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

AdST~ NGUYEN TAI CHUNG # ? 9 gmi PHIfODG TRiHH HE PHIfOnC TRinH BAT PHIIDnG TRinH PHlJdNG PHAP XAY DL/NG BE TDAN CAC DANG TCAN, CAC PHLfdNG PHAP GIAI CAC OE THI HOC SINH GICI GJUCC EIA, OLYMPIC 30/4 TAI LIEU BDI DL/QNG HOC SINH KHA GICI fx TAI LIEU ON LUYEN THI BAI HOC TAI LIEU THAM KHAC CHD GIAO VIEN NHA XUAT eXNTraNG HdP THANH PHD HO CHI MINH SANG TAO VA GIAI PHUONG TRINH, HE PHLfdNG TRINH, BAT PHaONG TRINH NGUYIN TAI CHUNG Chiu trach nhiem xuS't ban 4 ; NGUYEN THI THANH HlJdNG Bien tap : QUOC NHAN Si^abanin . : HOANG NHlTX Trinh bay : C6ng ty KHANG VIET Bia : C6ng ty KHANG VIET NHA XUAT BAN TONG H0P TP. HO CHf MINH NHA SACH TONG HOP 62 Nguyen ThI Minh Khai, Q.l DT: 38225340 - 38296764 - 38247225 Fax: 84.8.38222726 Email: tonghop@nxbhcm.com.vn Website: www.nxbhcm.com.vn/ www.fiditour.com Tong phdt hanh CONG TY TNHH MTV DjCH Vg VAN HOA KHANG VIET (^Dia chi- 71 Oinh Tien Hoang - P.Da Kao - Q.1 - TP.HCM 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: CONG TY CO PHAN THL/ONG MAI NHAT NAM 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 H6 Chi Minh cap ngay 19/03/2013 In xong va nop luU chieu Quy II nam 201 3 LMnoidau 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 giao vien cung chi biet suti tarn cac bai tap c6 trong sach giao khoa, sach tham 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 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 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, DHKHTN 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 am 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 tu phia cac ban la dieu chiing toi luon mong dpi. Hy 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. Tac gia Thac sy: NGUYEN TAI CHUNG Nha sach Khang Viet xin trdn trgng gi&i thi?u tai Quy dgc gia va xin idng nghe moi y kieh dong gop, de cuon sach ngdy cang hay hem, bo ich hon. Thuxingici ve: Cty TNHH Mpt Thanh Vien - Dich Vu Van Hoa Khang Vi?t. 71, Dinh Tien Hoang, P. Dakao. Quan 1, TP. HCM Tel: (08) 39115694 - 39111969 - 39111968 - 39105797 - Fax: (08) 39110880 Hoac Email: khangvietbookstore@yahoo.com.vn ChiMng 1. PhUcfng phdp sang tdc va giai phUcfng trinh, hf phUOng trinh, bd'i 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 li Lagrange 63 1.8 Phu'dng phdp hinh hpc 69 1.9 PhiTdng phap ba't dang thtfc 82 1.10 PhiTdng phap tham bien 95 ChUcfng 2. PhUcfng phdp da thiic va phUcfng trinh phdn thitc hOu ti. 116 2.1 Cdc dong nha't thiJc bo sung 116 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 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 ding 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 307 4.7 Phi/dng phap bien