Rèn luyện kỹ năng sáng tạo và giải phương trình, hệ phương trình, bất phương trình: Phần 1

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Phần 1 tài liệu Sáng tạo và giải phương trình, hệ phương trình, bất phương trình cung cấp cho người đọc các kiến thức: Phương pháp sáng tác và giải phương trình, hệ phương trình, bất phương trình; phương pháp đa thức và phương trình phân thức hữu tỉ. Mời các bạn cùng tham khảo.

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Nội dung Text: Rèn luyện kỹ năng sáng tạo và giải phương trình, hệ phương trình, bất phương trình: Phần 1

AdST~ NGUYEN TAI CHUNG gmi # ? 9 / 01 oc PHOfI DG TRiHH iH Da hi nT HE PHIfOnC TRniH uO ie iL BAT PHIDnG TR Ta s/ up ro /g om P H l J d N G PHAP XAY DL/NG B E TDAN .c CAC DANG TCAN, CAC PHLfdNG PHAP GIAI ok 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 bo T A I LIEU B D I DL/QNG H O C S I N H K H A GICI ce fx T A I L I E U O N L U Y E N T H I B A I HOC fa T A I LIEU T H A M K H A C C H D GIAO VIEN w. ww 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
SANG T A O V A GIAI P H U O N G T R I N H , LMnoidau 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? N G U Y I N TAI C H U N G Ai la nguai dau tien nghi ra cac bai tap do? Nghl nhu the nao? Ngay ca nhieu giao vien cung chi biet suti tarn cac bai tap c6 trong sach giao khoa, sach tham Chiu trach nhiem xuS't ban 4 ; khao khac nhau, chua biet sang tac ra cac de bai tap. Mpt trong nhimg each do N G U Y E N THI THANH HlJdNG 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 01 Bien tap : QUOC NHAN tinh diing dan ciia no. ' • oc Si^abanin . : HOANG NHlTX Nhu chiing ta da biet phuong trinh, h$ phuong trinh c6 rat nhieu dang va iH Trinh bay : C6ng ty K H A N G V I E T phuong phap giai khac nhau va rat thuong gap trong cac ky thi gioi toan ciing Da Bia : C6ng ty K H A N G V I E T nhu cac ky thi tuyen sinh Dai hpc. Nguoi giao vien ngoai nam dupe cac dang hi phuong trinh va each giai chiing de huong dan hpc sinh can phai biet each xay nT NHA XUAT BAN TONG H0P TP. HO CHf MINH dung nen cac de toan de lam tai li^u cho vi|c giang day. Tai lifu nay dua ra uO mpt so phuong phap sang tac, quy trinh xay dimg nen cac phuong trinh, he ie NHA SACH TONG HOP phuong trinh. Qua cac phuong phap sang tac nay ta ciing rut ra dupe cac iL 62 Nguyen ThI Minh Khai, Q . l phuong phap giai tu nhien cho cac dang phuong trinh, hf phuong trinh tuong Ta D T : 38225340 - 38296764 - 38247225 Fax: 84.8.38222726 ling. Cac quy trinh xay dyng de toan dupe tnnh bay thong qua nhiing vi du, s/ Email: tonghop@nxbhcm.com.vn cac bai toan dupe xay dung len dupe dat ngay sau cac vi du do. Da so cac bai up Website: www.nxbhcm.com.vn/ www.fiditour.com toan dupe xay dung deu c6 loi giai hoac huong dan. Quan trpng hon niia la ro mpt so luu y sau loi giai se giiip chiing ta giai thich dupe "vi sao lai nghl ra loi /g Tong phdt hanh giai nay". om Nhu vay cuon sach nay se trinh bay song song hai van de: Phuong phap .c sang tac eae de toan va Cac phuong phap giai ciing nhu phan loai cac dang ok CONG T Y TNHH MTV toan ve phuong trinh, hf phuong trinh. Diem moi la va khac bi?t ciia cuon bo DjCH V g VAN HOA KHANG V I E T sach nay la quy trinh sang tac mpt de toan moi (dupe trinh bay thong qua cac ce vi du) va each thiie chiing ta suy nghl, tim ra loi giai mpt bai toan (dupe trinh fa ( ^ 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 bay thong qua eae luu y, chii y, nhan xet ngay sau loi giai cac bai toan). Ngoai w. Fax: 08. 3911 0880 ra cuon sach nay con danh ra mpt ehuong (ehuong 5) de trinh bai cac bai toan ww Email: khangvietbookstore ©yahoo.com.vn phuong trinh, he phuong trinh, bat phuong trinh trong cac de thi Dai hpc 1 Website: www.nhasachkhangviet.vn trong nhiing nam gan day. Tot nhat, doe gia tu minh giai cac bai toan eo trong sach nay. Tuy nhien, de In ian thLT i, so lUdng 2.000 cuon, kho 1 6x24cm. thay va lam chii eae ky xao tinh vi khac, cac bai toan deu dupe giai san (tham 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 chi la nhieu each giai) voi nhiing miie dp chi tiet khac nhau. Npi dung sach da Dia chi: 006 L6 F, KCN Tan Binh, P. Tay Thanh, Q. Tan Phu, Tp. Ho Chi Minh c6' gang tuan theo y chii dao xuyen suo't: Biet dupe loi giai ciia bai toan chi la So DKKHXB: 1 55-1 3/CXB/45-24ArHTPHCM ngay 31/01/201 3. yeu cau dau tien - ma hon the - lam the nao de giai dupe no, each ta xir ly no, Quyet dinh xuat ban so: 296/QD-THTPHCM-2013 do NXB Tong Hop nhiing suy lu^n nao to ra "c6 ly", cac ket lu^n, nhan xet va luu y tir bai toan Thanh Pho H 6 Chi Minh cap ngay 19/03/2013 dua ra... In xong va nop luU chieu Quy II nam 201 3
