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TON THAN (Chu bien)

NGUY~N ANH HOANG- D~NG VAN QUAN

CAc cHuvEN oE CHON LOC . , . TOAN

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Phdn cong hien S0(_/11 :

Ph fin D~i so : D~ng Van Quim

Phan Hlnh h<;>c : Nguyen Anh Hoang

LOINOIBAU

D~ giup cac em h<;>c sinh h<;>c t~p tot m6n Tm\n a Trung h<;>c ca sa (THCS) hi~n nay va a Trung hQC ph6 thong (THPT) sau nax, chung t6i bien SO<,tn b¢ sach gam 8 cuon : "Cac chuyen cte' ch9n I(Jc Toan 6, 7, 8, 9 r~p mr)r \'(J t~p hai ".

Moi cuon trong b¢ sach c6 cac chuang tuang t'rng v6i cac chuang trong Sach giao khoa Toan. Cac chuang deu duqc viet theo nhiing chuyen de (co ban va nang cao) rna cac tac gia cho rang d6 Ia nhiing chuyen de can thiet cho vi~c hQC va hi~u sau kien thtrc cua chuang.

Moi chuyen de gam ba phan:

A. Kie/1 tlurc cdn nh(J': Phan nay dua ra nhiing kien thll'c ca ban, nhiJng kien thlrC b6 sung can thiet d~ CO th~ giai dL!<;1C CaC bai t~p, cac d~ng toan cua chuyen de.

B. Mr)t so' vi d~:~: Phan nay trlnh bay nhiing vf dl;l ch<;>n l<;>c minh ho<_~

cho nhiing d~ng toan di~n hlnh cua chuyen de v6i each trlnh bay IO'i giai chu~n mt,rc kern theo nhiing nh~n xer, hru y, hlnh /u~n, ... ve phuang phap giai, ve cac sai lam h<;>c sinh c6 th~ mac, ve vi~c tim toi them cac each giai khac, .... Nhieu vf d~:~ a phan nay duqc trfch trong cac de thi h<;>c sinh gioi Toan 6 THCS, trong cac de thi v?w lop I 0 THPT chuyen.

C. Bai r~p: Phan nay dua ra h~ thong cac bai t~p duac phan lo~i theo cac d(;lng roan d~ h<;>c sinh de Slr d~mg. H~ thong cac bai t~p nay kha da d<_~ng, bao gam cac bai t~p ca ban va cac bai t~p nang cao cho h<;>c sinh kha, gioi. Nhieu bai duqc trfch tlr cac de thi hQC sinh gioi Toan 6 trong va ngoai nu6c. Moi cuon sach deu cung cap m(>t so luqng lon cac bai t~p voi huang cHin giai kha chi tiet, minh ho~t cho phuang phap gi~ti cac d ~lng toan, cac chuyen de da de cap.

Cuoi sach Ia phan Huung ddn gidi - Dclp so' cho cac bai t~p a cac chuyen de. Qua nhiJng htrong dan giai Cl;l th~ , hQC sinh se nam fO han each giai cho moi d~ng roan.

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4

Cac kien thl'rc trong moi cu6n sach duqc sap xep tude den kh6, duqc trlnh bay dO'll giim, de hi~u, dap Lrng cho nhieu d6i tuqng h9c sinh.

Cac tac gia cua b¢ sach Ia nhfrng thiiy co giao da c6 nhieu kinh nghi~m trong vi~c giang d<;ly, boi duong h9c sinh gioi Toan a THCS, d6 Ia cac thay co giao: PGS.TS. NGND Ton Than (Chu bien b¢ sach), NGVT Bui Van Tuyen, NGVT Nguyen Ng9c DC;lm, Ths. Nguyen Due Truong, Ths. Nguyen Due Tan, Ths. Nguyen Anh Hoang, Ths. D~ng Van Quan, Ths. Ph(;lm Th! L~ Htmg.

M~c du da c6 nhieu c6 gtmg, song b¢ sach kh6 tranh khoi nhfrng thieu sot. Cac tac gia rat mong nh~n duqc thu g6p y cua cac em h9C sinh, cac thay co giao va cac b~c ph1,1 huynh. M9i y kien g6p y xin gi!i ve theo dja chi :

Ban Toan - Tin, Nha xudt hdn Giao d~tc Vi¢t Nam - 1878 Gidng V 6-He/ N()i.

