y2) + (x2-y2) + s i n 2 a ^ = ^3 + X 9.1/2 + 27?/ + 27 - 2= y + 3 ^ X • = y + 5.
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