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 ChiMng 1. PhUcfng phdp sang tdc va giai phUcfng trinh, hf phUOng trinh, bd 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 1.1 PhiTdng phdp he so bat djnh 3 / chiing toi luon mong dpi. H y vpng rang tren buoc duang tim toi , sang tao 01 1.2 Phifdng phdp duTa ve h$ 5 toan hpc, ban dpc se tim dupe nhiing y tuong tot hon, mai hon, nham bo sung oc cho cac y tuong sang tao va loi giai dupe trinh bay trong quyen sach nay. 1.3 Phufdng phap diTa phiTdng trinh ve phifdng trinh ham 15 iH 1.4 Mot so phep dSt an phu cd ban khi giai h$ phufdng trinh 26 Da Tac gia 1.5 PhiTdng phdp cpng, phufdng phdp the 35 hi 1.6 PhiTdng phdp dao an. Phifdng phap hiing so bien thien 54 nT Thac sy: NGUYEN T A I CHUNG uO 1.7 PhiTdng phdp sijf dung dinh l i Lagrange 63 ie 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 iL 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 Ta Thuxingici ve: 1.10 PhiTdng phap tham bien 95 s/ Cty T N H H M p t Thanh Vien - Dich V u Van Hoa Khang Vi?t. ChUcfng 2. PhUcfng phdp da thiic va phUcfng trinh phdn thitc hOu ti. 11 up 71, D i n h Tien Hoang, P. Dakao. Quan 1, TP. H C M ro Tel: (08) 39115694 - 39111969 - 39111968 - 39105797 - Fax: (08) 39110880 2.1 Cdc dong nha't thiJc bo sung 116 /g Hoac Email: khangvietbookstore@yahoo.com.vn om 2.2 PhiTdng trinh bac ba 117 2.3 Phu'dng trinh bac bon 127 .c 2.4 PhiTdng phdp sdng tdc cdc phiTdng trinh da thiJc bac cac ok 137 bo 2.5 PhiTdng trinh phan thiJc hi?u ti 149 ce ChUcfng 3. PhUcfng trinh, bdtphUcfng trinh chiia can thiic 158 fa w. 3.1 Phep the trong doi vdi phiTdng trinh 3/A(X) ± }JB{X) = 3^C(x) .... 158 ww 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 Chi:fc?ng 1 4.4 Phi/dng phdp dilng tinh ddn dieu cua ham so' 271 4.5 He lap ba an (hodn vi vong quanh) 277 Phi:fdng phap sang tac va giai / 4.6 SuT dung can bac n cua so phuTc de sang tac va giai he phiTdng trinh ... 01 oc 307 phifcfng trinh, he phi:fcfng iH 4.7 Phi/dng phap bien doi ding thiJc 314 trinh, bat phi:^dng trinh Da 4.8 MotsohekhongmaumiTc 317 hi ChUctng 5. Cdc bai todn phUcmg trinh, h^ phUcfng trinh, bat phUcfng trinh trong nT dethidt^ihQc 328 uO ie 5.1 Phtfdng trinh, bat phi/dng trinh chiJa can 328 Trong chitcJng nay ta se trinh bay nipt so phUdiig phap cO ban va mot so iL 5.2 He phiTdng trinh dai so 332 phUdng phap dac biet di giai va sang tac phitdug tiinh, lie phUdng trinh, Ta bat phUdng trinh. Co mot so vi du, bai toan c6 sii dung den kien thi'tc cua 5.3 PhiTdng trinh liTdng gidc 337 plntdug trinh da thi'tc bac ba, ban doc c6 the xcni bai i)liitdiig tiiiih bac ba d s/ 5.4 Phi/dng trinh, bat phi/dng trmh c6 chlJa cdc so n!,Pn, A^, C\5 chUdng 2 (chi can c6 kign thv'tc ve lUdng giac la co the hieu bai phUdng trinh up bac ba) trudc khi xem cac bai toan, vi du nay. - . , • 5.5 PhiTdng trinh, ba't phiTdng trmh mu 368 ro /g 5.6 PhuTdng trinh, ba't phifdng trinh logarit 373 1.1 PhifcTng phap he so bat dinh om 5.7 H? mu va logarit 387 .c 5.8 Phtfdng phap dilng dap ham 392 PhUdng ])hap he so bat djnh la chia khoa giup ta i)han tich, tim dudc l a i giai ok cho nhieu locii phUdiig trinh. Chung ta se Ian lUdt tini hieu phUdng phap nay bo thong qua cac bai toan va cac km y ngay sau do. ce Bai toan 1. Giai phiCdng trinh 2^^ - l l x + 21 - 3^4.x- - 4 = 0. fa w. Giai. Tap xac dinh D = E. Plntdug trinh da cho tu'diig ditdng vdi •^ikf •'A/nh 6" ww ^ ( 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. wid ;:,v»fb im V >. 3