Hi Y9ng rang, b¢ sach se Ia tai li~u tham khao thiet tht,tc, hfru fch d6i v6i cac em h9c sinh THCS, cac thay co giao d(;ly Toan va b(;ln d9c yeu thfch Toan.

cAc TAc GIA


'?fhuong Ill PHliCJNG TRiNH B~C NHAT "" ""?

MOT AN .

Chuyen de'J

PHUONG 1RINH BAC NHAT M6T AN

A. KIEN THUC CAN NHO

1. Kien thuc co ban

1.1. Mo dau ve phuong trinh

Phuang trlnh mt)t dn Ia ph11ang trlnh c6 d<;tng P(x) = Q(x) (x Ia ~n), trong d6 ve tnii P(x) va ve phai Q(x) Ia hai bieu thCrc cua cung m¢t bien X.

- So Xo gQi Ia nghi¢m cua phvang trlnh neu P(xo) = Q(xo) Ia m¢t d£ng thCrc

dung.

- M¢t phvang trlnh c6 the c6 m¢t nghi~m, hai nghi~m .. .. , nhung cling c6 th~ khong c6 nghi~m nao (v6 nghi~m). Gidi phuang rrinh Ia tim tat ca cac nghi~m (ho~c tim t(ip nghi¢m) cua phvang trlnh d6.

- Hai phvang trlnh dvqc gQi Ia ruung duang neu chung c6 tap nghi~m bang nhau (ke ca bang t~p rong). Quy uic bien m¢t phvang trlnh thanh m¢t phvang trlnh t11ang d11ang voi n6 dvqc gQi Ia quy t:ic hie/1 dO'i tUO'ng duang.

1.2. Phuong trlnh b:)c nhat m<)t ~n

a) Dfnh nghla: Phvong trlnh d<;tng ax+ h = 0, voi a, h Ia hai soda cho va

a* 0, dvqc gQi Ia phuang trlnh h(ic nhdr mt)t a'n.

h) Hai quy rae hien d6'i tUO'ng duang:

- Quy tac chuyen ve: Trang m¢t phvong trlnh, ta c6 the chuyen m(>t h<;tng tu tlr ve nay sang ve kia va d6i dau h<;tng tlr d6.

- Quy de nhan voi m¢t so: Ta c6 the nhan (ho~c chia) d hai ve cua m(>t phvang trlnh voi (cho) cung m¢t so khac 0.

c) Cach gidi phuung trinh h(ic nhat mc)t a'n:

5

Ta c6: ax+ b = 0 <=>ax= -b (quy tac chuy~n ve) -

<:::> X=-~ (chia hai ve cho a 1:- 0). a

V~y phucmg trlnh b~c nhat m(>t ~n ax+ h = 0 Juon c6 m(>t nghi¢m duy nhat Ia

b X=-- .

a

2. Kien thuc Nang coo

a) Phucmg trlnh c6 d{,lng !J~Ic nhdt mr)t dn: ax+ h = 0.

- V6'i a 1:- 0, phucmg trlnh c6 nghi¢m duy nhat Ia x = -~. a

- V6'i a= 0, phucmg trlnh c6 d<;~ng: Ox=- b

• Neu b = 0 thl phltcmg trlnh c6 v6 s6 nghi¢m.

• Neu b 1:- 0 thl phl((:mg trlnh v6 nghi¢m.

b) V6'i phtrang trlnh chua tham s6 m, gidi va hi¢n lucJn phtrong trlnh Ia gi<ii

phuang trlnh d6 tuy theo cac truang hqp ve gia tri cua m.

B. MOT so vi ov Dc;mg 1. Xet xem m¢t so c6 Ia nghi~m cua phuong trinh hay khong

Vi d~;~ 1. Hay xet xem X = - 3 c6 phai Ia nghi¢m cua moi phuang trlnh sau hay khong '?

6

a) 2x - 5 = -14- x :

6 c) --5=2x+l;

X

Gidi

2 b)-x-7=-3x •

3 I

a) Thay X = - 3 vao phtrcmg trlnh, ta duqc:

2.(-3)-5=-14-(-3)

<=> - I I = - I I: Ia m(>t dlmg thuc dung.

