Sáng tạo và giải phương trình, bất phương trình, hệ phương trình, bất đẳng thức - Tài liệu ôn thi Đại học môn Toán: Phần 1

510.76 T103L NGUYEN TRUNG KI£N TAI LIEU ONTHIDAI HOC /cirxg b o o v o gioi PHl/ONG TRlNH X BAT PHl/ONG TRlNH H£ PHl/ONG TRINH X BIT DANG THtfC • D^nh cho hQC sinh Idp 12 chiicfng t r i n h chuan n^ng cao • On tap nang cao k i nang 1km hki m Bien so^n theo npi dung vk cau true di t h i cua B Q G D & D T +6x^-2x + 3-(5x-l)Vx^+3=0 V-4x^+18x-20 + ^"'-^^^SV^ 2x2-9x + DVL.013496 Chung minh: — + — > (1) x 4y NHA XUATBAN DAI HOC QUOC 6IA HA NOI TAI LIEU O N T H I D A I H O C Joqg boo VQ gioi X PHl/ONG T R I N H BAT PHl/ONG T R I N H ' ; HE PHUWNG TRINH BAT D A N G THUt • Danh cho hoc sinh Idp 12 chiicrng t r i n h chuan va nang cao • O n tap va nang cao k i nang lam bai • Bien soan theo noi dung va cau true de t h i ciia Bo GD8fDT - I THi; VIEW T I f y ' H B I N H THUAN' OVL t 0[}{] '4 / -y • • • -' Phan 1: Ldi noi dHu PHL/ONG PHAP GlAl PHLTONG TRINH, P h u o n g t r i n h , bat p h u o n g t r i n h , h f p h u o n g t r i n h , bat d n g thuc la m ^ n g kien thuc quan t r o n g t r o n g c h u o n g t r i n h toan t h o n g Dac bi^t cac bai toan ve p h u o n g t r i n h , bat p h u o n g t r i n h , h ^ p h u o n g t r i n h , bat d a n g thuc t h u o n g BAT PHLfONG TRlNH VO TY xuyen xua't hien t r o n g cac k y t h i chQn h p c s i n h gioi, cOng n h u T u y e n sinh dai hQC va l u o n gay k h o k h a n cho hoc sinh N h S m g i i i p cac e m hpc sinh T H P T cung n h u cac e m hpc sinh chuyen Toan NHOTNG KI^N T H L f C B6 T R O C H O GlAl P H l / O N G T R I N H V O i t I CO m p t tai l i f u m a n g t i n h h f t h o n g de o n l u y f n , nang cao k i e n thuc k y nang G i a i p h u a n g trinh bac 4: giai toan de d a t ket qua cao nha't cac k y t h i H g c sinh g i o i , k y t h i T u y e n a) PhuoiTg trinh dang: x^ = ax^ + bx + c sinh dai hpc, c u n g n h u t h i vao cac l o p chuyen chpn, toi bien soan cuon: "Tai P h u o n g phap: Ta t h e m bot vao v e m p t l u p n g : 2mx^ + m ^ k h i d o p h u o n g li^u on thi Dai hoc - sang tao va giai phuong trinh, bai phuong trinh, h$ t r i n h t r o t h a n h : (x^ + m)^ = (2m + a)x^ + bx + c + phuong trinh, bai dang thitc" Ta m o n g m u o n ve'phai c6 dang: ( A x + B)^ N p i d u n g cuon sach d u p e chia l a m phan: 2m + a > ,Phan : P h u o n g phap giai p h u o n g t r i n h , bat p h u o n g t r i n h v t y A = h^- 4(2m + a)(c + m^) = ^ Phan 2: P h u o n g phap giai h^ p h u o n g t r i n