D o do (2) i = 2 => X = 3. B a i t o a n 3 . Gidi phuang trinh 4 \ / l - x = x + 6 - 3 \ / l - x^ + 5s/TTx. L U L U y . De c6 (1) ta can t i i i i a, (3,7 sao cho X V3 - l l x + 21 = a(4x - 4)^ +/3(4a; - 4) + 7 2x2 - l l x + 21 = IGrtx^ + (4/^ - 32rv);r + (16^ - 4 / ^ + 7) Dap so. X = - - — - . J f 16a = 2 , , fl 7 \ ^ { 4/3-32a = -11 ^{a;(i\-i) = I 1 6 a - 4 / 3 + 7 = 21 B a i t o a n 4. Gidi phuang trinh 4 + 2\/l - x = - 3 x + SVxTT + V l - x^. dung phuong phap he so bat dinh cho t a 15i giai bai toan m o t each rat tijt D a p so. PhUdng t r i n h c6 tap nghiem S — I 0; — ; — I . tah s v , ;: i; / 01 nhien va ro rang. 25 2 j oc B a i t o a n 2. Giai phiCcfng trinh iH - x + 3 = 2\/r^-/m^+3\/r^. (i) B a i t o a n 5. Gidi phuang trinh lOx^ + 3 x + 1 = ( 6 x + l ) V x 2 + 3. (*) Da 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 : G i a i . Dat u = 6 x + 1, ?; = \/.x2 + 3. Ta c6 hi = i(6x nT (1 + x ) + 2(1 - x) - 2v^r^ + \/l + .T - 3\/l- x2 = 0. (2) 10x2 + 3 x + 1 + 1)2 + (x^ + 3) - - = — + ^2 _ 9 uO 4 4 4 4 5, Dat u = ^/^+x,V = Vl - X {u >0,v>0), t a duoc ie Thay vac (*) : + 1,2 _ 5 = ^ i K : ^ (u - 2t;)2 = 9 0 ^ , ^ ( • a - 2v){u - + 1) = 0 ^ [ « Z 1 ^ 0 x2 + 3 = ( 3 x - l ) 2 ^x^l. s/ • ^/3^ up • Vdi u - 2t; = - 3 , t a c6 ro L u ^ i y. D l ( 2 ) , t a t i m a, f3 sao cho /g CO om - x + 3 = a ( l + x) + / 3 ( l - x ) o { ^ ; ^ ^ 3 ^ ^ { g = i .c Doi v 6 i bai toan t o n g quat : Giai phvfdng t r i n h Vay p h i M n g t r i n h c6 tap nghiem 5 = < 1; ——^ I. , ok bo p{x) = as/1 - X + bVl + X + cVl - x^, Lvfu y. Phudng phap he so b a t dinh de giai he phudng t r i n h se ditdc de cap ce trong phan phan t i c h t i m Idi giai cac bai toan c i i a bai 1.5 : PhUdng phap Ta bieu dien p{x) theo 1 - x , 1 + x va dat cong, phiTdng phap the (d trang 35). fa u = - / I + X, i; = \ / l - (u > 0, i; > 0 ) . w. X 1.2 Phifcfng phap difa ve he. ww 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. Ta sc. sang tdc mM phUdng trinh duclc gidi hhng phiMng phdp he so bat dinh nhu sau : Ta c6 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 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$ {a-b+ l ) ( 2 a - 6 + 3) = 0 ^ 20^ + 6^ - 3a6 + 5 a - 46 + 3 = 0. 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 Tii day lay a — ^Jl + x vd b = \/l - x ta diidc 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 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 2x + 2 + 1 - X - 3 \ / l - x2 + 5VI+X - 4s/l-x + 3 = 0. 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 Rut gon ta duac bai toan sau. 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 160). 4 5
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. y = X y - X = 0 5x + 2 2(y - x ) 5(.T2 - y2) ^ 2 = - 5 ( x + y) ^ B a i t o a n 6. Gidi phUdng trlnh x + 3 (2 - 3x^)^ = 2. Giai. D a t , = 2 - 3 x ^ . 1 ^ CO he ^[^ZlZ^. Vdi y = X , thay vao (1) ta dUdc 5x2 _2x -\=i) ^ x = 1 ± \/6 (1) t n r (2) t a fhrac • Vdi y = — . thay vao (1) ta du'dc >;V;!; .• o / y = X 01 X - y = Q 1 - 3x x - y = 3{x'^ - y^) ^ 3(x + y) = l ^ - l ^ = 5x2-l.=.25x2 + l ( , x - l = 0 ^ x = - ^ = ^ ^ y = —^—• oc 25 iH 1 ± v/O - 1 ± 72 Vc'ri y = x, thay vao (1) ta diWc Sx^ + x - 2 = 0 x G |~^' PhUdng t r i n h da cho c6 bon nghic'm Da 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 _ ^ hi 1 - 3x Vdi y = , thay vao (2) ta dudc 3 vdi a, h t h n sau. K h i do t i i u dUdc he nT ay + 6 = 5.r2 - 1 uO 1 - 3x 1 ± v/21 { + b + i = hx'- , = 2 - 3x^ | Z 9 Vfw 2 1-V21 1 + V21 I s/ 10a/; = 0 ' I « - - X = -1, X = - , X = — — , X = . 3 6 () ta C O phep dat 2y — 5x'^ - 1. f\ . -jj.,., up B a i t o a n 8. GidirphtMng trinh 5{5x~ - 17)2 - 343x - 833 = 0. ro 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 /g phitdng t r i n h da cho vc phiWng t r i n h da thiitc bac bon, sau do Ijien doi thanh om Y tvtdng. D a t ay + b^5x^- 17 (a ^ 0). Klii do . , (x + l ) ( 3 x - 2)(9x2 - 3x - 5) = 0. jay + b = 5.7:2 _ ^7 , ,, .c \ 5 ( a y + 6)2 - 343x - 833 - 0. (*) • ' ' ok Vay ncu k h i sang tac de toan, t a c6 y lam cho plnMng t r i n h khong v6 nghiem Tir (*) ta CO bo 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. ce 5(ay)2 + lOa^y + ^2 - 343x - 833 = 0 ^ x = 5(ay)2 + 10a6y + ^2 - 833 fa V i d u 2. Xet mot phucing trinh bac hai c6 cd hai nghiem Id so v6 ti „ * ^ 5a-^y2 + i0ft2.;;.y-|.^2^j_y33^ w. 5x'^ - 2x - 1 = 0 ^ 2x = 5x2 _ Suy ra ax + b = + b. (**) ww 5x2- 1\ Ta hy vong c6 ax + h = by^ - 17, ket hdp vdi (**) suy ra Hi v, 2x = 5 - 1. Ta CO bdi todn sau. . 2 5 a ' ^ y 2 + l ( ) o 2 . 6 y + 6 2 . a - 833a , •M! I - o/y-l7= 46 B a i t o a n 7. Gidi phUdng trinh 8x - 5 (5x2 _ _ _^