Yay x =- 3 Ia nghi~m CLia phltang trlnh.

b) Thay X=- 3 vao phuang trlnh, ta duqc:

2 -.( - 3) - 7 = -3.(-3) 3

<=> - 9 = 9: Ia m(>t d~ng thuc sai.

V~y X=- 3 khong Ia nghi~m cua phli'O'ng trlnh.

c) Thay X=- 3 vao phll'ang trlnh, ta dl.I'Q'C:

6 --5 = 2.( - 3)+ I -3

<=> - 7 =- 5: Ia m(>t d~ng thuc sai.

V~y X =- 3 khong Ia nghi¢m cua phuang trlnh.

d) Thay X=- 3 vao phLrang trlnh, ta duqc: ..,

(- 3 r - 4 = ·2. (- 3) + I I

<=> 5 = 5: Ia m(>t dtmg thuc dung.

Y~y X =- 3 Ia nghi~m cua phuang trlnh ..

Nhdn xet:

De xet xem m(>t so c6 Ia nghi~m cua phl1ong trlnh hay khong, ta thay s6 d6 vao phl1ang trlnh. Neu ket qua Ia m(>t dlmg thuc dung thl soda cho Ia nghiem; tnti l<:ti, soda cho khong phai Ia nghi¢m.

Vi dl,l 2. Tim gia trj cua m, biet rang X = 5 Ia nghi¢m cua phuang trlnh:

2x + m2 ( x-1) = 19 .

Gidi

VI x = 5 Ia nghiem CLta phuang trlnh 2x + m 2 (x - I) = 19, nen:

2.5 + m 2 ( 5 - I) = 19

')

<=> I 0 + 4.m- = 19

.., 9 <=> m- = -

4

<=>m = +{i=+~ -~4 - 2 °

Yay m E {-~·~ l. . 2 '2 J

Dc;mg 2. Giai phuong trinh duo dUQC ve dc;mg ax + b = 0

Vi d1,1 3. Giai cac phuang trlnh sau:

3x-2 4 - 7x a)-- = --; b) 2x(x - 5)+21=x(2x+l) - 12.

5 3

7

Gidi

3x -2 4-7x a) -- = -- <=> 3(3x- 2) = 5(4- 7x)

5 3

<=> 9x - 6 = 20 - 35x

<=> 9x + 35x = 20 + 6

<=> 44x = 26

13 <=>x=-.

22

13 V~y phvong trlnh c6 nghi~m duy nhat Ia x =

22.

b) 2x(x - 5)+21 =x(2x+l) - 12

<=> 2x 2 -I Ox+ 21 = 2x 2 + x -12

<=> 2x 2 -1 Ox- 2x 2 - x = - 12 -2 1

<=> - II X=- 33

<=>X= 3.

Y~y phvong trlnh c6 t~p nghi~m Ia S = 131.

Vi d~;~ 4. Giai cac phvO'ng trlnh sau:

8

a) X + 98 + X + 96 + X + 65 = X + 3 + X + 5 + X + 49 ; 2 4 35 97 95 51

X - 9J X - 86 X - 78 X - 49 b) ----+--+--+-- = 4.

37 42 50 79

Nhtjn xet:

Neu thlJC hi~n phep quy dong mau so thl ta thay phep tfnh kha cong kenh va sol6'n. Neu quan sat quy luat Clla cac c~p so a ti:r va mau cua d.c phfm so thl ta thay nhu sau:

- 6 cau a), tong cua so tren tlr va so dv6'i mau cua de phan so bimg nhau :

98 + 2 = 96 + 4 = 65 + 35 = 3 + 97 = 5 + 95 = 49 + 51.

- 6 cau b) , hi~u cua so tren tlr va so dv6'i mau cua cac phan so bang nhau:

-91 - 37 =- 86-42 =- 78- 50=- 49- 79.

D~ khai thac quy lu~t nay, ta c<)ng them I vao m8i phan so 6 cau a) va trlr m8i phan so 6 cau b) di I dan vi.

Gidi

a) Phuong trlnh da cho wang ducmg v6'i:

( X ~98 +I)+ ( X :96 +I ) + ( X ;565 +I J =

= (X 9~ 3 + I)+ (X 9~ 5 + I) + ( X ; ~9 + I)

X + 100 X + I 00 X + I 00 X + I 00 X + I 00 X + I 00 <=> + + = + +--

2 4 35 97 95 51

<=> (x + 100) -+- + - - - - --- =0 (I I I I I I ) 2 4 35 97 95 51

<=>X+ ] 00 = 0

<=>X=- 100.