doi ding thiJc 314 4.8 MotsohekhongmaumiTc 317 ChUctng 5. Cdc bai todn phUcmg trinh, h^ phUcfng trinh, bat phUcfng trinh trong dethidt^ihQc 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 368 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 Chi:fc?ng 1 Phi:fdng phap sang tac va giai phifcfng trinh, he phi:fcfng trinh, bat phi:^dng trinh 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 lai 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^^ - llx + 21 - 3^4.x- -4 = 0. Giai. Tap xac dinh D = E. Plntdug trinh da cho tu'diig ditdng vdi •^ikf •'A/nh 6" ^(4x-4)2-I(4x-4) + 12-3x/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 Do do (2) <=> i = 2 => X = 3. LULU y. De c6 (1) ta can tiiii a, (3,7 sao cho - llx + 21 = a(4x - 4)^ +/3(4a; - 4) + 7 <!=>2x2 - llx + 21 = IGrtx^ + (4/^ - 32rv);r + (16^ -4/^ + 7) 16a = 2 , , fl 7 \ f 16a = 2 ^{ 4/3-32a = -11 ^{a;(i\-i) = I 16a-4/3 + 7 = 21 dung phuong phap he so bat dinh cho ta 15i giai bai toan mot each rat tijt nhien va ro rang. Bai toan 2. Giai phiCcfng trinh -x+3 = 2\/r^-/m^+3\/r^. (i) Giai. Tap xac dinh D = [-1; 1]. PhUdng trinh (1) viet lai nhu sau : (1 + x) + 2(1 - x) - 2v^r^ + \/l + .T - 3\/l - x2 = 0. (2) Dat u = ^/^+x,V = Vl - X {u >0,v>0), ta duoc + 2^2 - 2u + It - 3uv = 0 {u^ - 2uv) + [u - 2v) - {uv - 2^^) = 0 ^(•a - 2v){u - + 1) = 0 ^ [ « Z 1 ^ 0 • ^/3^ Lu^i y. Dl CO (2), ta tim a, f3 sao cho -x + 3 = a(l + x) + /3(l-x)o{ ^;^^3^ ^{ g = i Doi v6i bai toan tong quat : Giai phvfdng trinh p{x) = as/1 - X + bVl + X + cVl - x^, Ta bieu dien p{x) theo 1 - x, 1 + x va dat u = -/I + X, i; = \/l - X (u > 0, i; > 0). Khi do dirdc phitdng trinh doi vdi u, v c6 thg phan tich ditdc. Vi du 1. Ta sc. sang tdc mM phUdng trinh duclc gidi hhng phiMng phdp he so bat dinh nhu sau : Ta c6 {a-b+ l)(2a - 6 + 3) = 0 ^ 20^ + 6^ - 3a6 + 5a - 46 + 3 = 0. Tii day lay a — ^Jl + x vd b = \/l - x ta diidc 2x + 2 + 1 - X - 3\/l - x2 + 5VI+X - 4s/l-x + 3 = 0. Rut gon ta duac bai toan sau. 4 Bai toan 3. Gidi phuang trinh 4\/l - x = x + 6 - 3\/l - x^ + 5s/TTx. X V3 Dap so. X = — J Bai toan 4. Gidi phuang trinh 4 + 2\/l - x = -3x + SVxTT + Vl - x^. Dap so. PhUdng trinh c6 tap nghiem S — I 0; —; — I. tah sv, ;: i ; 25 2 j Bai toan 5. Gidi phuang trinh lOx^ + 3x + 1 = (6x + l)Vx2 + 3. (*) Giai. Dat u = 6x + 1, ?; = \/.x2 + 3. Ta c6 10x2 + 3x + 1 = i(6x + 1)2 + (x^ + 3) - - = — + ^2 _ 9 4 4 4 4 5, Thay vac (*) : + 1,2 _ 5 = ^iK:^ (u - 2t;)2 = 9 <^ u - 2u = ±3. • Vdi w - 2D = 3, ta CO v 1 + 6x - 2v/x2 + 3 = 3<^3x-l = \/x2 + 3 -'^'^i'l'S 3x - 1 > 0 ^ , x2 + 3=(3x-l)2 ^x^l. • Vdi u - 2t; = -3, ta c6 Vay phiMng trinh c6 tap nghiem 5 = < 1; ——^ I. , Lvfu y. Phudng phap he so bat dinh de giai he phudng trinh se ditdc de cap trong phan phan tich tim Idi giai cac bai toan ciia bai 1.5 : PhUdng phap cong, phiTdng phap the (d trang 35). 