llTT 1 llTT llTT G i a i . D a t 7y = 5x^ + 17, t a c6 h? phitdng t r i n h COS = 4 cos-* 18 - 3 cos 18 ' 6 J7y = 5x2 _ 17 7y = 5x2 _ 17 (1) 137r 137r 137r 7x = 5 y 2 _ 1 7 . (2) COS — — = 4 cos'' 18 — 3 cos 1^5,y2 - 343a; - 833 = 0 6 18 • TT llTT 137r x = y r = cos ——, X = C O S la tat ca cac nghiem ciia phuong Lay (1) t r i t (2) t a c6 7{y - x) = 5(x + y ) ( x - y) ^ 18' • 18 ' 18 5x + 5y = - 7 . t r i n h (4) va cung la t a t ca cac nghiem cua phUdng t r i n h da cho. 7±\/389 * Neu X = •(/, thay vao (1) : Sx^ - 7x - 17 = 0 x = L t f t i y. Phep dat 6y = 8x3 _ ^ dUdc t u n ra n h u sau : Ta dat 10 * N i u 5x + 5j/ = - 7 , ket hdp (1) t a c6 . / ay + 6 = 8x3 - V3 ^jj^ 01 ^^^^ ^ oc Ket hdp v6i phudng t r i n h da cho c6 he iH ay + 6 = 8x3 - v/3 Da ' 7 ± \/389 - 3 5 ± 5 v ^ l 162x + 27V3 = a3y3 + 3a26y2 + Safety + ^3 Ket luan: Phudng t r i n h c6 tap nghiem 5 = 10 50 J • hi 8 73 nT Can chgn a va 6 sao cho : 162 o? 27\/5 - 63 V i d u 3. Ta Ax^ - 3x = ~
D a t y = l o g n ( l O x + 1), Ivhi d o I P = 1 0 x + 1. K e t Vay ta thu dUdc hai todn sau. , hdp v d i phUdng t r i n h d a cho, t a c6 he | JJy ^ ^ | ^ "» " • - - w., Bai t o a n 10. Gidi phuong trmh ; V L a y (1) t n r (2) tlieo ve t a d i t d c • f-: , A. • (x^ + 9 x - 4 5 ) V 8 1 (x^ + 9 x - 4 5 ) = 1215 + 81X. (1) I F - ir^ = l O y - l O x 0. hi x^ - + 9 x ^ 9)/ = 3?/ - 3 x ^ - + 1 2 ( x - ?y) = 0 / 1 nT . ^ ( x - y)(x^ + xy + + 12) = 0
L t f t i y- D o i vdi phu'dng t r i n h - JJx) + "\/b + / ( x ) = c, t a co each giai : 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 Dat u = 'ija - fix), v = '^h + f(x), dan den he { ^ H ^+,n"s'^['^ Nhit vay da cho t a c6 ho 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 4u = J 3 0 + -y/xT3Q 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 (1) 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 4x = A / 3 0 + - v / w + SO. tam rang se dan tdi phitdng t r i n h da thite bac 4 co i t nhat m o t nghiem dep. V i d u 8. Vd'i. x = - 2 thi 2 u. K h i do / chdn CO mot nghiem dep x = - 2 ) .sau. 1 "i;.,/ .;- .f '-,1,; 01 B a i t o a n 14. Gtdi phiMng trinh 2 v^3x - 2 + 3^6 - 5x = 8. oc 4u = \ / 3 0 + + 30 > ^ 3 0 + - \ / u + 30 = 4x =^ u > x =^ x = ?i. iH G i a i . Dieu kien x 0. K h i do 5 Da Vay t i t he (1) t a c6 a; = u va 4x = ^ 3 0 + - y x T S O . (2) hi Z t'0, 7 ^ox I"2 — + Sv'^ = 5(3x - 2) + 3(6 - 5x) = 8. . Dat . = \V^FT30, t i t (2) ta c6 he | ^J I (3) nT M a t khac t a lai co 2u + 3r - 8 = 0. Vay t a co he uO Gia sii x> V. K h i do ie 4v = Vx + 30 > VtTTSO = 4a: 4u > 4a: =^ V > a; =^ u = X . {^t +'fv= 8^ =^ + 3 ( ^ ^ ) ' = 8 ^ 15..^ + 4^2 - 32z. + 40 = 0 iL Ta , f T> 0 1 + \/l921 Phu'dng t r i n h nay c6 nghiem d u y nhat u = - 2 nen Vay r = .x va 4.T = ^ | J g p ^ ^ ^, ^ 30 — . PhUdng s/ 1 + 71921 v'Sx - 2 = - 2 X- = -2. up t r i n h da cho co nghiem d n y nhat x = 32 B a i t o a n 1 5 . Giai phiMng trinh ro V i d u 7. Vdi X = 8 thi ^/x-\-8-\- \ J x - l — 3, ia c6 bai todn {ch&c chan co /g mot nghiem dep x = 8) sau. 1 + \ / l - x 2 [ V ( l + :r)-* - ^ ( 1 - x)'A^ =2+ yjl - xK ffif, t uM.. om .c 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 ^ = vdi a > 0, 6 > 0. ok 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 K h i do a' + l? = 2. T a co he sau ( \ . . S "" ' bo \l + ab{a-^ -b^) = 2 + ab. (2) , u + r = 3 {V = 2) — u ce (1) =^ {a + bf = 2+ 2ab^ s/lT^=-^{a + b) [do a,b>0). ^ i 0/2--,,2)(„2 fa U.,V>{) + ,,2)^15 V2 u4-t;'*-15 [u,(;>0 { w. u = 3- u ft; = 3 - u Ket-hop (2) t a co ' . ,| ww 1 1 ' ( ' -7= (a + b){a - b){a^ + b^ + ab) = 2 + ah => ^ ( a ^ ~ h'-) = I. < 3 0 < i< < 3 v2 v2 u2 + (3 - uf = 5 ^ ' ( 2 u - 3 ) ( 2 u 2 _ 6 u + 9) = 5 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 ^ ro < 1/- < 3 ^ ro < u< 3 ^ \ 4 u ^ - 18u2 + 36u - 32 = 0 ^ \ = 2 - 2 a 2 - - = 2 + y 2 ^ a 2 = l + 4 = ^ l + a ; = l + ^ ^ x = 4=- T i t do t a t h u dudc 1 = 2 {^ + f=p ^ x = 8 (thoa m a n V2 s/2 V2 , Vay phitdng t r i n h co nghiem d u y nhat x = — . vj / j dieu kien). Vay phitdng t r i n h da cho co nghiem d u y nhat x = 8. 12 13
B a i t o a n 16. Gidi phuang irinh \/\/2 1 -x + = 71 + 7) = - v2 n + J' = - ^ < 8 - vfei h"^^ 8+ vfei G i a i . D i n i kien 0 < x < \/2 - 1. Dfit \/\/2 - 1 - x = u va ^ = v. K h i do 18 18 •, !i i 0 < u < \/s/2-l va - 1. N h i t vay t a c6 he 8 - yi94 = 0 (1) Vay w, f la nghiem ciia 18_ Do (2) v6 nghiem u = —^ V 8 + = 0. (2) ( 1 3' 18 / .u:^ + v^ = v/2 - 1 - V + 7-4 = v / 2 - 1 . nen nghiem duy nhat ciia phudng t r i n h la 01 / oc 1 Ttr phudng t r i i i h thi'i: hai, ta co -2 + ^ 2 ( 7 1 9 4 - 6 ) + ^ ^ iH / Da 1 \ 2v + + i ; ' = \/2 - 1 hi v/2 v/2 1.3 Phifcfng phap difa phifdng trinh ve phifdng t r i n h nT ham uO ie 1.3.1 Phu'dng phap giai. iL Dita vao ket ciua : Neu ham so y = f{x) ddn dieu tron khoang (a; b) va Ta 1 ± - 3 x,ye (a; b) t h i ,72 s/ /(^) = /(y) a; = 7^ up t a CO the sang tac va giai dUdc nhien phitdng t r i n h hay va kho, thudng gap ro trong cac k}' t h i hoc sinh gioi. D6 van dung dildc phitdng phap nay, t a thirdng /g bien ddi phiWng t i i n h da cho thanh phitdng t r i n h ham f {
