Bai Giang Toan Hoc 236

http://baigiangtoanhoc.com 14 ch trng tm n thi i hc 2013

Bi ging c cung cp c quyn bi http:// Baigiangtoanhoc.com. Bin son : Nguyn ng Dng- GV chuyn SP-Chuyn gia luyn thi i Hc.

1

CH 10 : MT S PHNG PHP GII PHNG TRNH

BT PHNG TRNH V T.

PHN A: PHNG TRNH

Dng I: Phng php bin i trc tip

Sau khi tm tp xc nh ca phng trnh ta dng php bnh phng hoc lp phng hai v

ca phng trnh ta c phng trnh tng ng howacj phng trnh h qu. Gii phng

trnh ny ta tm c nghim thuc tp x nh.

V d 1 : Gii phng trnh

5 1 3 2 1 0x x x 1

Gii

iu kin : 1x

Phng trnh 1 5 1 3 2 1 5 1 3 2 2 3 2 1 1x x x x x x x x

2 2

2

2 2 3 2 1 2 4 3 2 1 11 24 4 0 2

11

x

x x x x x x x xx

2x . Vy nghim ca phng trnh l: 2x .


3 3 31 2 3 0x x x 2

Gii

Phng trnh 2

3 3 3 3 331 3 2 1 3 1 3 1 3 3 2x x x x x x x x x x

3 33 33 1 3 1 3 3 6 1 2 3 2x x x x x x x x x *



2

3 2

1 2 3 2 2 1 3 2 0 2x x x x x x x x x

.Thay 2x

vo phng trnh 2 thy ng. Vy nghim cu phng trnh l : 2x .

Ch : Php bin i ra phng trnh * l php bin i khng tng ng nn khi tm

c 2x ta phi th li.

Bi tp : Gii cc phng trnh sau

1. 5 5x x . p s: 5x .

2. 2 2 8 3 4x x x . p s: 4x , 7x .

3. 10 3 4 2 2x x x . p s: Phng trnh v nghim.

4. 3 31 1 2x x . p s: 0x .

Dng II: Phng php t n ph

Nhiu phng trnh nu dng phng php bin i trc tip th li gii phc tp hoc khng i

n kt qu. khc phc tnh trang ta c th dng phng php t n ph. Phng php

t n ph gn cc bc c bn sau:

Bc 1: t n ph, xc nh iu kin ca n ph.

Bc 2: Chuyn phng trnh cho v phng trnh n ph. Gii phng trnh n ph tm

nghim thch hp.

Bc 3: Tm nghim ca phng trnh ban u theo h thc khi t n ph.

Ta thng gp cc dng t n ph c bn sau y:


23 1 4 4 3 2x x x x 1

Gii

iu kin : 1 3x

t 3 1t x x vi 0t 2 22 2 4 3t x x , thay vo phng trnh 1 ta c :



3

2 22

2 2 2 6 0 3

2

t

t t t tt

22 4 3 2 2t x x x (Tha mn).

Vy nghim ca phng trnh l: 2x .

Ch : Dng tng qut ca v d 1 l phng trnh: a cx b cx d a cx b cx m

(vi , , , ,a b c d m l hng s , 0d c ).

Cch gii: t t a cx b cx vi 0t .


22 2 2 4 2 2x x x x 2

Gii

iu kin : 2x

t 2 2x x t vi 0t 2 22 2 4t x x , thay vo phng trnh 2 ta c :

21

2 0 2 2 2 2 4 2 02

tt t t x x x

t

2x .

Vy nghim ca phng trnh l: 2x .

Ch : Dng tng qut ca phng trnh 2 l phng trnh:

2 0m f x g x n f x g x m f x g x k vi 2 2 0m n

Cch gii : t t f x g x .


2 32 3 2 3 8x x x 3

Gii

iu kin : 2x



4

Phng trnh 3 2 22 2 4 2 2 3 2 2 4x x x x x x *

Do 2x khng phi l nghim 2 0x , chia 2 v phng trnh * ta c :

2 22 4 2 42 3 2 0

2 2

x x x x

x x

. t

2 2 4

2

x xt

x

vi 0t , ta c phng trnh sau

22 2

22 4

2 3 2 0 2 2 6 4 012

2

tx x

t t t x xxt

3 13x .