1.2 Phifcfng phap difa ve he. Dg giai phUdng trinh bang each dua ve he phUdng trinh ta thutdng dat an phu, phep dat an phu nay cimg vdi phUdng trinh trong gia thiet cho ta mpt h$ phitdng trinh. Sau day ta se trinh bay phuldng phap sang tac (thong qua cac VI du), phUdng phap giai (thong qua Idi giai cac bai toan va quan trong hdn niia la cac hru y sau Idi giai). Cac phudng phap sang tac ciing nhiT phifdng phap giai cac phUdng trinh bang each dUa ve he con dildc de cap rat nhiiu 6 sau bai nay (chang han bai 3.2 d trang 160). 5 Vi du 1. Xet I y ~ 2 ^ 3^^ ^ x = 2 - 3 (2 - 3x•'^)^ Ta c6 hdi todn sau. Bai toan 6. Gidi phUdng trlnh x + 3 (2 - 3x^)^ = 2. Giai. Dat, = 2-3x^.1^ CO he ^[^ZlZ^. (1) tnr (2) ta fhrac x-y = 3{x'^ - y^) ^ X - y = Q 3(x + y) = l ^ y = X 1 - 3x y = —^—• Vc'ri y = x, thay vao (1) ta diWc Sx^ + x - 2 = 0 x G |~^' 1 - 3x Vdi y = 3 1 - 3x , thay vao (2) ta dudc = 2 - 3x^ <^ 9x2 - 3x - 5 = 0 X 1 ± v/21 PhUdiig trinh da cho co bon nghiem 2 1-V21 1 + V21 X = -1, X = -, X = ——, X = . 3 6 () Lxiu y. Tir Idi giai trcn ta thay iftng neu kliai Irien (2 - 3x'^)'^ tlii sc dira phitdng trinh da cho vc phiWng trinh da thiitc bac bon, sau do Ijien doi thanh (x + l)(3x - 2)(9x2 - 3x - 5) = 0. Vay ncu khi sang tac de toan, ta c6 y lam cho plnMng trinh khong v6 nghiem hiiu ti thi phildng phap khai trien dua ve phUdng trinli bac cao, sau do phan tich dua ve phu:dng trinh tich se gap nhieu klio khan. Vi du 2. Xet mot phucing trinh bac hai c6 cd hai nghiem Id so v6 ti 5x'^ - 2x - 1 = 0 ^ 2x = 5x2 _ 2x = 5 5x2- 1\ - 1. Ta CO bdi todn sau. Bai toan 7. Gidi phUdng trinh 8x - 5 (5x2 _ _ _^ Giai. Dat 2y = 5x2 _ ^ YAn do 2y = 5x2-1 . ^ r 2y = 5x2-1 (1) 8x - 5.42/2 = -4 ^ \x = 5y2 - 1. (2) 6 Lay (1) tru' (2) tlico ve ta du'dc y = X y - X = 0 2 = -5(x + y) ^ 2(y - x) 5(.T2 - y2) ^ Vdi y = X, thay vao (1) ta dUdc 5x2 _2x -\=i) ^ x = 5x + 2 1 ± \/6 • Vdi y = —. thay vao (1) ta du'dc >;V;!; .• o -l^ = 5x2-l.=.25x2 + l(,x-l=0^x=-^=^^ PhUdng trinh da cho c6 bon nghic'm 25 1 ± v/O -1 ± 72 5 5 Luti y. Phep dat 2y = 5x2-1 chrdc tini ra nhu sau: Ta dat n:y-\-b = 5x2 _ ^ vdi a, h thn sau. Khi do tiiu dUdc he ay + 6 = 5.r2 - 1 { + b + i = hx'- , 8x - 5 («y + bf = -4 ^ \.T + 4 - 5^2 = 5a2y2 + i{)aby. C fl _ _5_ _ ^+1 r , _ . Dg he tren la he doi xftng loai TT thi < g ~ 5„2 ~ 4 _ 5/^2 => | Z 9 Vfw I 10a/; = 0 ' I « - - ta CO phep dat 2y — 5x'^ - 1. f\ . -jj.,., Bai toan 8. GidirphtMng trinh 5{5x~ - 17)2 - 343x - 833 = 0. Y tvtdng. Dat ay + b^5x^- 17 (a ^ 0). Klii do . , jay + b = 5.7:2 _ ^7 , , , \5(ay + 6)2 - 343x - 833 - 0. (*) • '' Tir (*) ta CO 5(ay)2 + lOa^y + ^2 - 343x - 833 = 0 ^ x = 5(ay)2 + 10a6y + ^2 - 833 „ * ^ 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 5a'^y2+l()o2.6y+62.a- 833a , • M! I - o/y-l7= 46 <^343.5y2 5831 = 5a''.y2 + l()a2.6y + 62,„ g33„ _|. 