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 hiiu y. X e t h a m so /(x) = 5 ^ . 8 " ^ , Vx ^ 0. K h i do / ( 3 ) = 500 v a ,. ' 1 (1) ^ (5.T - 6)2 - ^ fix) = 5 ^ 8 ^ . In 5 + ^ . 5 ^ 8 ^ . In 8 > 0, Vx ^ 0. ' " ^ f{5x-6) = f{x), v d i f{t) = t^- (2) 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 . 1 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) T a c o f'{f) = 2t + _ , > 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)". 01 2v/rn:(t-i) b) T u d i i g t i t c a u a ) . '' oc tit (2) CO 5x - 6 = X X = 1,5. Phifdng t r i n h c6 nghiem duy nhat x = 1,5. iH 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 ) . Gidi Da 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 phuang trinh 32^'-^+2 - 3^'+^^ + x^ - 3x + 2 = 0. ' ' (1) hi G i a i . PhUdng t r i n h (1) viet lai tiino -.fu.:': nT G i a i . Dieu kien x > 1. Dg thay x = 1 khong l a nghigm ciia phirong t r i n h nen uO ta chi xet x > 1. T a c6 3 2 x ^ - . + 2 _ 3 x 3 + 2 x ^ _ (23,3 _ ^ + 2) + (x^ + 2x) (2) ie ^/^T6 + x^ = 7 - v / ^ ^ ^ s/^Te + + \ / x - 1 = 7. (*) ^ 3 2 x 3 - x + 2 ^ (2x^ - X + 2) = 3 ^ ' + 2 . ^ (^3 ^ 2x) h)\ iL (2x^ - X + 2) = / (x^ + 2x) , v d i /( 1. K h i do Ta s/ 1 H a m so / d o n g b i e n t r e n E v i f'{t) = 3 * h i 3 + 1 > 0 , G E . Vay f'(t) = - I , + 2t + > 0 , V i > 1. up X = -2 (3) ^ 2x^ - X + 2 = x^ + 2x
Dong nhat he so vdi ve trai cua (1) ta dildc Vi /'(/:) = — ^ + 1 > 0 , > 0 neii / (long bicii ticii (0; +oo), tit (3) c6 /.In 3 -12u = -36 : *' \2 ^ 6w2 _ 1 = 53 u = 3. s 2x - 1 = 3(a; - 1)' ^ 2Q - 2/3 = - 8 ^\ S 3 1 [ -a + ft + 'y = 5 I 7 = 1- Giai. Dicu kien x > 1. Neu 2 3x - 5 > - - . Ta co Da hi fix) (3x - 5)2 + 3x " 5 = X - 1 + v/x - 1 ': : nT Dang tong quat log„ —T-T- = h(x) dUdc giai titdng tu. Ta thitcing dung phudng ^ / ( 3 x - 5) = / ( ^ F ^ ) , v6i J{t) = e + t uO phap he so bat dinh nhir tren dfi dua ve mot trong cac tnrdng hdp 0 ^ Ta ^ 1 (3x - 5p = X - 1 ^ Bai toan 23. Giai phucing trinh Sx^ - SGx^ + 53x - 25 = \/3x - 5. (1) s/ up Giai. Ta CO 3 1 ro (1) ^ 8x^ - 36x2 + 54x - 27 + 2x - 3 = 3x - 5 + N/3X^ Neu l < x < - = > 4 - 3 x > - - thi /g ^ (2x - 3)^ + 2x - 3 = 3 x . - 5 + s / 3 ^ ^ om (1) 0, Vi e R nen / dong bien tren R, vay tir (2) ta c6 ok 4 - 3x = i / x - 1 ^do f { t ) dong bien tren ( - ^ ; +c)o)^ • bo 2x - 3 = ^ (2x - 3)^ = 3x - 5 4 ce r 4 - 3x > 0 " - 3 25 - \ / l 3 «>x =
Dong nhat he so t a dUdc: /(3x - 3) = / ( ^ / 9 ( : : 3 x 2 + 21.7?T5)) vdi i{t) = + 27t. ^^ ^ 3 x - 3 = > y 9 ( - 3 x 2 + 21x + 5) o 3x^ - 6x2 _ - 8 = 0. / - 1 8 w - 8 = -28 ^ j i i = — f . . s 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). L i i u y. N h a n 9 cho 2 ve c i i a (1) t a dUdc • Neu 771 = 1 t h i fit) = t'^ + t. Ta can diia (1) ve dang 27X''' - 54x2 273. 153 = 2 7 ^ 9 ( - 3 x 2 + 21 + 5 ) . ' (2) (3x - uY + 3x — u = {x — 1) + \/ X — 1 - - < ( ^ x > - . Con 1 < x < - t h i sao ? L a i de y rang ham so up la - + 3 i . Viec nhan chi ddn gian la k h i i mau so. ro 9 Si'ssi;: ,,, f;,? bac 2 cung c6 cai hay cua no, do l a ( - / ) 2 — t r e n , dua vao he so bac cao /g 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 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 trinh dUa (1) ve dang om 30.7:2011 ^ 4^^,^2010 ^ 30y4022 ^ 4y2012 , , „ .c ( u - 3x)2 + u - 3 x = x - l + \/x - 1 ok ^Six^ + x ( ~ 6 7 / - 4) + + 77, + 1 = V x - 1. 1627/2 + 27 ^ 3 = (8x^-^3) ^ (2) bo ce D6ng nh^t he s6 : { "2*^J " "f^ ^ 7i = 4. 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 1 fa 3 ™2011 „ K i e m t r a lai : T a c6 x < - do do chon u = 4. Den day bai ( 1 ) ^ 3 0 . ^ + 4 . - = 30^2011 + 47/. (3) z z w. y y 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 ww dung ham so, nhat l a doi vdi ham bac chan. Ta cuiig c6 the giai bai toan Xet ham so f{t) = 30 Q nen tren bang each dat ^x-\ 3?/ - 5 de dUa ve he doi xi'tag loai 2. ham / dong bien tren M . Do do t i f (3) t a c6 B a i t o a n 25. Giai phuong trinh ' x \, , X o • Mki f ( - ) = f i y ) ^ - = y^x = y\ 3x^ - 6x2 _ 3^. _ 17 = 3 y 9 ( - 3 x 2 + 21x + 5). (1) \y/ y Vay (!)