V ~Y phucmg trlnh c6 tap nghiem Ia S = (- 100 }.

b) Phtrcmg trlnh da cho tuang dtrang v6'i:

(~- l ) + ( x - 86 - l ) + (~- l )+( x - 49 - l) = o 37 42 50 79

X - 128 + X - 128 + X - 128 + X - 128 = O 37 42 50 79

<=> (X - 128). - +-+-+- = 0 ( I I I I J

37 42 50 79

<=>X- 128 = 0

<=>X= 128.

Y~y phucmg trlnh c6 nghi~m duy nhat x = 128.

Dc;mg 3. Xet xem hoi phuong trinh c6 tuong duong hay khong

Vi d1,1 5. Hai phucmg trlnh sau c6 wang duang kh6ng? VI sao?

a) 2x=6 va (x-3)(x2 +1)=0 ;

b) X 2 - 4 = 0 va X + 2 = 0.

Gidi

a) Phuong trlnh 2x = 6 c6 t~p nghi~m Ia S1 = ( 3}.

9

Phtrang trlnh (x -3)( x2 + 1) = 0 <=> , <=> x = 3, nen c6 t~p nghi~m [

X- 3 = 0

' x-+1 = 0

VI 51 =52 nen hai phuang trlnh da cho Ia wang tLrang.

b) Ta c6: x~- 4 = 0 <=> (x- 2)(x + 2) = 0 <=> <=> , [x - 2 = 0 [x=2 x+2=0 x =-2

~ T~p nghiem Ia 53 = 1- 2; 2).

PhLrang trlnh x + 2 = 0 <=> x =- 2, c6 t~p nghi~m Ia 54 = 1- 2).

VI 53 -:F- 54 nen hai phuang trlnh da cho kh6ng ttrang duang.

Nhan xet:

6 cau b) ta c6 the gi~ti thfch nhLr sau: VI x = 2 Ia nghi¢m dm phuang trlnh ' ., x-- 4 = 0 (do 2-- 4 = 0) nhung kh6ng Ia nghi ~m cl:1a phLrang trlnh x + 2 = 0

(do 2 + 2 "j:. 0). nen hai phuung trlnh kh6ng tuang duang.

Vi d1,1 6. Tim m M hai phuang trlnh sau tuang duang:

10

X- l1l = 0 (I) Va ITIX- 9 = 0 (2)

Gidi

PhLrang trlnh (I): x - m = 0 c6 nghi~m duy nhat Ia x = m. VI hai phuang trlnh tuang duang nen X = m cling Ia nghi¢m cua phuang trlnh (2), tlrc 1~1 : m.m = 9

' ') <=> Ill-= 3- <=> Ill= ± 3.

Thv lai:

- Voi m = 3. ta c6 phuang trlnh (I): X- 3 = 0 va phuang trlnh (2): 3x- 9 = 0 c6 cung t~p nghiem Ia j3). V~y m = 3 tho:. man.

- Voi m =- 3, ta c6 phuang trlnh (I): x + 3 = 0 v ~1 phLrang trlnh (2):

(- 3)x- 9 = 0 c6 cLmg tap nghiem Ia 1- 3). V~y m = - 3 thoa man.

V~y c6 hai gia trj cuam thoa man yeu du Ia - 3 va 3.

Nh(in xet:

UJi giiii tren dl,l'a vao tfnh chat: Hai phuang trlnh tuang duang khi va chi khi ITIQi nghi~m cua phuang trlnh nay cling Ia nghi~m CLia phuang trlnh kia. Ngoai each lam nay ta c6 th~ giai thfch nhu sau:

VI (2) Ia phuang trlnh c6 d~mg b~c nhat nen d~ (2) tuang duang voi (I), tCrc Ia

c6 m(>t nghiem duy nhat bang m, thl (2) phcli Ia phuang trlnh bac nhat (m -:F- 0)

va c6 nghi¢m X = m. v u di~m cua each nay Ia chung ta khong phai thl'r l~i cac

gia tri cua m rna chi can kifm tra dieu ki~n m =;t:. 0.

Dc;mg 4. Giai va bi~n lu¢n phuong trinh ax + b = 0

Vi dt,t 7. Giai va bi~n lu~n phucmg trlnh: (m -3)x = m2 -3m.