3435. f 343 = a'* r - 7 ' Dong nhat he so ta ditcJc I al^h = 0 1 / = (1 "^''^^ ''"^ t 833a+ 3436=-5831 Idi giai sau. 7 Giai. Dat 7y = 5x^ + 17, ta c6 h? phitdng trinh J7y = 5x2 _ 17 1^5,y2 - 343a; - 833 = 0 7y = 5x2 _ 17 (1) 7x = 5y2_17. (2) Lay (1) trit (2) ta c6 7{y - x) = 5(x + y)(x - y) ^ * Neu X = •(/, thay vao (1) : Sx^ - 7x - 17 = 0 x = * Niu 5x + 5j/ = -7, ket hdp (1) ta c6 . x = y 5x + 5y = -7. 7±\/389 10 Ket luan: Phudng trinh c6 tap nghiem 5 = '7± \/389 -35 ±5v^l 50 J • 10 Vi du 3. Ta CO Ax^ - 3x = ~ <^ Qx = 8x^ - V^. Vdy xet fSx^ - v/3\ -V3 { >1296x + 216v/3 = 8 (Sx^ - ^/fj ^ 162x + 27^ = (Sx^ - Vs^ . Ta CO hai todn sau. Bai toan 9. Gidi phUOng tnnh 162x + 27\/3 = (8x^ - s/zf . Giai. Dat 6y = 8x^ - \/3. Ta c6 h^ 6y - 8x3 - v/3 162x + 27\/3 = 216?y3 r 6?y = 8x3 _ ^3 \x = 8?y3 - v/3 (2) Lay (1) tilt (2) theo ve ta dudc 6(y - x) = 8(x3 - 2/3) ^ (a; _ [g (x^ + xy + y^) + 6] = 0. Vi x^ + xy + y2 > 0 nen 8 (x^ + xy + y^) + 6 > 0. Do do tit (3) ta dUdc x = Thay vao (1) ta dUdc 6i = 8x3 - \/3 <^ 4x3 - 3x = ^ 4^3 _ 3^, = cos ^. a a Sii di^ng cong thiic cosa = 4cos3 - - 3cos - , ta c6 cos-=4cos 18-3COS-, 8 llTT 1 llTT COS = 4 cos-* 18 6 18 137r 137r COS —— = 4 cos'' 18 6 18 TT llTT 137r r = cos ——, X = COS 18' • 18 ' 18 - 3 cos — 3 cos llTT 18 ' 137r 18 • la tat ca cac nghiem ciia phuong trinh (4) va cung la tat ca cac nghiem cua phUdng trinh da cho. Ltfti y. Phep dat 6y = 8x3 _ ^ dUdc tun ra nhu sau : Ta dat ay + 6 = 8x3 - V3 ^^^^ ^ ^jj^ Ket hdp v6i phudng trinh da cho c6 he ay + 6 = 8x3 - v/3 162x + 27V3 = a3y3 + 3a26y2 + Safety + ^3 Can chgn a va 6 sao cho : 8 73 162 o? 27\/5 - 63 3a26-3a62 = 0 Vay ta c6 phep dat 6y = 8x3 _ ^ Vi du 4. Xet tarn thiic bdc hai luon nhdn gid tri duang : x'^ + 2. Khi do / 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. 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 o dong bien k{x) sao cho k{x) = x <^ g{x) — 0, muSn vdy ta xet v. . ^ + ax - 15 = X <!::^ ^ + (Q - l)x - 15 = 0. Do do chon a sao cho a - 1 = 2 a = 3, khi do k{x) = — + 3x - 15 fd x3 ^ A;(x) = y tuang dudny U(H .• — + 3x - 15 = y <^ x^ + 9x - 45 = 3y. TH phiiang tnnh cuoi ndy thay x bdi y ta thu duac he doi xilng loai hai x3 + 9x - 45 = 3y y3 + 9y - 45 = 3x. Til: he tren, sU dung phep the ta thu duoc phuong trinh x3 + 9x - 45 + 9 /x3 + 9x-45\ - 45 = 3x 0fiOJ i.' I 9 <^ (x'' + 9x - 45)^ + 81 {x^ + 9x - 45) = 1215 + 81x. Vay ta thu dUdc hai todn sau. , Bai toan 10. Gidi phuong trmh ; V (x^ + 9x-45)V81 (x^ + 9x-45) = 1215 + 81X. (1) ji Giai. Tap xac dinh M. Dat + 9x - 45 = 3y. Ket hdp v6i (1) ta c6 he :^i^:,^.