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Lª ThÞ Ph−¬ng Hoa
Tr−êng THPT Tam D−¬ng II -- 1 -- http://ebook.here.vn
1.ph−¬ng tr×nh1.ph−¬ng tr×nh1.ph−¬ng tr×nh1.ph−¬ng tr×nh –bÊt ph−¬ng tr×nh c¬ b¶nbÊt ph−¬ng tr×nh c¬ b¶nbÊt ph−¬ng tr×nh c¬ b¶nbÊt ph−¬ng tr×nh c¬ b¶n
a.ph−¬ng tr×nh c¬ b¶n:
D¹ng ph−¬ng tr×nh:



≥
≥
⇔=
)()(
0)(
)()( 2
xgxf
xg
xgxf (nÕu g(x) cã TX§ l R)
b.BÊt ph−¬ng tr×nh c¬ b¶n:
D¹ng 1:









≥
≥



<
≥
⇔>
)()(
0)(
0)(
0)(
)()(
2
xgxf
xg
xg
xf
xgxf
D¹ng 2:
( )
( )
( ) ( )




<
≥
>
⇔<
xgxf
xf
xg
xgxf
2
0
0
)()(
Chó ý: Khi hÖ chøa tõ hai biÓu thøc c¨n bËc hai trë lªn , ®Ó cã thÓ ®−a vÒ d¹ng c¬ b¶n
, ta l m nh− sau:
+ §Æt mét hÖ ®iÒu kiÖn cho tÊt c¶ c¸c c¨n ®Òu cã nghÜa .
+ ChuyÓn vÕ hoÆc ®Æt ®iÒu kiÖn ®Ó hai vÕ ®Òu kh«ng ©m .
+ B×nh ph−¬ng hai vÕ .
+ TiÕp tôc cho ®Õn khi hÕt c¨n .
bµi tËp ¸p dông
B i 1.1: Gi¶i c¸c ph−¬ng tr×nh sau:
)1(3253.1 −=+ xx
)2(632.2 xx −=+
Gi¶i1:
Ph−¬ng tr×nh ® cho t−¬ng ®−¬ng víi:








=
=
⇔
=+−
≥
2
7
2
014154
2
3
2 x
x
xx
x
Gi¶i2:
3
113
6
03314
6
2
=⇔



=∨=
≤
⇔



=+−
≤
x
xx
x
xx
x
B i 1.2 Gi¶i ph−¬ng tr×nh sau
)1(1266.1 2
−=+− xxx (§H X©y Dùng -2001).
Gi¶i:

1
1
2
1
)12(66
2
1
22
=⇔




=
≥




⇔
−=+−
≥
x
x
x
xxx
x
B i 1.3 Gi¶i ph−¬ng tr×nh
321 =++− xx
Gi¶i:Ph−¬ng tr×nh ® cho t−¬ng ®−¬ng víi hÖ:
2
)4()2)(1(_
41
4)2)(1(
1
2
=⇔






−=−−
≤≤
⇔
−=+−
≥
⇔ x
xxx
x
xxx
x
B i 1.4: Gi¶i ph−¬ng tr×nh
231 −=−−− xxx
3
326
3
326
3
326
43
0883
43
6524
3
231
3
22
+
=⇔





−
=∨
+
=
≤≤
⇔



=+−
≤≤
⇔



+−=−
≥
⇔



−+−=−
≥
x
xx
x
xx
x
xxx
x
xxx
x
--
xxxx −+=−+ 1
3
2
1 2
(§HQG H Néi 2000)








−=−
≤≤
⇔
−+=−+−+
≤≤
22222
3
2
3
2
3
2
10
21
3
4
3
2
3
2
1
10
xxxx
x
xxxxxx
x



=
=
⇔



=∨=
≤≤
⇔



=−−−
≤≤
⇔
1
0
10
10
0)1(
10
22
x
x
xx
x
xxxx
x
( ) 3428316643 −=−−+ xx
( ) 2
2
2
2
4
3
3428316643
4
3
=⇔






=
≥
⇔




−=−−+
≥
x
x
x
xx
x

B i 1.7: Gi¶i bÊt ph−¬ng tr×nh:
27593137 −≤−−− xxx (§H DL Ph−¬ng §«ng -2001)
§iÒu kiÖn:
5
27
≥x
BÊt ph−¬ng tr×nh ® cho t−¬ng ®−¬ng víi:





−+−≤−
≥
93275137
5
27
xxx
x
( )( ) ( )( )
23
59
65762229
044345859
23
5
27
23275932
5
27
275932368137
5
27
2
≤≤
+
⇔





≥+−
≤≤
⇔





−≥−−
≥
⇔





−−+−≤−
≥
⇔
x
xx
x
xxx
x
xxxx
x
B i tËp l m thªm:
B i 1: (PP B§ T§)
2 2
2 2
2
2
1. 3 2 2 1; 2. 3 9 1 2
3. 4 6 4; 4. 2 4 2
5. 3 9 1 | 2 |; 6. 2 3 0;
7. 1 1;
x x x x x x
x x x x x x
x x x x x
x x
− + = − − + = −
− + = + + + = −
− + = − − + =
+ + =
B i 2: (PP B§ T§)
1. 3 6 3;
2. 3 2 1 3;
3. 3 2 1;
4. 9 5 2 4;
5. 3 4 2 1 3;
6. 5 1 3 2 1 0;
x x
x x
x x
x x
x x x
x x x
+ + − =
− + − =
+ − − =
+ = − +
+ − + = +
− − − − − =
7. 3 4 4 2 ;x x x+ + + =
8. 5 5 10 5 15 10;x x x− + − = −
9. 4 1 1 2 ;x x x+ − − = −

2
10. 3 2 1 2;
11. 1 5 1 3 2
x x x
x x x
− + − + + =
− − − = −
12. 1 9 2 12x x x+ − − = −
2 2
13. 5 8 4 5x x x x+ − + + − =
2 2
14. 3 5 8 3 5 1 1x x x x+ + − + + =
2 2
15. 9 7 2 5 1 3 2 1x x x x x+ − − = − − − − −
2 2 2
2
16. 3 6 16 2 2 2 4
3 1 1 4 2
17.
3 9 9
x x x x x x
x
x x x
+ + + + = + +
+
= + +
2
18. 1 2 5x x x− = − −
19. 11 11 4x x x x+ + + − + =
20. 1 1 8x x x+ − = − +
--------------------------------------------------------------------------
2.ph−¬ng ph¸p §Æt mét Èn phô
D¹ng 1: Gi¶i ph−¬ng tr×nh:
( ) ( ) 0=++ CxfBxAf
Ph−¬ng ph¸p gi¶i : §Æt ( ) ( ) ( ) 2
0 txfttxf =⇔≥= ;
Ph−¬ng tr×nh ® cho trë th nh : ( )002
≥=++ tCBtAt
L m t−¬ng tù víi bÊt ph−¬ng tr×nh d¹ng: ( ) ( ) 0≥++ CxfBxAf
D¹ng 2:Gi¶i ph−¬ng tr×nh:
( ) ( )( ) ( )( ) 0)(2 =++++ CDxgxfBxgxfA
(Víi ( ) Dxgxf =+ )( )
Ph−¬ng ph¸p gi¶i :
§Æt ( ) ( ) ( ) ( )xgxfDtttxgxf 20)( 2
+=⇔≥=+
Ph−¬ng tr×nh ® cho trë th nh : ( )002
≥=++ tCAtBt
L m t−¬ng tù víi bÊt ph−¬ng tr×nh d¹ng:
( ) ( )( ) ( )( ) 0)(2 ≥++++ CDxgxfxgxfA
bµi tËp ¸p dông:
B i 2.1: Gi¶i c¸c ph−¬ng tr×nh
)1(75553,1 22
+−=+− xxxx

)2(3012.2,2 22
=++ xx (§H DL Hång l¹c-2001)
Gi¶i1: )1(75553,1 22
+−=+− xxxx
§Æt )0(552
≥=+− ttxx
Ph−¬ng tr×nh ® cho trë th nh:








±
=
=
=
⇔




=+−
=+−
⇔


=
=
⇔=+−
2
215
4
1
455
155
2
1
023 2
2
2
x
x
x
xx
xx
t
t
tt
Gi¶i2: )2(30122,2 22
=++ xx
§Æt )0(122
>+= txt



−=
=
⇔=−+
)(7
)(6
0422
Lt
tmt
tt
VËy 626122
±=⇔=+ xx
--------------------------------------------------------------------------
B i 2.2: Gi¶i c¸c ph−¬ng tr×nh
)1(4
2
47
.1
2
x
x
xx
=
+
++
(§H §«ng ®«-2000).
)2(4324.2 22
xxxx −+=−+ (§H Má -2001)
Gi¶i2:
§Æt )0(4 2
≥−= yxy