Gidi

Ta c6: ( m- 3) x = m 2 - 3m <=> ( m- 3) x = m ( m- 3).

+ Neu m - 3 * 0, tuc m * 3, thl phuang trlnh c6 nghi¢m duy nhat Ia:

m(m - 3) X= =111.

m - 3

+ Neu m - 3 = 0 tu·c m = 3, thl ta c6 phu·ang trlnh: O.x = 0. dung v6'i mQi x.

V~y, neu m =;t:. 3 thl phuang trlnh c6 t~p nghi¢m Ia l m l;

neu m = 3 thl phuang trlnh c6 t(tp nghi~m Ia JR.

Vi dt,t 8. Cho phu0'11g trlnh m ( m - I) x = m 2 +3m+ 2 ( x + I).

Tim m de phv0'11g trlnh (I):

a) C6 nghi~m duy nhat.

b) Yo nghi~m.

Gidi

Ta c6 ( I ) <=> ( m 2 - m) x - 2x = m 2 + 3m + 2

<:::> (m 2 - m-2)x = m2 + 3m+2

<=> (m + l)(m - 2)x = (m + l)(m + 2) .

a) Phu0'11g trlnh (I) c6 nghi¢m duy nhat khi va chi khi:

(m+l)(m-2)=t:.0

<=> m =t:. - I va m =;t:. 2.

b) Phuang trlnh (I) v6 nghi~m khi va chi khi:

{

( m + I) ( m - 2) = 0 <=> 111

= 2

.

(m+l)(m+2)=t:.O

(I)

Nh(tn xet: Phuang phap giai bai toan tim m de phuang trlnh th6a man dieu ki¢n ve so nghi¢m:

II

- Bie·n doi dua ve phucmg trlnh ve d;:tng b~c nhat: ax + b = 0 (*)

- Ap dyng mc;>t trong cac ket qua sau:

- Phucmg trlnh (*) c6 nghi¢m duy nhat <=>a :t 0.

{a =0

- Phuong trlnh (*) v6 nghiem <=> b * 0

Ph ' h (*) ' A A ' h' A r a= 0 - ucmg tnn · co vo so ng ·~m <=> l b = 0

c. BAI TAP

3.1. Xet xem X = 4 c6 Ia nghi~m cua moi phucmg trlnh sau hay kh6ng?

a) 2(3x-1) - 7=15 - (x - 4);

b) x(3-4x)-5=1-x 3.

3.2. nm m M X= I ,5 Ia nghi¢m cua phucmg trlnh:

m 2 (2x --3)-4x+m =5.

3.3. ChCrng minh rang phucmg trlnh: 2mx- 5 =-X+ 6m-2 lu6n c6 mot nghiem

X kh6ng phy thu<';>c vao m.

3.4. Giai cac phuang trlnh sau:

a) 5 - (x - 6) = 4(3 - 2x);

b) 3- 4x(25- 2x) = 8x2

+ x- 300.


a) 3x + 2 _ 3x +I = 2x + ~. 2 6 3'

b) X_ 2x - 5 +X+ 8 = 7 +X - I . 5 6 3 '

c) 5x+2 _ 8x-l = 4x+2 _5

.

6 3 5


a) x + I + x + 2 + x + 4 + 6 = 0 . 15 7 4 '

12


b) X+ 14 + X+ 27 + X+ I 05 = X+ 200 + X+ 187 + X+ I 09 ;

200 187 I 09 14 27 I 05

) X - 342 X- 323 X - 300 X - 273 I 0

c + + + = . IS 17 19 2 1

3.7. Gi iii cac phuang trlnh sau:

a) X+ 97 + X - 63 = X - 7 + X - 77 ; 125 35 2 1 49

b) x + 2 + 2x + 45 = ~x +8 + 4x +69 . 13 15 37 9

3.8. Giai phuang trlnh:

_x_ + X+ 2 + X -t--4 + ... + X+ 12 = 7. 2000 2002 2004 2012

3.9. Tlm m de hai phuang trlnh sau nrang duang:

2x+3=0 va (2x + 3)(mx- 1) = 0 .

3.10. Gi ii i va bi~n lu ~n cac phuang trlnh sau:

a) (1-m)x=m 2 - l;

b) ( m2- Sm + 6) x = m2

- 9.