Xet ham so dac trung f{t) = t'^ + t, t eR. Ta co / ' ( i ) = Si^ + 1 > 0, vay ham Then l)ni loan 9, cJ trang 8, cac nghiem cua (4) la cos-;^. cos^-i^. c o s i ^ - lo 18 18 so dong bien tren R nen (*) a + 1 = Ta co he sau : 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 ^ , Sii dung phep the ta co ; ;; . ^3 _ _ 1)2 = l 0, V x e uO 2\/jr'^ + 3a;2 + 9.7: - v/3 - 2x ^ 7 - ^ - ' + 6 log7 7^-^ - (6x - 5) + 6 logy (6x - 5) 4 ie / 4 \ \/543 - \/27 ^r~^ + 6 (x - 1) = (6x - 5) + 6logy (6x - 5) i : '-ih ; / , mat khac / nen - la iL Vay ham / rlong bion trcn Uj ~ 9 3 ^7^-^ = 1 + 2 logy (6x - 5)^ . •[ ', x < Ta , , , , , , ^, nghiem duy nhat cua phitdng trinh da cho. Ta duoc bai toan sau. s/ Bai toan 28 (De thi hoc sinh gioi cac trtrdng Chuyen khu vu'c Duyen Bai toan 30. Gidi phuang trinh 7'^''^ = 1 + 2 logy (6x - 5 ) ^ up (1) Hai va Dong Bang B a c B o nam 2010). Gidi phiMng trinh ro Giai. Dieu kien .x > | . Ta co /g 2.x'* - x2 + v/2x^ - 3 x + 1 = 3 x + 1 + v/x2 + 2 om 6 Hu'dng dSn. Tap xac dinh D = M. Bien d5i phildng trinh ve • (1) 0, Vx G R. Vay ham / CO do thi luon luon loin nen cat true Ox tai khong qua hai diem, suy Bai toan 29. Gidz phuang trinh (x + 5)\/x + 1 + 1 = \/3x + 4. 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}. Giai. Dicu kiOn x > -1. Dtlt a = 4- 1. = v / 3 x T 4 . Klii do x = _^ Cach 2. Ta co / ' ( x ) = 0 7^-' = — x = xo = 1 + logy (6. logy e). V i 3a2 + 1 = 6'^ Ta c6 he | ^^^2^^^" "^3^ " ^ Cong ve fheo ve ta co 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) = + b. (*) / ' ( x ) < 0, Vx G ( - 0 0 ; xo) ; / ' ( x ) > 0, Vx € (XQ; + 0 0 ) . , 22 23
Vay ham / nghich bien tren {-OO;XQ) va dong bien tren ( x o ; + o o ) , do do X - 1)2 ^ 8 logi = x2 - 18x - 31. (4) C O khong qua 2 nghiem. Vay x = 1, a; = 2 la l a t ca cac nghiem ciia (4). 5 2 ^ ^ - - ' i 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 Ta diCcfc bdi toan sau. ^ 1 ' ••• l=a(x-l)+/3(6x-5)=.{ - + 6/?j0^ ^ { ? = T.' B a i t o a n 32. Gidi phuang trinh 8 l o g i -~—-j- = x'^ ~ ISx - 3 1 . Cac phiTdng t r i n h long quat 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,. 01 ' '' " ' - klog,jj{x) = h{:r.) (vdi a > 1, A; > 0 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 trmh oc a-''^^) + klog^gix) = hix) (v6i 0 < a < 1, A: > o) iH 9x2 + (ix + 3126 x^ - 2x = 1 + ^ / ^ + in ^,^3^^^ . , ^ • • Da Dudc giai tudng t i f n h i l tren bang phitdng phap he so bat d i n h , phan tich hi = ^(x) - / c / ( V ) , (doi vdi (*)) Htfofng d a n . PhiJdng trinh viet lai '• ' '' "' lifii| ham f = / ( ^ ~ 1)' ^" ^-'^ Ta s/ 1 or-i , f /^V^ f^V~" 2x2-2x-l 5(t*^ + 3125) = Si** + 5*^ > 6^56^30 = 30 < 1 + 3125 up ro 1\ l-x nen ham c6 / ' ( < ) > 0, suy ra ham / dong bien tren E. -i • = 2x^-2x-l. /g 2x(v/3)' 2x - 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 om Ta CO bai toan sau. 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 .c trinh ok 1\ B a i t o a n 31. Gidi phudng trinh 2x (v/3) ^ - 2x ( - = 2x2 _ 2x - 1. bo (6^' - 3^) (19-^- - 5^) (10^ - r) + (15^ - 8^) (9^ - 4^') (5^ - 2^) = 2 3 F . (1) ce 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 G i a i . Ta c6 cac nhan xet sau : fa l + V^ l-\/3 N h a n xet 1. Vdi a > 6 > c > 1 t h i > 6^ neu x > 0 va < 6^ neu x < 0. w. X = — - — , X = — - — . N h a n xet 2. Vdi a > 6 > 0 cho tritdc t h i ham so / ( x ) = - 6^ xac djnh ww dong bien va lien tuc tren tap D = fO; + 0 0 ) do V i d u 3. Xet ham so nghich bien tren khodng (0; + 0 0 ) la f{t) = l o g i t — t. / ' ( x ) = a"" In a - 6^ in 6 > 0, Vx > 0. '' ' Tii phuang trinh ham j (\{x - = f {2x + \) ta cd \ / 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. logi (\{x-\f 'Ux - if] = l o g i (2x + 1) - (2x + 1) Ta se ap dung ba nhan xet tren de giai bai toan nay. Ngu X < 0 t h i 's.ji. - •< 'i t , ^ 3 + l o g ! ((x - 1)-) - l o g i (2x + 1) = (\{x - lf \ (2x + 1) (6^ - 3^) (19x - 5^) (10^ - 7^) + (15^ - 8^) (9^ - 4^) (5^ - 2^) < 0, 2 2 \ / 24 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 +-V = 2x, uv = x2 - y2, ^2 + 1-2 = 2(x2 + y^). + iv ) (2) 4r^ - ,-2 Thay vao he ta ditdc | ^f+^;^ I T i t (1) suy ra u 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 3 / dong bien t r e n D = (0; +oo) va thav vao (2) dUdc : 01 9 _ 4\5 _ 2\ oc 4v'^ - t;2 5(1) = ( 2 - 1 ) ( { ^ + 3^;2 = 4v^ 4v^ + 8t'2 - 12u = 0 7 iH 7) [3 3) _ 14 3 7_b3_ {4V^ + 8v -12) =0