Vy nghim ca phng trnh l : 3 13x .

Ch : Dng tng qut ca phng trnh 3 l : a.f . . . 0x b g x c f x g x vi

0abc .

Cch gii :

- Nu 0 0f x g x , kim tra trc tip.

- Nu

0 0g x g x

f x a b cf x f x

, t

g xt

f x .


23

6 9 6 96

xx x x x

4

Gii

iu kin : 9x

Phng trnh 4 6 9 3 9 3 23x x x

, t 9t x vi 0t

2 9x t . Khi ta c phng trnh: 26 3 3 32t t t *

Vi 3t th * 24 73

12 32 08 25

t xt x

t x

.

Vi 0 3t th * 2 4 2 13t t x .



5

Vy nghim ca phng trnh l : 13x , 25x , 73x .

Ch : Dng tng qut ca phng trnh 4 l phng trnh:

2 22 2x a b a x b x a b a x b cx m ,vi , , ,a b c m l hng s, 0a .

Cch gii : t x b , 0t .


1. 2 23 21 18 2 7 7 2x x x x

HD: t 2 7 7t x x , vi 0t . p s : 6x , 1x .

2. 1 4 1 4 5x x x x

HD: t 1 4t x x , vi 0t . p s : 0x , 3x .

3. 23 2 1 2 3 5 2 4 9x x x x x

HD: t 3 2 2t x x , vi 0t . p s : 2x .

4. 22 3 1 2 2 5 3 3 16x x x x x

HD: t 2 3 1t x x , vi 0t . p s : 3x .

5. 2 3 32 2 5 1x x x

HD: 2

2 2

2

1 12 1 2 1 5 1 1 2 2 5

1 1

x x xx x x x x x

x x x

.

t 2 1

1

x xt

x

vi 0t . p s :

5 37

2x

.

6. 5

2 2 1 2 2 12

xx x x x

HD: t 1t x , vi 0t 2 4

1 12

tt t

. p s : 3x , 1x .

Loi 2 : Dng n ph a v phng trnh tch


2 32 2 5 1x x x 1

Gii



6

iu kin : 1x

Phng trnh 2 2 22 1 2 1 5 1 1x x x x x x . t 2

1

1

x u

x x v

,0

0

u

v

Ta c : 2 22

2( ) 5 2 2 02

u vv u uv u v v u

v u

.

Vi : 2 22 1 2 1 4 5 3 0u v x x x x x V nghim.

Vi : 2 22 2 1 1 5 3 0v u x x x x x 5 37

2x

.

Vy nghim ca phng trnh l: 5 37

2x

.


1. 2 27 25 9 2 35 7 2x x x x x

HD: 2 27 25 9 7 2 2 35x x x x x

23 11 22 7 2 5 7x x x x x

2 23 5 14 4 5 7 5 5 14x x x x x x . t t 5u x , 0u ;

2 5 14v x x , 0v . p s: 3 2 7x , 61 11137

18x

.

2. 2 47 10 14 5 4x x x

HD: 2 2 2 2( 2 2) 5 ( 2 2)( 2 2) 6( 2 2) 0x x x x x x x x .

t:t 2 2 2u x x 0u ; 2 2 2v x x , 0v . p s: 5 7

3x

.

Loi 3 : t n ph khng hon ton


2 21 2 3 1x x x x 1

Gii



7

t : 2 2 3x x t vi 0t 2 2 2 3x t x , Khi phng trnh 1 tr thnh :

2 21 2 2 1 2 1 0 1 2 0x t t x t x t x x t t 2

1

t

t x

.

Vi : 2 22 2 3 2 2 1 0 1 2t x x x x x .

Vi : 21 2 3 1t x x x x Phng trnh v nghim.

Vy nghim ca phng trnh l: 1 2x .