3.11. Cho phuang trlnh (4m 2- 25 )x- 5 =2m .

a) Giai phuang trlnh vai m = 5.

b) Tlm m de phuang trlnh v6 nghi~m.

3.12. Cho phuang trlnh (4m 2- 9 )x =2m 2 + m- 3. Tlm m de phuang trlnh:

a) C6 nghi~m duy nhat.

b) C6 v6 so nghi~m.

3.13. Gi ii i phuang trlnh an X sau :

a+b - x b+c - x c+a-x 4x 1 ---+ + + = .

c a b a+b+c

3.14. Giiii phuang trlnh an X sau :

X -a X- b X-C 2 2 2 --+--+-- =-+-+-.

be ca ab a b c

13

Chuyen de'2

PHUONG TRINH TICH

A. KIEN THUC co BAN

- PhL!Cmg trlnh tfch Ia phvong trlnh c6 d<;1ng: A(x).B(.r) ... C(x) = 0 (I)

- D~ giai phuong trlnh (I) te~ giai cac phvong trlnh A(.r) = 0; B(.r) = 0: ... ;

C(x) = 0, roi lay hqp tat ca cac nghi¢m cua chung.

Trong Chuyen de nay ta thvong v~n dyng cac phuong phap phan tfch eta thuc

th~mh nhan tl.r de bien d6i phtrong trlnh da cho ve phuong trlnh tfch.

B. MOT so vi ov Dc;mg 1 . Giai phuong trinh tich

VI dl,l 9. Giai cac phuong trlnh sau:

a) (2x-7)(4-5x)=0;

b) 8x(3x-5)=6(3x-5).

Gidi

[

X =7_

[?x - 7 = 0 2

a) (2x-7)(4 - 5x)=O¢:::> - _ ¢:::> 4 - Sx - 0 4

X= ·-5

V A I ' h ' A I .. I ' s {7 4 } ':lY p1trong tnn co t(;lp ng11~m a = l;S .

b) 8x (3x -5) = 6(3x- 5) ¢:::> 8x (3x- 5) -6(3x- 5) = 0

[

5 X=-

3x- 5 = 0 3 ¢:::> (3x -5)(8x -6) = 0 ¢:::> rl ¢:::>

8x -6 = 0 3 X=-

4

VA h ' h ' A h" A I' s {5 3} ~~y p Lrong tnn co t~tp ng 1~m a = "3; 4" .

14

Luu j: Khong dvqc chia hai ve cua phvang trlnh cho 3x - 5, vi nhv v~y se lam mat

nghi¢m X =% cua phuang trlnh. Neu muon chia ta phai xet tnrang hqp 3x - 5 = 0

trtr0c.

Vi dt,~ 10. Gi <l i de phLtO'ng trlnh sau: ')

a) x- - 6x + 9 = 49;

b) (x2

+3x+2t =(x 2 -x-2t .

Gidi

a) X 2 - 6x + 9 = 49 <::? (X- 3 )2

= 72 Q (X - 3? -72 = 0

<::? (X- 3 -7)( X- 3 + 7) = 0 <::? (X-) 0) (x + 4) = 0

<::?[X-) 0 = 0 <::?[X= J0 x +4= 0 x=-4

V ~y phvO'ng trlnh c6 t~p nghiem Ia S = l-4; I 0 ).

b) (x 2 +3x+2t = (x 2 - x-2)2

<=?(x 2 t3x+2)2

- (x 2 -x-2t =0

<::? [ (X 2

+ 3X + 2)- (X 2

-X - 2) J [ (X 2

+ 3x + 2) + (X 2

-X - 2) J = 0

( 1 ) 2 [X = 0 <::? ( 4x + 4) 2x- + 2x = 0 <::? 8x ( x + I) = 0 <::? x = _

1

V ~y phVO'ng trlnh c6 t~p nghi¢m Ia S = l- I; 0 ).

Nlujn xet:

C6 th~ giai cac phVO'ng trlnh tren theo bien d6i sau:

Dc;mg 2. Giai phuong trinh do thuc bc;lc coo quy ve phuong trinh tich

Vi dt,~ 11. Giai cac phVO'ng trlnh sau:

a) (12x 2 -3)(x+3)+(2x 2 +7x+3)(x-3)=0;

~ ') b) x· -3x- -6x+8=0.