Khi 7/.^?; = 1 t a c6 1' = —r, thay vao {v.ny = - 5 t a ditdc u-v= ^ ~ ^ _ 2/- 1 = 2 (x - y) x+1 y + 1 (x + 1) ( y + 1 ) 1 x - l y - 1 x y - ( x + y) + l uv — x + l ' y + 1 (x+1)(y+1) 5 V-5 , _ x y - (x + y) + 1 2xy + 2 '^""^ (x + l ) ( y + l ) (x + l ) ( y + l ) Do do X = —-—z ) y = -. Cac nghiem ciia h§ 1^ ' ^ f " \ - u v ^ \ ^2/ - (x + y) + 1 ^ 2 (x + y) 2 ' " 2 (x+1) (y+1) (x + 1) ( y + 1 ) ,. / •i> ( S I ofiv "/,fii! 01 " - ^ _ X- y , x,,ff,ij 11 oc 1 — uz) X + y , (x; y) = , (x;y) = iH x - y ^r:^g''\-^i-iu..W.a Da B a i t o a n 4 0 . Giai he phuong trinh < 1 - xy 3 - X hi 3^ + y ^ 1 - 2y '.\> g i i b - J i i - i nT B a i toan 38. he phuang trinh | ^3 :t | ^ 2 ^ ?! 17^ 5. V i . Ui*o.. 1 + xy 2 - y • Jiftwrfs ,ijtjt.3,i VB uO Hifdfng d a n . D a t u = x + y, v = x - y. K h i do Hu'dng d a n . Dtit x = ^^—4- V = - — r - K h i do ie u+1 t; + 1 iL x-y u — V X + y uv — \ Ta I - xy u + v' I + xy uv -\-1 s/ Thay \ao he t a dvtdc { JJ3 | J^3 ^ 35 3u - 3 2w - 2 1 -3x 1 - 1 - u+\ 4 - 2 ^ 1 - 2 y v+l up ^-v 3-x " - 1 2it + 4 ' 2 - y V - 1 u + 3' ro B a i t o a n 3 9 . Giai he phUOng trinh | ^2 3y2^_""i^ 3- 2 - u +1 V+1 /g Ta t h u dUdc he om I Hu'dng d a n . .c C a c h 1. D a t D a t u = x + y, v = x - y. K h i do u - V _ 4 - 2u kb .QJTi ok u + •() ~ 2u + 4 uv — \ — V bo '* ' \+ - UV = 1 ^ \. uv + 1 7' + 3 ce
n _ y x'^ + 1 uv = xy -\ 1 4.4 P h e p d a t a — x + i , v = y + -• V x' y'^ + 1 ' xy xy y • X / 1A K l i i do {x + y) 1+ — = 4 \ B a i t o a n 4 1 . Giai he phudng trinh • 1 x2 + u + t) = (x + y) f l + —) 1^-11 = ( x - y ) H xy + h — = 4. V xy xy xy uv = x y H 1-2 = 4 / Hifcing d a n . D a t u = x + i , u = y + ^, t a t l m dudc he { ^ 4 . xy V xyy 01 oc B a i t o a n 4 2 . Giaijie | ^ ^2) _^ 3.2^2^ ^ 208x2y-. f (x + y) 1 \ iH phmng Irinh 1+ — =4 B a i t o a n 46. Gidi he phiMng irinh \ V xyy Da xy + — = 2 / Hvfdng d a n . De thay (x; y) = (0; 0) la nghieni cua h§. T i e p tlieo xet xy 0. ^- xy hi Ho titdng ditrmg vrJi nT Hifctng d a n . D a t u = x + - i , v = y + - , t a t h u dudc he | " , + 2 ' r i y uO X • u t — 4t. ie 1 \ iL = 208. B a i t o a n 4 7 . G?d? he phuong trinh Ta 1 V xyJ 4 s/ Dat u = X + ^ , v^y^-K t a t l m diOTc he | "2"t|.\,2 ^ 2 1 2 . up y (x2 + 1) = 2x (y2 + l ) Hu-dng d a n . Dat u = x + ^, i ; = y + ^, t a t h u dUdc he | 4("2"^_j_^]2y f 45. ro B a i t o a n 4 3 . Gidi he phMng trinh \ .2 _^ 2^ ^^ = 16. /g om / •! •2\ 1\ 25 B a i t o a n 48. Gidi he. phudng trinh .c Uvtdng d a n . Dieu kieu x y ^ 0. Dat u = x + - , 7; = y + ^, t a t l m ditcfc he ^ y ok bo Hifdng d a n . Dat u = x + J , ^ = y + ^ , t a t h u dudc he | + J J ^ j I ^35 ce + i;2 = 20. { fa xy (2x + y - 6) + y + 2x = 0 = 1 w. ^^2 _|_ y2^ ^2 _)_ ^ _ g B a i t o a n 4 4 . Gidi he phmng trinh ww 2u + w = 6 r 2x'^y + y^x + 2y 4- x = Qxy Hifdng d i n . D a t u = x + - , v = y + - , t a t l m dUdc he u2 + t;2 ^ 8. y X B a i t o a n 4 5 . G i d i he phUdng trinh I J_ + 4_ E = 4. I xy X y 1-4.5 M o t s o p h e p d a t §n p h u k h a c . \ / 1 ( X + - = 4 y + - Cac phep dat fiu p h u r a t da daug va phong phii. Ta can kliai thac cac dac H i f d n g d a n . He phimng t i i i i h tucing diMiig V ..V/ fiipm rieng, cac t i n h chat dilc biet cua tftng he phUdng t r i n h de dita ra phep 2 (x+ - = 6. x)