2 210 3 1 1 6 3x x x x 2

Gii

t : 2 3t x , 0t 2 2 3x t . Thay vo phng trnh 2 v thu gn ta c

2 21 6 9 3 2 0t x t x x , Xem y l phng trnh bc hai n t 9 0t nn phng

trnh c hai nghim phn bit :

1 6 32 3

2

1 6 31 3

2

xt x

xt x

.

Vi : 2

2

26 40

2 3 3 2 3 38

8 12 1 0

xt x x x x

x x

.

Vi : 2

2

1

1 3 3 1 3 13

4 3 1 0

xt x x x x

x x

.

Vy nghim ca phng trnh l : 6 40

8x

, 1x .


1. 2 23 1 3 1x x x x



8

HD: t 2 1t x . p s : 2 2x .

2. 2 4 4 1 2 1 4x x x x x

HD: t 24 4t x x t 2 1 4 1 2t x x x

2

2 2 1 4 1 2 4 3 1 2tt t x x x x x .

p s : 20 8 13

9x

.

3. 2 22 4 2 2 8 4x x x x

HD: 2

2 2 2 22 4 2 2 8 4 2 4 2 2 8 4x x x x x x x x

2 25 8 16 4 2 2 4 0x x x x * . t 2

2 2 42 42

tt x x

, thay vo

phng trnh * c : 25 8 2 16 32 0t x t x . p s :Phng trnh v nghim.

Loi 4 : t n ph chuyn phng trnh v t v h phng trnh v t


32 3 2 3 6 5 8 0x x 1

Gii

iu kin: 6

5x , t :

3 3 2

6 5

x u

x v

, vi 0v , ta c h phng trnh 3 2

2 3 8

5 3 8

u v

u v

*

**

T *8 2

3

uv

, thay vo ** thu gn ta c : 22 15 26 20 0u u u 2u

3 3 2 2 3 2 8 2x x x ( Tha mn iu kin).

Vy nghim ca phng trnh l : 2x .


2 24 5 1 2 1 9 3x x x x x 2

Gii



9

iu kin:

1

4

1

x

x

, t : 2

2

4 5 1

2 1

x x u

x x v

, vi 0

0

u

v

, ta c h phng trnh

2 2

9 3u v x

u v u v

2 2

2 2

04 5 1 2 1

561 4 5 1 1 2 1

65

xu v x x x x

u v xx x x x

.

Vy nghim ca phng trnh l: 0x , 56

65x .


3 2 33 3 3 5 1 3x x x x 3

Gii

Phng trnh 3 3 31 3 3 5 2x x . t : 3 3 5 1x y ta c:

3

3

1 3 5

1 3 5

x y

y x

Tr v theo v ca hai phng trnh ta c : 2 2

1 1 1 1 3 0x y x x y y

3 3 21 3 5 3 4 0 1x y x x x x x , 2x .

Vy nghim ca phng trnh l : 1x , 2x .


1. 2 24 5 1 4 5 7 3x x x x

p s : 5 13

8x

.

2. 31 1

12 2

x x

HD: t 31 1

,2 2

x u x v . p s : 17

2x ,

1

2x .

3. 3 2 242 1 17 1x x

HD: t 3 22 1 ,x u 24 17 x v . p s : 1x .

4. 22 3 1 1x x x



10

HD: t:2 2

2

1 2

1 31

x a a b

a ab ax b

. p s : 24

25x ,

3

2x .

5. 23 1 4 13 5x x x

HD: 2

3 1 2 3 3x x x . t : 3 1 2 3x y

2

2

2 3 3 1

2 3 2 1

y x

x y x

p s: 15 97

8x

,

11 73

8x

.

6. 3 23 3 5 8 36 53 25x x x x

HD: 33 3 5 2 3 3x x x .t : 3 3 5 2 3x y ,p s

5 3, 2

4x x

.

Loi 5 : t n ph l lng gic ( Phng php lng gic ha)

V d : Gii phng trnh

2 21 1 1 2 1x x x 1

Gii

iu kin: 1 1x

t : sin ;2 2

x t t

, ta c phng trnh : 1 cos sin 1 2cost t t

3 3 22 os 2cos sin sin

2 2 2 2 2

t t t tc (v ;

2 2t

nn os 02

tc )

16

1

22

xt

xt

Vy nghim ca phng trnh l : 1x , 1

2x .