15

Gidi

a) Ta c6 phuong trlnh tuong duong:

3 ( 2x - I)( 2x +I)( x + 3) + ( 2x +I)( X+ 3 )(X- 3) = 0

<=> (2x+l)(x+3)[3(2x-l)+(x--3)]= 0

I x=--

2

<=> (2x+l)(x+3)(7x - 6) = 0<=> x= - 3

6 X=-·

7

V~yphuongtr1nhc6tapnghi~m laS= {- ±;-3;~} · b) x 3 - 3x 2 - 6x + 8 = 0 <=> x 3

- x 2 - 2x 2 + 2x- 8x + 8 = 0

<=> x 2 (X- I)- 2x (X - I)- 8 (X- I)= 0

<c> {x -t)( x2 -2x - 8) = 0<c> (x -t){x+2){x-4) = O<c> r:: ~2 l X= 4

V ~Y phuong trlnh c6 t~p nghiem Ia S = 1- 2; I; 41.

Vi dl;l12. Giai phuong trlnh 64x 3 = (x- 2)3 + (3x + 2)3. ( 1)

16

Gidi

Phuong trlnh ( 1) ttrong duong vai

(4x)3 =[Cx-2)+(3x+2)][cx - 2)2 -(x-2)(3x+2)+(3x+2)2]

<=> (4x)3 =4x(7x2 +12x+l2) <=> 4x[(4x)2 - (7x 2 +12x+12)] = o

x=O

<=> 4x(9x2 -12x-12)=0 <=> 12x(3x+2)(x-2)=0<=> 2

X=--3

x=2

V~y phuong trlnh c6 t~p nghi~m laS= { 0; - ~;2}.

Nhf!,n xet:

VI ( x- 2) + ( 3x + 2) = 4x, nen phuang trlnh c6 d~ng:

(a+ b )3

= a3 + b3

.

Ap d1,mg htmg dlmg thuc: (a+ b )3

= a3 + b3 + 3ab( a+ b), ta suy ra:

3ab(a+b)=O, tuc Ia: 3(x-2)(3x+2)(4x)=O. Th~t nhanh g<;m!

D<;mg 3. Giai phuong trinh bang phuong phap d¢t cin phl:J D(ir d/1 ph1.J Ia phuang phap dtmg m¢t ~n moi (sau gQi Ia d/1 p/11~) thay cho m¢t bi~u thuc cua a'n cu. Gicii bai toan btmg phucmg phap d~t fin ph~:~ thtr<'mg c6 cac buoc sau:

- Bien doi phuang trlnh de lam xuat hi¢n cac nh6m h<:mg tu chua fin giong nhau .

- D~t nh6m h<;tng tl'r giong nhau b<lng ~n moi. Thay vao phuang trlnh da cho ta duqc m¢t phuang trlnh theo a'n moi (dan gian han phuang trlnh ban dau ho~c da biet each giai).

- Giai phuang trlnh theo ~n moi .

- Yoi moi gia tri tim duqc cua a'n moi, thay vao bi~u thuc d~t ~n ta tim dtrqc cac gia tri tuang Lrng cua ~n ban dau.

Vi d~:~ 13. Giai phuang trlnh (x - I )(x + 2)(x - 6)(x - 3) = 34.

(De'rhi vao !tip 10 chuyen ngi/, DHNN-DHQC HN , 2000)

Gidi

Ta c6 phuang trlnh:

[ (X - I ) (X - 3) J [ (X + 2) (X - 6) J = 34

<=> (X 2 - 4 X + 3 )(X 2 - 4 X - 12) = 34

D~t t=x2 -4x+3,tac6: t.(t - 15) = 34

<=> t 2 - 15t - 34 = 0 <=> ( t + 2) ( t - 17) = 0 <=> [t = -

2

. t = 17

- Voi t = -2, ta c6: x2- 4x + 3 =- 2 <=> x2

- 4x + 4 =- I

<=> (x - 2)2 = - I: v6 nghi¢m.

17

- V 6'i t = I 7, ta c6: x 2 - 4 x + 3 = I 7 <=> x 2

- 4 x + 4 = I 8

<::::> (X - 2)2

= 18 <::::> X - 2 = ± J18 <::::> X = 2 ± J18. V~y phuang trlnh c6 t~p nghi~m Ia S = 12- Jl8; 2 + Jl8 1.