=−+
=−+
⇔



+=+
=+
23
42)(
32
4 222
xyyx
xyyx
xyyx
yx

Gi¶i hÖ ®èi xøng n y ta ®−îc nghiÖm:








+
−=
=
=
⇔


=∧=
=∧=
3
142
2
0
02
20
x
x
x
yx
yx
Gi¶i1:§iÒu kiÖn: 0≥x §Æt )0( ≥= ttx
04874 234
=+−+− tttt
Gi¶i ph−¬ng tr×nh bËc 4 :
XÐt t=0 kh«ng l nghiÖm
XÐt t ≠ 0 ,chia hai vÕ cho t2
v ®Æt )22(
2
≥+= u
t
tu
Ta ®−îc ph−¬ng tr×nh 


=
=
⇔


=
=
⇔


=
=
⇔=+−
4
1
2
1
3
)(1
0342
x
x
t
t
u
Lu
uu
B i 2.3: Gi¶i c¸c bÊt ph−¬ng tr×nh sau
123342.1 22
>−−++ xxxx (§HDL Ph−¬ng §«ng -2000)
2)2(4)4(.2 22
<−++−− xxxxx (§H QG HCM -1999)
Gi¶i1:
§iÒu kiÖn: 13 ≤≤− x
§Æt: )0(23 2
≥−−= txxt
BÊt ph−¬ng tr×nh ® cho trë th nh:
2
5
0
0
2
5
1
0
0532 2
<≤⇔




≤
<<−
⇔



≤
>++−
t
t
t
t
tt
Thay v o c¸ch ®Æt: 13
0
4
13
2
13
2
≤≤−⇔




≥++
≤≤−
x
xx
x
Gi¶i2:
2)2(4)4(.2 22
<−++−− xxxxx
§iÒu kiÖn: 40 ≤≤ x
§Æt: 042
≥+−= xxt

Thay v o BPT § cho v gi¶i ra ta ®−îc 1>t
Thay v o c¸ch ®Æt ta ®−îc: 3232 +<<− x
7
2
1
2
2
3
3.1 −+<+
x
x
x
x (§H Th¸i Nguyªn -2000)
3)7)(2(72.2 ≤−++−++ xxxx
Gi¶i1: BiÕn ®æi bÊt ph−¬ng tr×nh ® cho trë th nh:
( )
09
2
1
3
2
1
2
9
2
1
12)
2
1
(3
2
2
2
>−





+−





+⇔
−








++<+
x
x
x
x
x
x
x
x
§Æt: 2
2
1
≥⇒+= t
x
xt
BPT ® cho trë th nh:






+>
−<<
⇔>+⇔
>⇔




>−−
≥
7
2
3
4
7
2
3
40
3
2
1
3
0932
2
2
x
x
x
x
t
tt
t
Gi¶i 2:
§Æt )0(72 ≥−++= txxt
VËy
2
9
)7)(2(
2
−
=−+
t
xx
BÊt ph−¬ng tr×nh ® cho trë th nh:



=
−=
⇔




≤−++
≤≤−
⇔
≤≤⇔≤−+
7
2
9)7)(2(29
72
3001522
x
x
xx
x
ttt
B i tËp.B i tËp.B i tËp.B i tËp. Gi¶i c¸c PT sau:

B i 1:
2 2
2 2
2 2
2
1. 3 5 5 5 7;
2. 2 12 30;
3. 13 7;
4. ( 5)(2 ) 3 3 ;
x x x x
x x
x x x x
x x x x
− + = − +
+ =
− − − + =
+ − = +
2
6. ( 4)( 1) 3 5 2 6;x x x x+ + − + + =
2 2
11. 2( 2 ) 2 3 9;x x x x− + − − =
2 2
12. ( 3) 3 22 3 7;x x x x− + − = − +
( )( ) 2
15. 1 2 1 2 2 ;x x x x+ − = + −
( )2 2
16. 2 2 2 3 9 0;x x x x− + − − − =
2 2
17. 3 15 2 5 1 2;x x x x+ + + + =
B i 2:
2 2
5. 3 3 3 6 3;x x x x− + + − + =
2 2
7. 5 2 2 5 9 1;x x x x+ + + + − =
9. 1 4 ( 1)(4 ) 5;x x x x+ + − + + − =
2 2
10. 4 2 3 4 ;x x x x+ − = + −
2 2
13. 2 5 2 2 5 6 1;x x x x+ + − + − =
2 2
14. 3 2 2 6 2 2;x x x x+ + − + + = −
2 2 2
18. 4 1 2 2 9;x x x x x x+ + + + + = + +
2 2 2
8. 4 8 4 4 2 8 12;x x x x x x+ + + + + = + +
2 2
19. 1 2 1 2;x x x x− − + + − =
2 2
20. 17 17 9;x x x x+ − + − =
22
21.1 1 ;
3
x x x x+ − = + −
24 4
22. 16 6;
2
x x
x x
+ + −
= + − −
2
23. 3 2 1 4 9 2 3 5 2;x x x x x− + = = − + − +