1. 3 32 21 1 1 1 2 1x x x x



11

HD: t os 0;x c t t . p s : 2

2x ,

1 2 2 2 1

2x

.

2. 1 2 1 2

1 2 1 21 2 1 2

x xx x

x x

HD: t 2 os 0;x c t t . p s : 0x .

3. 2

32

9

xx

x

HD: t 3

0; ,cos 2

x t tt

. p s : 3 2x .

Dng III: Dng biu thc lin hp

Nhn cc biu thc trong phng trnh vi biu thc lin hp th xut hin nhn t chung v ta

chuyn c phng trnh cho v phng trnh tch n gin hn. Ta thng s dng cc

biu thc sau:

A BA B

A B

;

A BA B

A B

2A BA B

A B

;

2A BA B

A B

3 3

3 32 23

A BA B

A AB B

; 3 3

3 32 23

A BA B

A AB B

3

3

3 2 23

A BA B

A AB B

;

3

3

3 2 23

A BA B

A AB B

Sau y ta xt mt s v d


22 5 2 1 2 13 0x x x 1

Gii

iu kin: 5

2x . Khi hng trnh 21 2 5 1 2 1 2 2 9 0x x x

2 3 2 32 3 3 0

2 5 1 1 1

x xx x

x x

1 12 3 3 0

2 5 1 1 1x x

x x



12

3

1 13 0

2 5 1 1 1

x

xx x

*

Vi : 5

2x ta c :

1 13 3

2 5 1 1 1x

x x

nn phng trnh * v nghim.



2 212 5 3 5x x x 2

Gii

Nhn xt : 2 2 512 5 3 5 0

3x x x x ,

Phng trnh 2 2 212 4 3 2 5 3x x x

2 2

2 2 2 2

4 4 2 23 2 0 2 3 0

12 4 5 3 12 4 5 3

x x x xx x

x x x x

2 2

2

2 23 0

12 4 5 3

x

x x

x x

*

2 2 2 2

5 2 2 2 23 0

3 12 4 5 3 12 4 5 3

x x x xx

x x x x

* v nghim.



235 1 9 2 3 1x x x x 3

Gii

iu kin : 1

5x , khi phng trnh 3 23( 5 1 2) 9 2 2 3 5x x x x



13

2 33

5 1 11 2 5

5 1 2 9 2 9 4

x xx x

x x x

2 33

5 11 2 5 0

5 1 2 9 2 9 4

x xx x x

2 33

5 5 1 5 11 2 0 1

5 1 2 9 2 9 4

xx x x

x x x

.



233 1 2 19 8 2 5x x x x 4

Gii

iukin:1

3x ,phng trnh 3 2 32 2 1 3 1 2 2 19 8 0x x x x x x

222

2 23 3

2 142 0

1 3 1 2 2 19 8 (19 8)

x x xx xx x

x x x x x x

2 2

2 23 3

1 2 142 0

1 3 1 2 2 19 8 (19 8)

xx x x x

x x x x x x

0

1

x

x

v

2 23 3

1 2 142 0

1 3 1 2 2 19 8 (19 8)

x

x x x x x x

vi

1

3x .



2 22 9 2 1 4x x x x x 5

Gii

Ta thy : 2 22 9 2 1 2 4x x x x x



14

4x khng phi l nghim

Xt 4x , khi phng trnh 5

2 2

2 2

2 84 2 9 2 1 2

2 9 2 1

xx x x x x

x x x x

Vy ta c h:

2 2

2

2 2

02 9 2 1 2

2 2 9 6 82 9 2 1 4

7

xx x x x

x x xxx x x x x

Th li tha mn phng trnh 5

Vy phng trnh c 2 nghim 8

0;7

x x .

Ch : V d 5 l phng php a v h tm

Nu phng trnh v t c dng A B C , m : A B C

dy C c th l hng s ,c th l biu thc ca x . Ta c th gii nh sau :

A BC A B

A B

, khi ta c h: 2

A B CA C

A B


1. 23 1 6 3 14 8 0x x x x

HD: 3 1

5 3 1 03 1 6 1

x xx x

. p s: 5x .