h Phan 3: P h u a n g phap h a m so'trong cac bai toan chua t h a m so' Vi Phan 4: P h u o n g phap h a m so c h i i n g m i n h bat dang thuc va t i m a) GTLN, G T N N 1: Giai cac phuang trinh: x+V W ^ =6 b)2x2-6x-l =V i ; ^ T r o n g m o i p h a n toi l u o n c6' gang h^ t h o n g p h u o n g phap, p h a n tich, d j n h h u o n g each giai, cuoi m o i p h a n deu c6 bai tap ren l u y ^ n de cac e m hpc sinh thusuc Giii: a) D i e u k i ^ n : < x < Dat y = T o i h y v p n g cuo'n sach se la tai li^u b o ich cho cac e m hpc sinh hpc tot m o n Toan va dat ket qua cao t r o n g cac k y t h i ' ' >0 P h u o n g t r i n h t r o thanh: y2 + ^/^^^ ^ ^ j y ' " l O y ' - y + 20 = (0 < y < Vs Mac d i i da co gang d a n h nhieu t a m huyet cho vi?c bien soan cuon sach > ^f,, song thie'u sot la d i e u k h o n g the tranh k h o i Rat m o n g s u d o n g g o p phe b i n h Xet p h u o n g t r i n h : y4 - I0y2 - y + 20 = o y" = 10y2 + y - 20 ciia ban dpc de Ian tai ban sau d u p e hoan thi^n h o n Ta t h e m vao v e ' p h u o n g t r i n h m p t l u p n g : 2my^ + m^ C u o i cung toi x i n g u i l a i cam o n sau sac den ban be, d o n g n g h i f p, cac dien dan toan da cung cap m p t so'tai l i ^ u quy gia de hoan thi^n cuon sach K h i d o p h u o n g t r i n h t r o thanh: y'^ + 2my2 + m^ = (10 + 2m)y^ + y + m^ - 20 T a c gia Nguyen Trung Kien T a c o Ayp = l - ( m - ) ( + 2m) = « > m = - Ta Viet lai p h u o n g t r i n h thanh: y ^ - y + (-] =y +y+7«> y-2 =0 Tdt lieu ou tin h,u tao vd ^fiat fi, oai fi, ne yrrmn^i si'iii^ « ( y - y - ) ( y + y - ) = 0=*y = -i + Vi7 (TM) => -nguyen X = irung IVICTI II-V17 a) Dieu kifn: x > — Binh phuong hai ve'ta thu duoc phuong trinh: b) Dieu ki^n: x > — o x ^ - x ^ +8x2 + x - l = < » x ^ - x ^ = - x - x + l Dat y = V4x + > thi phuong trinh da cho c6 dang: Ta tao vetrai dang: (x^ - 3x + m)^ = y ' ' - 2 y - y + 77 = » y"* = 22y2 + y - 7 Tuc la them vao hai ve mpt lup-ng la:(9 + 2m)x2 - m x + m2 phuong trinh Ta them vao ve phuang trinh mot luong: 2my^ + m^ tro thanh: (x^ - 3x + m)^ = (2m + l)x2 - (6m + 2)x + m^ +1 Khi phuang trinh tro thanh: Ta can A\p = ( m + ) - ( m + I ) ( m + ) = o m = y^ + 2my^ + m^ = (22 + 2m)y^ + 8y + - 77 Phuang trinh tro thanh: (x^ - 3x)2 = (x -1)^ Ta CO Ayp = - 4(22 + 2m)(m2 - 77) = o m = - x = + V3 Ta viet lai phuong trinh thanh: y'* - 18y2 + 81 = 4y^ + 8y + o (y^ - y =:> m = ? b) x^ = (1 - V5^)(2x - sVx + 3) x - x - l = V4x + Giai: _ -3 + N/2T (Thoa man) ,_-3-V2T ^ Ayp =(am + c)2 - 2m + — + b (m^ +dj = v > V i