Bai toan 50. Gidi he phUdng trinh x = 4- y/W va y = 3 + / l O 4 + v'lO va y = 3 - ^10. Giai. Bien d6i he da rho, ta tlni ditrJc X = ' V' ''•'' + x/ - xy{x + y) = 3 Thii lai, thay tat ca deu thoa man. He phirdng trinh da cho c6 4 nghiem la x^ + xy{x + y) = 15. (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. / 01 Bai toan 52 (HSG tinh Ha Tinh, nam hoc 2010-2011). Gidt he do. ta oc CO . 2y = 1 y = 21 3 + iH X = x-'^ + =9 >^ < . y. = 3 _ 1 x' + i/V 2x = 4 x2 + y2 xy{M '+y) = 6 • =2 Da .X = 2 hi y = l- y he CO nghiem (x; y) = (1; 2), (x; y) = (2; 1). nT Giai. Difiu kien: xy ^ 0, x^ + y^ 7^ 1. Dat a = x^ + y^ - 1, 6 = 06 7^ 0. He uO Bai toan 51 (HSG Hai Phong, bang A, nam hoc 2010-2011). Gidt da cho trd thanh ie /x + - + ^ x + y - 3 = 3 3 2_ iL he •phuanc) trinh sau V 26 + 3 6 ~ 2x + y + -1 = rt = 26 + 3 Ta a-2b =3 6=-lvaa = l s/ y h = 3 vh a = 9 up Giai. Dieu kieii y 7^ 0, x + - > 0, x + y > 3. Dat 1 • V6i a= l,h= - 1 , ta c6 x^ + y^ = 2 va x = ~y, ta tini ditdc hai nghiem la ro /g a= ^ x + i , 6= v^x + y - 3, a , 6 > 0. om (x;y) = ( l ; - l ) , ( x ; y ) = ( - l ; l ) . ^ , Vdi n = 9,b = 3, ta c6 x^ + y^ = 10 va x = 3y, ta tini dUdc hai nghiem la .c He da oho viet lai la { "J'^^t a = 2 va 6 = 1 (.x;y)=:(3;l),(x;y) = ( - 3 ; - l ) . ok 5 a = 1 va 6 = 2. Thii lai ta deu thay thoa man. Vay he dii cho co 4 nghiem phan biet la bo • V6i a = 2 va 6 = 1, ta c6 ce (x;y) = ( l ; - l ) , ( x ; y ) = = ( - l ; l ) , (x;y)= (3;l),(x;y) = (-3; - 1). ,,,, '= 2 va Jx + y - i = l < ! = > x + 1- = 4 v a x + y = 4 fa xJr- ' x^ + y2 + x^ = 4y - 1 y V Bai toan 53. Gidi he phuang trinh x + y = w. X = 3 va y — I +2 x2 + l ww f x 2 - 8 x + 15 = 0 X+ = 4
= 7 ^' iifii-f. oc 1 Dat u = X + y + _^ ^ ^, = x - y, dieu kien u > 2. Thay vao (1), ta ditoc (sin65"; cos65"), (sin 185"; cos 185"), (siii305"; cos 305"), iH (sin85"; cos85"), (sin205"; cos 205"), (sin325"; cos.325"). . •„ • ^, , Da { a + 6 =^3"" ^^^"i^^ ^) = 1)' '^"y '•'^ (^; 2/) = (1; 0). hi nT B a i t o a n 55. Crdi he pliMng trlnh sau ( + '^-V ^^^J' " 2 Si! 1.5 Phu'dng phap cong, phifdng phap the. - | uO \ + 3x^y - + y^ =0. (2) ie G i a i . Xet x = 0 => y = 0. Vay (0;0) la mot nghiem ciia he. Xet x ^ 0, chiii Day la phitdng phap cd ban nhat. T i t bai hoc "vfl long" ve he phudng t r i n l i iL hai ve cua (1) cho x, hai ve ciia (2) cho x^, ta dudc da C O phitdng phap nay. T u y nhien phitdng phap nay van thilclng xuat hien ci Ta nhrtng ky t h i Idn, nhifng ky t h i chi danh cho nhitng hoc sinh xuat s;lc. Sach + - +y = 3 s/ X (x+^)+y = 3 nao viet vc he phitdng t i i n h cung c6 pluldng phap nay, do vay sau day t a chi up trinh bay mot so bai toan kho va d i sau lidn vao viec phan tich k y thuat giai x2 + ^ + 3y = 5 x'' cung nhif ky thuat sang tac. ro /g Dat 2 = X + - , t a t h u ditdc he om V i d u 1. Xuat phdt tit mot bien doi tiMng dudng do ta chon '"^ , (x - 2)'* = (y + 3)^ ^ x'^ - 6x^ + 12x = y'* + 9y^ + 27y + 35. (1) .c / f + y = 3 y\) = ( 4 ; - l ) y;z) = \l-2). ok I z' + y = b Khi (x; y) = (3; - 2 ) thi (1) dung. Cung vdi (x; y) = (3; - 2 ) thi , bo x - ' - y ^ = 35. -nf. ,4. (2) ce k h i (y; z) = (4; - 1 ) : I ^ ; I ^ _1 ^ { ^ 4 ^ 0 (vo nghiem). fa Tf( (1) vd (2) fa dmr 2x2 ^ 3,^2 ^ 4^,, (3) (l;2),tac6{;;;L2 w. Khi(y;z) = ^ { + I = 0 ^ { y = }. Tii (2) vd (3) /.a c6 bdi toan. sau. ww Vay he c6 hai nghiem (x; y) = (0; 0), (x; y) = ( 1 ; 1). B a i t o a n 57. Gidt he phuang trinh | l[2f~^yi _ cjy. (2) B a i t o a n 56. Gidi he phuang tfinh | + ^^^2^) = ^ G i a i . Nhan hai ve ciia (2) vdi - 3 roi cong vdi (1), ta dUdc G i a i . Do x^ + x/ = I nen ton Lai a e [O; 360"] sao cho x = s i n n , y = coso. x^ -x/ - 6x^ - 9y2 = 35 - 12x + 27y Do do (1) trcl tlianh - 6.r2 + 12.T - 8 = + 9.1/2 + 27?/ + 27 • \/2(sina - co.sa)(l + 2 sin 2a) = \/3 ^ (x - 2)^ = (y + 3)^ X - 2= y + 3 ^ X = y + 5. 34