2. 3 2 2 2 6x x x

HD: 3 2 6 3 2 4 0x x x . p s: 3x , 11 3 5

2x

.

3. 2 32 11 15 3 4 4x x x

HD:

2 33

12 33 2 5

4 4 2 4 4 4

xx x

x x

. p s: 3x .

4. 22 3 4 3 5 9 6 13x x x x



15

HD: 22 3

1 03 4 2 5 9 3

x xx x x x

. p s: 0x , 1x .

5. 1 1

21 3

xx

x x

HD: 1

1 2( 1)1 3

xx

x x

. p s: 1x .

Dng IV: S dng tnh n iu ca hm s v bt ng thc

Nhiu khi ta gp cc phng trnh m dng phng php : bin i trc tip, t n ph hay

nhn lin hp th cch gii s phc tp hoc khng i n kt qu cui cng. Khi phng

php dng hm s v bt ng thc li t ra hiu qu v cho li gii ngn gn, vic tm nghim

s d dng hn rt nhiu.

Loi 1 : Dng tnh n diu ca hm s

Cho hm s y f x n iu trn D khi ta c cc kt qu sau :

f x m c nghim duy nht.

f x f y x y .

0

0

f x mx x

f x m

l nghim duy nht.


3 23 4 2 3 2 3 1x x x x x 1

Gii

iu kin : 1

3x , khi phng trnh

331 1 1 3 1 3 1x x x x . Xt hm

s c trng : 3f t t t vi 0t , 2' 3 1 0f t t f t : l hm ng bin trn 0; .

Ta c : 1 3 1f x f x 20

1 3 1 01

xx x x x

x

.





16

2 23 2 9 3 4 2 1 1 0x x x x x 2

Gii

iu kin phng trnh cho c nghim l : 1

02

x , khi

Phng trnh 2 2

2 3 2 3 3 2 1 2 1 3 2x x x x .

Xt hm 22 3f t t t vi 0t 2

2

2 3' 2 0

3

tf t

t

vi 0t f t l hm

ng bin trn 0; . Ta c : 3 2 1 3 2 1f x f x x x 1

5x .

Vy ngim ca phng trnh l : 1

5x .


3 2 2 3 32 10 17 8 2 5x x x x x x 3

Gii

Nhn xt 0x khng phi l nghim ca phng trnh 3 nn chia hai v cho 3x ta c :

32 3 2

10 17 8 52 2 1

x x x x . t

1y

x vi 0y 3 2 238 17 10 2 2 5 1y y y y

3 2 232 1 2 2 1 5 1 2 5 1y y y y . Xt hm s : 3 22 ' 3 2f t t t f t t

f t l hm ng bin. Ta c : 2 23 32 1 5 1 2 1 5 1f y f y y y

217 97 17 97

16 128 17 6 0

17 97 17 97

16 12

y x

y y y

y x

Vy nghim ca phng trnh l: 17 97

12x

,

17 97

12x

.



17


1. 24 1 4 1 1x x

p s: 1

2x .

2. 6 8

63 2x x

p s: 3

2x .

3.

22 1

2 1 3 22

xx x

HD: Xt hm s

22 1

2 1 3 22

xf x x x

, vi

1 3;

2 2x

' 1 1( ) 2 2 1 1 02(2 1) 3 2 2 1 3 2

f x x xx x x x

nn phng

trnh cho c ti a 2 nghim. p s: 1

2x ,

3

2x .

4. 3 2 315 78 144 5 2 9x x x x

HD: 3 35 5 5 2 9 5 2 9x x x x . p s: 4x ,

11 5

2x

.

5. 3 38 4 1 6 1x x x

HD: 3 32 2 6 1 6 1x x x x . p s: os

9x c

,

5os

9x c

,

7os

9x c

.

Loi 2 : Dng bt ng thc

Xt phng trnh f x g x xc nh trn D nu

f x h x

g x h x

, vi x D th

f x h xf x g x

g x h x

.