^7j . / x^ + 9x - 45 = 3y (2) ••-•J\ + 97/- 45 = 3x. (3) ^ Lay (2) tiif (3) theo ve, ta dUdc x^ - + 9x ^ 9)/ = 3?/ - 3x ^ - + 12(x - ?y) = 0 .^(x - y)(x^ + xy + + 12) = 0 <^ X = y. Thay vao (2) ta diWc x^ + 9x - 45 = 3x <=> (x - 3) (x^ + 3x + 15) = 0 ^ x = 3. Phildng tiiiih da cho c6 nghiem duy nhat x = 3. LuTu y. Phcp dat x^ + 9x - 45 = 3y dittfc tun ra uhu sau: Ta dat x^ + 9x - 45 = ay (vdi a tini sau). Khi do x^ + 9x - 45 = ay ^ | xj^ + 9x— 45 = ay_ a'Uj^ + Slay = 1215 + Six \ + Slay - 1215 = 81x. , . , «3 81« 1215 81 „ „ Vay can chon a thoa man dieu kien — = — = —7- = — => a = J. Uo cto ••' • 1 9 45 a dat x^ + 6x ~ 45 - 3y, ta so thu ducJc mot ho doi xi'mg loai hai. Vi du 5. Chon mot phMng trmh chi c6 hai nghiem /d0 vd 1 IdlV = lOx+1. Tii phiCOng trinh nay ta thiel lap mot he doi xvCng loai hai, sau do lai quay ve phuang trmh nhu sau : f ll'- = lOy + 1 ^ I y = logn (lOx + 1) - = log,, (lOr + 1) \iry = 10x + l ^\lF=10y + l ^ 10 logiiuux+ij. Suy ra IF = lOlogn (10x + 1) + 1 ^ IF = 21ogii (10x+ 1)^ + 1. Ta c6 bdi todn sau. Bai toan 11. Gidi phuang trinh IV = 21ogii (lOx + 1)^ + 1. 10 Giai. Dicu kiCm > Dat y = logn (lOx + 1), Ivhi do IP = 10x+ 1. Ket hdp vdi phUdng trinh da cho, ta c6 he | JJy ^ ^ | ^ "» " • - - w., Lay (1) tnr (2) tlieo ve ta ditdc • f-: , A. • IF - ir^ = lOy - lOx <^ ir^ + lOx = IP + lOy, (3) Xet ham so /(/) = 11' + 10/ Ta c6 f'{f.) = 11' hi 11 +10> 0, V/. G K. Vay ham so / dong bien tren E. Ma (3) chinh la /(x) = /(y) nen x = y. Thay vao (1) ta difdc IL''= 10x+1 <^ IF - lOx-1 =0. .• (4) Xet h;\ so f/(x) = IF - lOx - 1 tren khoang ( ; +00 . Ta c6 V 10 J g\x) = ll-^nll - 10, g'\x) = lF(lnll)2 > 0. / 1 Vay ham so g c6 do thi luon 16m tren khoang ^-j^; +00 j , suy ra do thi cvia ham g va true hoanh co vdi nhau khong qua hai diem chung, suy ra (4) c6 khong qua 2 nghiem. Ma g{l) = 0, y(0) = 0 nen x = 0 va x = 1 la tat ca cac nghiem ciia (4). Nghiem ciia phUdng trinh da cho la x = 0 va x = 1. Vi du 6. Ta se su dung phuang phdp lap di sang tdc phuang trinh tit. he phuang trinh doi xvtng loai hai. Xuat phdt tit \^^Z ^^^4^30 •^^ ^^'^'"'(1 P^^'^P the ta diMc phuang trmh Ax = ^30 + |v/x + 30. Til phuong trinh nay ta lai, thu dtWc he doi xtCng loai liai ; .Zi nst,- 4 4M = ^/30+ -v/aM^ 4x= ^/30+ -v/^r+30. Tir he niiy, ticp tuc s'li dung phcp the ta thu diMc phuang trmh \ 4x = Ta CO bdi todn sau. Bai toan 12 (De nghi Olympic 30/04/2010). Gidi phuang trinh 4x = 30 + - W 30 + - ^30 + - \/^T30. 