du 2: Giai cac phuang trinh: a) - 6x^ + (9 + 2m)x2 - 6mx + m^ (Loai) 15-3^ r=> X= 2 Phuang trinh v6 ty ca ban: = g(x) g (x)>0 f(x) = g2(x) 7^ Vi du 1: Giai cac phuang trinh: a) 7x2 +2x + ^2x + l b) V2x +1 + = V4x + GiAi: + ( ^ - b ) [ ^ +^b + ^ = a-b^ + ( ^ / I - b ) ( V ^ + b) = a-b2 a) Phuong trinh tuong duang voi: +2x + = (2x + l ) x - l x = l i-, 3x2+2x-5 = b) Dieu ki^n: x > Binh phuong ve ta dugc: 3x +1 + l^lx^ +x = 4x + l^fhJTx x>-8 i 7x2-12x-64 = =X +8 o f / + Neu h(x) = C O nghiem x = XQ thi ta luon phan tich dugc , h(x) = (x-Xo)g(x) x>-8 Nhu vay sau buoc phan tich va riit nhan tu chung x - XQ thi phuong trinh 4(2x2+x) = (x +8)2 ban dau tro thanh: (x - Xo)A(x) = x=4 X - Xg = A(x) = Vi?c lai la diing ham so', bat dang thuc hoac nhirng danh gia co ban de ke't luan A(x) = v6 nghiem 16 x = —7 Doi chieu vai dieu kien ta tha'y chi c6 x = la nghi^m cua phuong trinh II MOT S DANG PHUONG TRJNH VO Neu phuong trinh c6 nghiem x ^ X j theo dinh ly viet dao ta c6 nhan tu • chung se la: x^ - (xj + X2 )x + Xj Xj THL/ONG GAP Ta thuong lam nhu sau: Giai phuong tiinh v6 ty bling phuong phap su dung bieu thuc lien hgp: Dau hi^u: + Muon lam xua't hien nhan tu chung ^ ( ( x ) ta tru di mot lugng ax + b + Khi ta gap cac bai toan giai phuong trinh dang: yfu^ + '^g(x) + h(x) = Khi nhan tu chung se la ke't qua sau nhan lien hgp ciia Ma khong the dua ve mpt an, hoac dua ve mot an thi tao nhihig phuong trinh bac cao dan den vi^c phan tich hoac giai true tiep kho khan + Nham dugc nghi^m cua phuong trinh do: bang thii cong (hoac su dung may tinh cam tay) Phuong phap: • Dat dieu kif n chat cua phuong trinh ( neii c6) Vi du: Doi phuong trinh: Vx^ +3 + ^fly^+V + 2x + Neu binh thuong nhin vao phuong trinh ta tha'y: Phuong trinh xac dinh voi moi x € R Nhung chua phai la dieu ki^n chat De giai quyet triet de phuong trinh ta can den dieu ki^n chat la: + Ta viet lai phuong trinh thanh: N / X ^ + - ^ x + = x - Dey rang: ylx^+S-^ll)^ + < do phuong trinh c6 nghiem 2x-3 x2+3x + ^ < va , >2 7x2 - + x - + 7x^-2+5 < 1; -< ^ li; , ^xA + -r l 1> !^ i j nen iicii , 1+V - X , Vx-2+1 1 N h a n xet: De d a n h gia p h u o n g t r i n h cuol cung v n g h i e m ta t h u o n g d u n g cac u o c l u g n g co ban: A + B > A v o i B > tir d o suy A + B>0 B>0 x - l Dat x - = t > = > x = t ^ + l -(2x + l) nen ta tach n o k h o i bieu thuc de cac thao tac t i n h toan dup-c d o n gian h o n V i d y 3: G i a i cac p h u o t i g t r i n h : b) V3x - - Vx + = a) 4Vx + + V l - x = x ^ + x + c) x2+7 '^^x d) 2(x + l ) x^+5x^+4x + x^ + x + 2x-ll ri = Vx +X + 72 Giai: 19 a) D i e u k i ^ n : - < x < — Ta n h a m duoc nghi em la x = l , x = - nen ta p h a n tich de tao nhan t u chung la: x^ + x - De l a m dug-c dieu ta th^c hien t h e m b o t nhan t u n h u sau: + Ta tao 4\Jx + - (ax + b) = cho p h u o n g t r i n h nhan x = 1, x = - la nghiem o 20 + Vl9-3x —X +— - ri3_x 3 -(x2+x-2 3 , 3s/l9r^-(13-x) 3Vx + - ( x + 5) -X' - x + 3N/X + 3+(X + 5) - x^ - x - =0 /2 ^ (x2-x-2) =0 -x-" - x + 3^3^19-3x + ( - x ) •^'•^^^ x'=+x-2 = 3>/x + + (x + 5) 3Vl9-3x +(13-x) n + =0 De thay v o i - < x < — t h i > 0, — — - — ^>0 3Vx + + ( x + 5) 3Vl9-3x +(13-x) N e n — 3N/XT3+{X + 5) 3\3^fl9^ + {13-x) + P h u o n g t r i n h da cho t u o n g d u o n g v o i x^ + x - = x=l x = -2 Vay p h u o n g t r i n h c6 nghiem la: x = 3, x = I' • N h a n xet: N e u da nha m dugc hai nghiem ciia p h u o n g t r i n h va d u doan dugc p h u o n g t r i n h chi c6 nghiem t h i ta c6 the giai theo m p t each khac n g i n g o n h o n n h u sau: Xet ham so f(x) = 4Vx + + V l - x - x^ - 2x - tren 19 Ta thay x = - ; x = - - k h o n g phai la nghiem, i3 -3-^ '3 Cty TNHH Tren -3; 19 ta -9 CO 5V3X-8 + (3x-4) ^ •-2x-2;f'(x) = -3x ' ^(x + f f'(x) = ^ -2/X + MTV DWH Khattg Viet 5V3x-8 + x - - ( x + + 5Vx + l ) < ^(19-3x) < : : > x - - V x - + —+ — + x + V x T T > 4 nen f(x) = c6 toi da hai nghiem tren '3) M a t khac: f ( - ) = f(l) = nen p h u o n g t r i n h c6 d i i n g hai n g h i f m la V3x - - — x=l x = -2 + + X + 45v'x + l > Dieu la hien nhien d u n g Vay p h u o n g t r i n h c6 nghiem la: x = 3, x = Chuy: b) D i e u k i ^ n : x > — N h u n g danh gia de ke't luan A ( x ) < t h u d n g la n h i r n g bat dang thuc khong P h u o n g t r i n h d u g c viet lai n h u sau: V X - - N / ^ chat nen ta luon dua ve dugc tong cac bieu thuc b i n h p h u o n g = 2X-11 N g o a i neu t i n h y ta c6 the thay: Ta n h a m d u o c nghiem x = 3, x = nen suy n h a n t u c h u n g la: 5\J3x - + 3x - < 9x + 63 + s V s i x + 81 N h u n g dieu la hien nhien d u n g x^ - l l x + 24 do: V x - < ] x + ; x - < x + 63 v o i m g i x > - Ta phan tich v o i n h a n t u 5\/3x-8 n h u sau: + ,,,, Tao sVSx - - (ax + b) = cho p h u o n g t r i n h n h a n x = 3,x = la N g o a i ta cung c6 the giai bai toan theo each d u n g h a m so cau a) nghiem Tuc la a,b can thoa m a n he: + 5V3x - + 3x - - ( x + + sVx + 1) < 3a + b = c) Dieu kien: x > ja=:3 8a + b - 20 ^ Ta n h a m d u g c x = 1; x = nen bie'n d o i p h u o n g t r i n h n h u sau: b = -4 T u