Ngoi bt ng thc C Si, ta thng p dng cc bt ng thc c bn sau :

2A A A . Du bng xy ra khi v ch khi 0A .

2 0A . Du bng xy ra khi v ch khi 0A .

A B A B . Du bng xy ra khi v ch khi 0AB .



18

u v u v . Du bng xy ra khi v ch khi u v v cng hng.


24 6 10 27x x x x 1

Gii

iu kin : 4 6x . Theo C Si ta c: 4 1 3

4 4 12 2

x xx x

;

6 1 5

6 6 12 2

x xx x

4 6 2x x .Du bng xy ra khi v ch khi

4 15

6 1

xx

x

.

Mt khc : 22 10 27 5 2 2x x x . Du bng xy ra khi v ch khi 5x .



2 2 219 7 22 28 13 43 37 3 3 3x x x x x x x 2

Gii

Nhn xt : 2A A A , du bng xy ra khi v ch khi 0A .

Ta c:

2

2 1 75 75 5 3192 4 4 2

x x x

.

2 2 227 22 28 2 1 3 3 3 3 3 3 3 3x x x x x x x .

2 2 2

2 1 7 7 7 713 43 37 3 2 3 2 3 2 3 22 2 2 2 2

x x x x x x x

.

Cng v theo v cc bt ng thc trn ta c :



19

2 2 25 3 7

19 7 22 28 13 43 37 3 3 3 22 2

x x x x x x x x

3 3 3x . Du bng xy ra khi v ch khi:

10

2

2 1 0 1

3 0 2

72 0

2

x

xx

x

x

.

Vy nghim ca phng trnh cho l : 1

2x .


2 22 2 2 2 2 2x x x x 3

Gii

Phng trnh 2 2

3 1 1 1 1 2 2x x . Xt : 1;1u x , 1 ;1v x

2;2u v 2

1 1u x , 2

1 1u x , 2 2u v . Phng trnh 3 tr

thnh u v u v khi u v v cng hng 1 1

01 1

x xx

.



1. 24 6 10 27x x x x

HD: 2VT ; 2VP . p s : 5x

2. 2 32 11 21 3 4 4x x x

HD: 3VP x . p s : 3x .

3. 2 2 2 217 1

13 6 10 5 13 17 48 36 36 8 212 2

x x x x x x x x

HD: 3

62

VT x ; 3

62

VP x . p s : 3

2x .

4. 21 3 2 1x x x x

HD: Xt : 1;1u x , 1 ;1v x . p s : 1x , 1 2x .



20

5. 4 3 316 5 6 4x x x

HD: 24 4 3VP x x . p s : 1

2x .

6. 34 1 8 1 1x x

p s: 1

2x .

PHN B: BT PHNG TRNH

Dng I: Phng php bin i trc tip

Ta thc hin theo cc bc sau y

Bc 1: Tm tp xc nh ca bt phng trnh.

Bc 2: Bnh phng, lp phng bt phng trnh cho bin i, ta c bt phng

trnh n gin hn v tm c nghim ca bt phng trnh ny.

Bc 3: Kt hp tp xc nh suy ra nghim ca bt phng trnh.

Ch : Cc bt phng trnh c bn

2

0

0

g x

f x g x f x

f x g x

2

0

0

0

g x

f xf x g x

g x

f x g x

0f x g x f x g x

V d 1 : Gii bt phng trnh

5 2 3 2x x x 1

Gii

iu kin : 5

2 *2

x , khi bt phng trnh 1 tng ng vi bt phng trnh



21

2

22

2 3 0

2 3 05 2 3 2 5 2 3 2 2 11 15 2 3

2 11 15 2 3

x

xx x x x x x x x

x x x

2

33

223

32 2

22 6 0

xx

xx

x x

, kt hp vi iu kin * , ta c: 2 2x .

Vy tp nghim ca bt phng trnh l 2;2D .


5 1 1 2 4x x x 2

Gii

iu kin : 2 *x , bt phng trnh 2 5 1 2 4 1x x x

2

2 05 1 3 5 2 1 2 4 1 2 4 2

1 2 4 2

xx x x x x x x

x x x

2 10 0 0 10x x x , kt hp vi * ta c : 2 10x .