11 Giai. Do x la nghifni thi x > 0. Dat u = 30 + ^x + 30, tit phitdng tnnh da cho ta c6 ho Gia s\t X > u. Khi do 4u = J30+ -y/xT3Q (1) 4x = A/30+ -v/w + SO. 4u = \/30+ + 30 > ^30 + -\/u + 30 = 4x =^ u > x =^ x = ?i. Vay tit he (1) ta c6 a; = u va 4x = ^30 + -yxTSO. Dat . = \V^FT30, tit (2) ta c6 he | ^J I Gia sii x> V. Khi do (2) (3) 4v = Vx + 30 > VtTTSO = 4a: 4u > 4a: =^ V > a; =^ u = X. , f T > 0 1 + \/l921 Vay r = .x va 4.T = ^ | Jgp^^ ^, ^ 30 — . PhUdng trinh da cho co nghiem dny nhat x = 1 + 71921 32 Vi du 7. Vdi X = 8 thi ^/x-\-8-\- \Jx-l — 3, ia c6 bai todn {ch&c chan co mot nghiem dep x = 8) sau. Bai toan 13. Giai phUdng trinh y/x + 8 + \/x - 7 = 3. Giai. Dieu kien x > 7. Dat u = ^x + 8 > 0 va u = v'x - 7 > 0. Ta c6 he u + r = 3 { V = 2) — u U.,V>{) ^ i 0/2 ,,2)(„2 + ,,2)^15 u4-t;'*-15 [u,(;>0 { u = 3- u ft; = 3- u 0 < i< < 3 (2u-3)(2u2_6u + 9) = 5 ^ ro < 1/- < 3 ^ ro < u < 3 ^ \4u^ - 18u2 + 36u - 32 = 0 ^\ = 2- Tit do ta thu dudc 1= 2 <=>{^ + f=p ^x = 8 (thoa man dieu kien). Vay phitdng trinh da cho co nghiem duy nhat x = 8. < 3 u2 + (3 - uf = 5 ^' 12 Ltfti y- Doi vdi phu'dng trinh - JJx) + "\/b + /(x) = c, ta co each giai : Dat u = 'ija - fix), v = '^h + f(x), dan den he {^H ^+,n"s'^['^ Nhit vay dang nay la j)hn'dng trinh vo ti, infi san khi dat an phu dita ve he, roi dimg phep the dan tdi phudng trinh da thitc, do do khi sang tac de toan ta phai dac biet chii y cac chi so can. Chang han d vi dn 7 thi m = n = 4 nen ta yen tam rang se dan tdi phitdng trinh da thite bac 4 co it nhat mot nghiem dep. Vi du 8. Vd'i. x = -2 thi 2<y3x -2 + 3^6 - 5x = 8, ta c6 bai todn {chac chdn CO mot nghiem dep x = -2) .sau. 1 "i;.,/ .;- .f '-,1,; Bai toan 14. Gtdi phiMng trinh 2 v^3x - 2 + 3^6 - 5x = 8. Giai. Dieu kien x <^. Dat u = ^3x - 2, v = ^/G - 5x > 0. Khi do 5 I"2 Z t' 7 ^ + Sv'^ = 5(3x - 2) + 3(6 - 5x) = 8. . — 0, ox Mat khac ta lai co 2u + 3r -8 = 0. Vay ta co he {^t + 'fv = 8^ =^ + 3 (^^) ' = 8 ^ 15 ^ + 4^2 - 32z. + 40 = 0 Phu'dng trinh nay c6 nghiem duy nhat u = -2 nen v'Sx - 2 = -2 X- = -2. Bai toan 15. Giai phiMng trinh 1 + \/l - x2 [V(l + :r)-* - ^(1 - x)'A^ =2+ yjl - xK ffif, t uM Giai. Dieu kien -1 < x < 1. Dat ^l + x = a, \/r^ = vdi a > 0, 6 > 0. Khi do a' + l? = 2. Ta co he sau ( \ . . S "" ' \l + ab{a-^ -b^) = 2 + ab. (2) , (1) =^ {a + bf = 2+ 2ab^ s/lT^=-^{a + b) [do a,b>0). V2 Ket-hop (2) ta co ' . , | 1 1 ' ( ' -7= (a + b){a - b){a^ + b^ + ab) = 2 + ah => ^(a^ ~ h'-) = I. v2 v2 Tit do ta c6 he | ~ ^2 !l 2^ Cong hai phitdng trinh ve theo ve ta co 2a2 =2+y2^a2 = l + 4=^l+a;=l + ^^x = 4=- V2 s/2 V2 , Vay phitdng trinh co nghiem duy nhat x = —. vj / j 13 Bai toan 16. Gidi phuang irinh \/\/2 1 - x + = v2 Giai. Dini kien 0 < x < \/2 - 1. Dfit \/\/2 - 1 - x = u va ^ = v. Khi do 0 < u < \/s/2-l va - 1. Nhit vay ta c6 he .u:^ + v^ = v/2 - 1 Ttr phudng triiih thi'i: hai, ta co u = —^ V ( 1 - V + 7-4 = v/2-1. 