2
24. 2 3 1 3 2 2 5 3 16;x x x x x+ + + = + + + −
25. 2 2 5 2 3 2 5 7 2;x x x x− + − + + + − =
( ) ( )
3 3
5 5
26. 7 3 8 7 3 7;x x
−
− − − =
2
27. 2 3 2 ;
2 3
x
x x
x
+ + =
+
4 2 2
28. 1 1 2;x x x x− − + + − =
2 2
29. 5 14 9 20 5 1;x x x x x+ + − − − = +
( )3 2
30.10 8 3 6 ;x x x+ = − −
3 2
31. 1 3 1;x x x− = + −
2
32. 1 ( 1) 0;x x x x x x− − − − + − =
§Æt Èn phô ®Ó trë th nh ph−¬ng tr×nh cã 2 Èn:
* L viÖc sö dông 1 Èn phô chuyÓn ®Ó chuyÓn PT ban ®Çu th nh 1 PT víi 1 Èn phô
nh−ng c¸c hÖ sè vÉn cßn chøa x
* PP n y th−êng ®−îc SD ®èi víi nh÷ng PT khi lùa chän 1 Èn phô cho1 BT th× c¸c BT
cßn l¹i kh«ng BD ®−îc triÖt ®Ó qua Èn phô ®ã hoÆc nÕu BD ®−îc th× c«ng thøc BD
qu¸ phøc tap.
* Khi ®ã th−êng ta ®−îc 1 PT bËc 2 theo Èn phô (hoÆc vÉn theo Èn x) cã biÖt sè ∆ l
1 sè chÝnh ph−¬ng.
B i 1:
2 2
1. 1 2 2 ;x x x x− = −
2 2
2. 1 2 2;x x x− = +
2 2
3. (4 1) 1 2 2 1;x x x x− + = + +
2 2
4. 4 4 (2 ) 2 4;x x x x x+ − = + − +
2 2
5. 3 1 (3 ) 1;x x x x+ + = + +
2 2
6. (4 1) 4 1 8 2 1;x x x x− + = + +
2
7. 4 1 1 3 2 1 1 ;x x x x+ − = + − + −

2 2
2 2
2
8. 2(1 ) 2 1 2 1;
9. 1 2 4 1 2 1;
10. 12 1 36;
1 1 1
11. 2 1 3 0;
x x x x x
x x x x
x x x
x
x x
x x x
− + − = − −
+ − = − − +
+ + + =
−
+ − − − − =
3.Ph−¬ng ph¸p §Æt hai Èn phô
( ) ( )( ) ( ) 0)( =+++ CxgxfBxgxfA nnn
(Víi ( ) Dxgxf =+ )( )
Ph−¬ng ph¸p gi¶i : §Æt:
( )
( )
Dvu
vxg
uxf nn
n
n
=+⇒




=
=
( )



=+
=+++
Dvu
CBuvvuA
nn
0
( ) ( )( ) ( ) 0)( =++− CxgxfBxgxfA nnn
(Víi ( ) ( ) Dxgxf =− )
Ph−¬ng ph¸p gi¶i : §Æt:
( )
( )
Dvu
vxg
uxf nn
n
n
=−⇒




=
=
( )



=−
=++−
Dvu
CBuvvuA
nn
0
B i 3.1: Gi¶i ph−¬ng tr×nh:
)x6)(2x(x62x −+=−++ (§H Ngo¹i Ng÷-2001)
Gi¶i :
§Æt )0v,u(
vx6
u2x
≥