Bi tp : Gii cc bt phng trnh sau

1. 22 16 7

33 3

x xx

x x

. p s: 10 34x .

2. 3 2 8 7x x x . p s :4 5

6 7

x

x

.

3. 1

3 12

x x . p s : 8 31

18

x

.



22

4. 2 4 3

2x x

x

. p s:

0

1 2

x

x

.

5. 251 2

11

x x

x

. p s:

1 1 52

1 52 5

x

x

.

Dng II: Phng php nhn lin hp

Ta cn ch cc cng thc sau:

A B

A BA B

2A B

A BA B

3 33 32 23

A BA B

A AB B

3

3

3 2 23

A BA B

A AB B

.


2

2

221

3 9 2

xx

x

1

Gii

iu kin :

99 2 0

29 2 3

0

x x

xx

*

Bt phng trnh

22

2

2 3 9 21 21 18 2 6 9 2 2 42

4

x xx x x x

x

79 2 4 9 2 16

2x x x . Kt hp vi * ta c tp nghim ca bt phng

trnh l: 9 7

;0 0;2 2

D

.




23

1 1x x x 2

Gii

iu kin : 1 1x

Bt phng trnh 2 1 1 1 1 1 1 2x x x x x x x x x *

Nu 0x th * lun ng.

Nu 0x th 2 2* 1 1 2 1 1 2 1 4 1 1x x x x x x lun

ng.

Nu 0x th * 1 1 2x x v l v 1 1 2x x lun ng.

Vy tp nghim ca bt phng trnh cho l: 0;1D .


1.

2

41 1

xx

x

. p s: 1 8x .

2. 2 1 2 2x x x . p s: 7 32 2x .

3. 21 1 4

3x

x

. p s:

10

2

10

2

x

x

.

4. 2 1x x x . p s: 2 3 3

3x

.

5. 2

1 2 12 9

xx

x

. p s:

450

8x .

Dng III: Phng php t n ph

Ta thc hin theo cc bc sau

Bc 1: Chn n ph thch hp, tm iu kin ca n ph.

Bc 2: Chuyn bt phng trnh cho v bt phng trnh n ph, gii bt phng

trnh n ph, tm nghim ca n ph.

Bc 3: T tp nghim ca bt phng trnh n ph suy ra tp nghim ca bt phng

trnh cho.



24


27 7 7 6 2 49 7 42 181 14x x x x x 1

Gii

iu kin : 6

7x

Bt phng trnh 2

1 7 7 7 6 7 7 7 6 182x x x x

t 7 7 7 6x x t vi 0t , ta c bt phng trnh 2 182 0 13t t t

2

6 612

77 7 7 6 13 12

1183 709849 7 42 84 7

x xx x

xx x x

66

7x .

Vy tp nghim ca bt phng trnh l 6

;67

D

.


5 1

5 2 422

x xxx

2

Gii

iu kin : 0x

t 1 1

2 . 22 2

t x xx x

2t ; 2 21 1

1 2 2 24 2

t x x tx x

Ta c : 2 22

15 2 2 4 2 5 2 0 2 21

22

t

t t t t t xt x

2 2 32

2 22 4 1 0

32 2 0 222

x x

x x

xx

Vy tp nghim ca bt phng trnh l : 3 3

0; 2 2;2 2

D

.



25


2

1 1 24

xx x 2

Gii

iu kin : 1 1x

t cos 0;x t t , ta c bt phng trnh : 2

223 2 2 1 2

4

xx

2 22 2

2 4 2os 7 sin2 2 1 os 2 2 2sin sin 14sin 32sin 17 04 4

c t tc t t t t t

2 2sin 1 sin 2sin 17 0t t t lun ng vi 0;t .



1. 3 1

3 2 722

x xxx

. p s:

8 3 7

2

8 3 70

2

x

x

.

2. 1 3

1 2

x x

x x

. p s:

1 0

1 2

x

x

.

3. 42 3 2 2 3 3 2 2x x x x . p s: 2

2 34

3 47

x

x

.