o n g t u v o i 5vx + - ( m x + n) = ta t h u dugc: 3m + n = 10 8m + n = 15 V x - - ( x - ) + (x + ) - V x + T = x ^ - l l x + 24 573X-8 +(3x-4) {x + 7) + 5^[x + l (x^ - n x + 24J Ta c6: k h i x = 2(x + ] ) = 2,khi x=3 n=7 X =0 -4x + x -4x + Vx3 + x + x " 2(x + l ) x2+7 2(x + l ) = nen ta t r u vao ve Vx^ + - V x x^-4x +3 r \ 2(x + l ) x''-4x + 3= (1) Vx^+3x + x - ( x + l ) (2) Giai (1) suy x = 1, x = -9 5V3x-8+(3x-4) (x + 7) + V ^ =0 Giai (2) ta c6: Vx-* + x + 2x = 2(x + 1) c:> Vx^ + x = < o x ^ + x - = : < = > x = l x ^ - 1 x + 24 = Ke't luan: P h u o n g t r i n h c6 nghiem la x = 1; x = -9 5V3x-8+(3x-4) (x + 7) + V x + T Ta xet A ( x ) = • m=l thi t h u dugc: J x + — - = ^, -2 x > X +1 + Vs - -;2 3x Dat t = (x^ - 7) => X = - 4x^)72x3 - + voi t > Ta nham d u g c t = N e n phan tich p h u o n g t r i n h n h u sau: a) Dieu k i f n: I x 1< — t ^ t T - 4t + 5t - + t t ^ T - 4t = D i i n g may t i n h bo t u i ta t h u dugc nghiem la: x^ - - , , X =1,618 Xj.X2=-l « t f - V r f - ) + ( t - l ) + 4t(V2t-l -1) = nen n g h l d e n n h a n t u c h u n g la »2t x-^ - x - t-1 + ( t - l ) + 4t 2t-2 U2t-l+lj =0 Xet Vs-Sx^ - ( a x + b) = cho p h u o n g t r i n h c6 n g h i e m X j =-0,618, X = 1,618 ta t h u duoc: axj + b = y - 3X] axj + b = ^ - 3x2 2t «(t-l) -0,618a + b = 2,61 l,618a + b = 0,38 7(t + 7)2 + ^ / t 7 + Ia = - -4(x^ - x - ) o(x^ - x - l ) ( x +l) = - ^ 3x^ + ( - x ) V8-3x2 +(2-x)^ + 5- 4t =0 721^ +1 2t 4t De thay v o i t > — t h i • + 5>0 7(t + 7)2 + ^ + V2F^+1 ' ^ ] b =2 P h u o n g t r i n h dug-c vie't lai n h u sau: x'' - 2x - = \/8-3x^ - (2 - x) «>(x^ - x - ) x + + i + Vs 15 P h u o n g t r i n h da cho t r o thanh: 2t\/t + - 3t - + t V t - l = Giai: x^+X2=l, + 3=0 Ta vie't lai p h u o n g t r i n h thanh: 2x(x-' - 7) - 3{x^ - 7) - = 4(7 - x^)^J2ix^ - ) x^-3x + l = \/8-3x^ d o ta thay +(2-X) + = Vay p h u o n g t r i n h ban dau t u o n g d u o n g v o i x^ - x - = x = x = -2 Vi du 4: Giai cac phuong trinh: Tu N/S-SX^ P h u o n g t r i n h v nghiem Ket l u a n : P h u o n g t r i n h c6 nghiem: x = — r ; x = l ; x = - b) 2x^ - 3x3 _ -,4^ ^ |g ^ = 0(x + l ) o ( x + l ) V - x ^ - X + X + = O ( X + ) V - X - X + X + 12 = D a t t = Vx^ +x + > P h u o n g t r i n h t r o thanh: a) +1 + •3x^ + ( - X ) (x + 3) + \ / x + x + -t-2 = ^ X (1) x + x + 3^ Giai (1): t =0 +(2-x) VB-SX^ N h u vay p h u o n g t r i n h c6 nghiem d u y nhat t = x = Vi du 5: Giai cac phuong trinh: a) 2V4x' - X + + 2x = 372x^ - x^ + 79x^ - 4x + b) V3x + - Vx + - X +1 =0 Tai l.cn 