1 \ 2v + + i;' = \/2 - 1 v/2 v/2 1 ± - 3 ,72 Bai toan 17. G'jdv phifdng trmh v/l - -x^ = Q - . Giai. Dieii kien | - ^' <^ 0 < x < 1. Dat u = sji: va v; = ^ - vdi 2 r 1 - x^ = 1 - «4 « > 0, v< Do do 1(1 Ta CO he U + V = - I («2+t.^)^-2»^.7>2^1 I [(i7,+ r)2-2n.r 2 \ 2 - - 2u v 3 - 2u2.,;2 ^ 1 W + U = - 2u^.i)-^ u.v =0 9 81 14 n + J' = - ^ < 8 - vfei h"^^ 18 71 + 7) = - 8+ vfei Vay w, f la nghiem ciia 8 - yi94 18_ 8 + 3' 18 nen nghiem duy nhat ciia phudng trinh la / 18 •, ! i i = 0 (1) = 0. (2) 1 -2 + ^2(7194-6)+^^ / Do (2) v6 nghiem 1.3 Phifcfng phap difa phifdng trinh ve phifdng trinh ham 1.3.1 Phu'dng phap giai. Dita vao ket ciua : Neu ham so y = f{x) ddn dieu tron khoang (a; b) va x,ye (a; b) thi /(^) = /(y) a; = 7^ ta CO the sang tac va giai dUdc nhien phitdng trinh hay va kho, thudng gap trong cac k}' thi hoc sinh gioi. D6 van dung dildc phitdng phap nay, ta thirdng bien ddi phiWng tiinh da cho thanh phitdng trinh ham f {<f){x)) = f (ipix)), trong do / la ham ddn dicu. Tfr day diui den mot phvtdng trinh ddn gian lidn 4>{x) = tpix). De giai dUdc cac bai toan bang phitdng phap nay thi nhftng kien thifc ve ham so nlut dao ham, xet sit bien thien va kl nang doan nghiem la cite ki ciuan trong, c6 nhitng bai doan dUdc dap so la da hoan thanh den hdn 90% Idi giai. Phitdng phap nay ditdc si't dung nhien, chang han d muc 3.6.3 d trang 191. Mot so tritdng hdp dac biet thitdng gap : • Neu / la ham ddn dieu tren khoang (a; 6) thi plutdng trinh /(x) = k {k la hang so) CO khong qua 1 nghiem tren khoang {n; h). • Neu f yk g \h hai ham ddn dieu ngitdc chieu tren khoang (a; b) thi phitdng trinh /(x) = g{x) c6 khong qua 1 nghiem tren khoang (o;6). • Neu ta thay cum tit "/ la ham ddn dieu tren khoang (a; 6)" bdi cum tit "/ la ham ddn dieu tren m5i khoang (a; 6), {c]d)" thi hai ket qua d tren se khong dung, ti'tc la plutdng trinh co thc^ sc c6 nhien hdn mot nghiem. Ban doc hay xein bai toan 20 d trang 16. Bai toan 18 (HSG Quang Ninh 2011). Gidi phieang tnnh 1 1 = + v^5x - 7 v/^^ = 0. (1) 15 [...]... Giai he VTTx^ = v/3x2 + 3 + l-l) (2x - ^ ) + (^ + 2x) _ 4x2 + 1 - 4x = 1 . — I - xy u + v' I + xy uv - -1 3u - 3 1 -3 x 3-x Ta thu dUdc he 1 - 3- u+ 4-2 ^ 1 -2 y " - 1 2it + 4' 2-y u + 1 1 - 2w - 2 v+l ^-v 2 - V - 1 V +. .x > |. Ta co 6 (1) <^ 7 ^-^ - 6 logy (6x - 5) = -6 (x - 1) + (6x - 5). (2) 7-^ -^ + 6(x - 1) = (6x - 5) + 61ogy (6x - 5) <^ 0 (7 ^-1 ) = (;6 (6x - 5), vcii (/>(«) = « + 6. (3x - uY + 27(3x - 77) = 9 (-3 x2 + 21x + 5) + 27 V9 (-3 x2 + 21x + 5) ^27x^ + x2 (-2 77/+ 27) + x(9772 ~ 108) + (-u^ - 2777 - 45) =27^9 (-3 x2 + 21x + 5). wi-iart.R f -2 717 + 27 = -5 4 D5ng