Nhdn xet:

C6 th~ ap dyng phuang phap tren d~ giai phuang trlnh t6ng quat sau day:

( x +a) ( x +b) ( x +c) ( x +d) = e , v6'i a + b = c + d.

Vi d~ 14. Giai phuang trlnh

18

(x2- 3x + 3)(x2

- 2x + 3) = 2x2.

(De' thi vao l6p /0 chuyen nf?ii, DHNN-DHQG HN, 2005)

Gidi

Nh~n thay X = 0 khong phai Ia nghi~m cua phuang trlnh, nen chia hai ve cua

phuang trlnh cho x2 ,_ 0, ta duqc:

x2

- 3x + 3. x2

- 2x + 3 = 2

X X

D~t t = x + l_ 3, ta c6: X

. ? [t =I t(t + I) = 2 <=> c + t - 2 = 0 <=> (t - I )(t + 2) = 0 <=>

t = -2

V ,. I . ' 3 3 I - O'I t = , taco: x +-- = X

2 [x =I => x -4x+3=0<=>(x-l)(x-3)=0<=> x=3

- V6'i t = - 2, ta c6: x +l-3 = -2 X

V ~Y phuang trlnh c6 t~p nghi~m Ia S = II; 31.

NhtJ_n xet:

C6 the ap dt,mg phtrong phap tren d~ giai phtrong trlnh t6ng quat sau day:

( x 2 +ax+ c) ( x 2 + bx +c) = dx 2 , v6'i a, b, c, d Ia cac soda cho.

Vi dt.Jl5. Giai phtrang trlnh (6-xt +(x - 8)4

= 16.

Gidi

Phuong trlnh (I) <=> ( x- 6 t + ( x- 8 t = 16

<=> (x-7+It +(x-7-lt = 16

D<;tt t = x - 7, ta c6 ( t + It + ( t - It = 16

<=> ( t4 + 4t3 + 6t 2 + 4t +I)+ ( t4 - 4t 3 + 6t 2

- 4t +I)= I6

<=> 2t4 + 12t2 + 2 = I6

<=> t4 + 6t2 -7 = 0

<=> (t2

- I)(t2 + 7) = 0

<=> t2 = I <=> t = ± I .

- V 6'i t = I , t.a c6 x - 7 = I <=> x = 8.

- V 6'i t = - I, ta c6 x - 7 = - I <=> x = 6.

V ~Y phtrong trlnh c6 t~p nghi~m Ia S = 16; 8}.

C. BAI T~P 3.15. Giai cac phtrong trlnh sau:

a) (x - 6)(2x - 5)(3x t 9) = 0;

b) 2x(x- 3) + 5(x- 3) = 0;

c) (x2- 4)- (x - 2)(3 - 2x) = 0.

3.16. Tim m de phtrong trlnh sau c6 nghi~m x =- 7:

(2m -5)x -2m2 +8 = 43.

3.17. Giai cac phtrong trlnh sau:

a) (2x - 1)2 - (2x + I )2 = 0;

b) _I (x-3)3 --

1-(x-5)3 =0.

27 125

(I)

I9

3.18. Giai cac phtrO'ng trlnh sau:

a) 125x 3 - ( 2x +I )

3 - ( 3x- I )

3 = 0;

b) (X - 3 )3

+ (X + I )3 = 8 ( x - I )

3 .


a) x 3 - Sx 2 + 8x - 4 = 0;

b) x4 -4x 2 +12x-9=0.


a) (x+l)(x+2)(x+3)(x+4)-24=0;

b) (x 2- 3x + 2)(x 2 + ISx +56)+ 8 = 0.

3.21. Giai phuO'ng trlnh sau:

(2x 2 -3x + 1)(2x 2 + Sx + l)-9x2 = 0.


a) (x+6)4 +(x+8)4 =272;

b) (5-x)4 +(2-x)4 = 17.

3.23. Giai phuO'ng trlnh


Sx3 + 6x2

+ 12x + 8 = 0 .

(Dei rhi \'ao Mp 10 chuyenngii', DHNN-DHQG HN, 2005)


( x 2 + I I x + 12 )( x 2 + 9x + 20 )( x 2 + 13x + 42) =

36 (X 2 + I I X + 30) (X 2 + I I X + 3 I ) .

20

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