4. 22

9 1 3 7 1 3 4x x x . HD: t 3 4t x . p s: 4

13

x .

5. 2

1 3 2 3 2 2x x x x . HD: t 1; 3u x v x . p s: 5x .

Dng IV: Phng php chia tp xc nh

Bt phng trnh c tp xc nh D , vic gii bt phng trnh trn tp D gp kh

khn, khi ta chia 1 2 3 .... nD D D D D .



26

Trn 1 2, ,.. nD D D ta gii bt phng trnh cho.

Kt hp nghim ca bt phng trnh trn 1 2, ,... nD D D suy ra nghim ca bt phng

trnh cho.


2 24 3 2 3 1 1x x x x x 1

Gii

Bt phng trnh 1 1 3 1 2 1 1x x x x x *

Trng hp 1: 1 0 1x x tha mn bt phng trnh *

Trng hp 2: 1x , bt phng trnh * 3 2 1 1x x x v nghim v :

3 2 1 0

1 0

x x

x

.

Trng hp 3: 1x , bt phng trnh * 1 1 1 2 1 3x x x x x

11 2 0 1

21 3 1 221 3 2 1 3 1 2

1 3 3

x xx x x x

x x x x xx x

Vy tp nghim ca bt phng trnh l : 1

; 12

D

.


2 2 23 2 4 3 2 5 4x x x x x x 2

Gii

iu kin: 4

1

x

x

Trng hp 1: 4x th 2 1 2 1 3 2 1 4x x x x x x



27

2 3 2 4x x x lun ng v : 2 4

3 4

x x

x x

4x .

Trng hp 2: 1x th 2 1 2 1 3 2 1 4x x x x x x *

Nu 1x * ng.

Nu 1x * 2 3 2 4x x x v nghim v 2 4

3 4

x x

x x

.

Vy tp nghim ca bt phng trnh l: 4; 1D .


1. 2 2 28 15 2 15 4 18 18x x x x x x . p s:

3

5

175

3

x

x

x

.

2. 2 2 23 2 6 5 2 9 7x x x x x x . p s:

1

2

5

x

x

x

.

3. 2 23 4 9x x x . p s: 3

13

6

x

x

.

4. 3 21 1 3 1 0x x x x . p s: 1x .

5. 2 2 28 15 2 15 4 18 18x x x x x x . p s: 17

3x .

Dng V: Phng php khng mu mc

Ta thng dng cc phng php sau

Phng php nh gi.

Dng bt ng thc.

Dng hm s.




28

3 32 8 7 8x x x x x 1

Gii

iu kin bt phng trnh c nghim 0x . Xt hm s 3 32 8 7f x x x x x x

2

233

1 1 3 7' 2 0

2 2 8 3 7

xf x

x x x x

f x l hm ng bin trn 0; m

1 8f ; 8 0 1f x x .



5 1 4 1 3x x x 2

Gii

iu kin 1

4x , bt phng trnh

1 12 5 3 4

x x *

Do 1 1 1 1

4 5 5 4 3 3 44

xx x x

* lun ng.

Vy tp nghim ca bt phng trnh l 1

;4

D

.


21 3 2 10 16x x x x 3

Gii

iu kin : 1x ; bt phng trnh 22

3 3 1 2. 3 1x x x x

Xt 3; 1 ; 1;1u x x v . 3 1u v x x ; 22

2 1 1u v x x



29

Bt phng trnh 3 .u v u v ; m .u v u v .u v u v ,u v cng hng

33 10 5

1 1 1 3

xx xx

x x

.

Vy tp nghim ca bt phng trnh l 5D .


1. 3 541 5 7 5 7 13 7 8x x x x . p s:5

37

x .

2. 22 7 2 7 35x x x x x .p s:

229

012

x

.

3. 21 2 1 2 2x x x . HD: 2

4 21 1 4 0x x . p s 0x .

4. 1 1x x x . p s: 1 0x .

5. 2

1 1 24

xx x . HD:

4 221 1 0

16

xx . p s: 1 1x .

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Bai Giang Toan Hoc 236