6nthUaihocshngtaovagtdtPl,batl'l,hel^l,bai VI -N^yen imngi^ien Xetham so f(t) = Vt^ + - Vt^ +15 - t + v a i t < - l Giai: a) Phuang trinh duoc viet lai nhu sau: Vl6x2 - 4x + + 2x = 3\/2x^ - Tac6:f'(t) = -3 + 3t^ + N/QX^ - 4x + ;.Vy " I « V l 6x2 - 4x + - Vl 6x2 - 4x + = 3\/2x2 - x^ - 2x > Suy dieu kien de phuang trinh c6 nghiem la 3^/2x^ - x'' - 2x > o 27(2x2 - x ^ ) > ^ x ^ » 35x3 - x < o X < X ?i Trucmg hop 1: x > Chia hai ve phuang trinh cho x ta thu dugc: X^ - - = t ^ Phuong trinh da cho tro thanh: Vt'^+15+2 = V t ^ + + t o V t ^ + - V t ^ + = t - De phuang trinh c6 nghiem ta can: t - > o t > - Nham duoc t = nen ta viet lai phuong trinh thanh: Vt^ +15 - = «(t^l) 't-Vl)(t2 +t +l) 15+4 De y rang: (t^+l ^2 + t ( t + l ) ( t + t + l )- Vt^78 + +1 Vt" +15 + ^2 +t+l - + 3t - =0 /3x + l + Vx + = Binh phuang ve ta thu dugc: 4x + + 2J(3x + l)(x + 3) = o J(3x + l)(x + 3) = - x o i ' ^ f ° * [x2-10x-3 = O X = 5-2N/7 Ket luan: Phuang trinh c6 nghiem la x = 1, x = - 2\l7 Nhan xet: V s x + T - + - V x + + - x = thi sau lien hgp phuang trinh moi thu dugc se la: 3x-3 Truong hgp 2: x < Chia hai ve phuong trinh cho x ta thu dugc: X^ Dat / - - I = t = ^ - - l = t = > t < - l X Nhan xet: Trong mpt phuong trinh c6 chiia nhieu dau ,' _ + Ta thay phuang trinh c6 n g h i f m x = l Neu ta phan tich phuong trinh nghiem nhat t = x = V Tom lai: Phuang trinh ban dau c6 nghiem la x = 0, x = 2x-2 +l-x =0o(2x-2) ' , - ± =0 V3x + l + V x + ^ W x +l+Vx +3 2j ;7X4.2=33/1:;.J9-i.4 ' %' ' gian nhat Vi^c la se giiip cau true phuang trinh mod d a phuc t^p hon 54 De don gian ta dat ^J- - = t < do f(t) > f ( - l ) = Suy phuong trinh v6 nghiem b) Dieu ki^n x e X Vt^+15 phu la mot bieu thuc chua 35 Xct X = thoa man phuong trinh Xet Vt^+8 X Phuong trinh da cho tro thanh: - V t " +15+2 = -Vt^ + 8+3tc^Vt^ + - V * ' ^15-3t + = + 1-x ^7" jj +4-4x = c ^ ( x - l ) ^ + L _ _ = V3X + 1+2 + Vx + \ V x + l + 2 + Vx + 3 ' R6 rang phuong trinh h? qua , + - = phuc t^p hon V3X + 1+2 + VX + phuong trinh ban dau rat nhieu + De y rang x thi VSx + l = Vx + nen ta se lien hgp tr\rc tiep bieu thuc: Tsx + l - Vx + Tm V!r:N TifvHBINH THUA;\ ... theo each: p= l - p + q = -4 p = 1, q = -3 p-2q = x2 - x + = p ( x - x + l ) + q ( x - ) : 1- 2 (x^ - x + l ) - ( x - ) N/X^ = 1- 5 .D3t t = x^-x + l ^ I • -2 1? ?? x-2 Vx^-x + l - Ta t^o phuong... 4Vx + + V l - x - x^ - 2x - tren 19 Ta thay x = - ; x = - - k h o n g phai la nghiem, i3 -3 -^ '3 Cty TNHH Tren -3 ; 19 ta -9 CO 5V3X-8 + (3x-4) ^ ? ?-2 x-2;f'(x) = -3 x ' ^(x + f f'(x) = ^ -2