=−
=+
2vu
08uv2)uv(
vuuv
vuuv
8vu
2
22
==⇔



=−−
+=
⇔



+=
=+
VËy:

2x2x62x =⇔=−=+
B i 3.2:Gi¶i ph−¬ng tr×nh:
13x22x 33
=+−+ (An Ninh-01)
Gi¶i :
§Æt:




=+
=+
v3x
u22x
3
3



−=
=
⇔


−==
==
⇔



=
=−
30x
5x
2u;3v
3u;2v
6uv
1vu
541xx56 44
=++−
§Æt:
)0uv(
v41x
ux56
4
4
≥




=+
=−



=
−=
⇔


==
==
⇔



=+
=+
40x
25x
2v;3u
3v;2u
97vu
5vu
44
B i tËp l m thªm: Gi¶i c¸c pt:
20 20
1. 6;
x x
x x
+ −
− =
42. 6 2 2(1 (6 )( 2);x x x x− + − = − − −
3
3
3
2 2
33
3. 2 1 1;
4. 9 2 1;
5. 9 1 7 1 4;
6. 3 10 5;
7. 9 ( 3) 6;
x x
x x
x x
x x
x x
− = − −
− = − −
− + + + + =
+ + − =
− = − +
3
3
4 4
2 2
8. 24 12 6;
9. 7 1;
10. 5 1 2;
11. 3 3 3 6 3;
12. 1 8 ( 1)(8 ) 3;
x x
x x
x x
x x x x
x x x x
+ + − =
+ − =
− = − =
− + + − + =
+ + − + + − =

3 3
3 3
2 3 3 2
2 23 3 3
(34 ) 1 ( 1) 34
13. 30;
34 1
14. 1 2 (1 ) 1;
15. 1 1 (1 ) 1 2 1 ;
16. 2 2 4;
x x x x
x x
x x x x
x x x x
x x x x
− + − + −
=
− − +
+ − − − = −
 + − − − + = + −
 
+ + + − − =
2 3 3 244 4 4
17. (1 ) (1 ) 1 (1 );x x x x x x x x+ − + − = − + + −
3 3
3 3
7 5
18. 6 ;
7 5
x x
x
x x
− − −
= −
− + −
2 2
3 3
sin cos
2 23 3 3
2 2
2 24 4
19. 7 2 3;
20. 81 81 30;
21. sin cos 4;
22. sin 2 sin sin 2 sin 3;
23. 10 8sin 8 s 1 1;
x x
tgx tgx
x x
x x x x
x co x
+ + − =
+ =
+ =
+ − + − =
+ − − =
4 4
1 1
24. cos2 cos2 2;
2 2
x x− + + =
3 3
3 3
3 3
4 4
3 3
25. 5 7 5 12 1;
26. 24 5 1;
27. 47 2 35 2 4;
28. 47 10 5;
29. 12 14 2;
x x
x x
x x
x x
x x
+ − − =
+ − + =
− + + =
− + + =
− + − =
3 3
4 4
30. 1 7 2;
31. 97 15 4;
x x
x x
+ + − =
− + − =
--------------------------------------------------------------------------
4.Ph−¬ng ph¸p Nh©n liªn hîp
D¹ng : Gi¶i ph−¬ng tr×nh:
( ) ( ) ( )xhCxgBxfA .=−
Víi ( ) ( ) ( )xhDxgBxfA .22
=−
Nh©n hai vÕ víi biÓu thøc: ( ) ( )xgBxfA +
Ta ®−îc ph−¬ng tr×nh ( ) ( ) ( ) ( )( )xgBxfAxhCxhD += ..
Nhãm nh©n tö chung v gi¶i hai ph−¬ng tr×nh:

( )
( ) ( )( )



=+
=
DxgBxfAC
xh 0
)1(
5
3
2314.1
+
=−−+
x
xx
(§H B−u ChÝnh-2001)
)2(62)22(3.2 ++=−+ xxx (§H Qu©n Sù -2001)
Gi¶i1: )1(
5
3
2314.1
+
=−−+
x
xx
§iÒu kiÖn:
3
2
≥x Nh©n hai vÕ víi biÓu thøc liªn hîp:
2314 −++ xx , Ph−¬ng tr×nh ® cho trë th nh:
( )
2
)(342
2
0684344
7
26
3
2
3
2
72623142
3
2
52314
3
2
2314
5
3
3
2
=⇔


=
=
⇔





=+−
≤≤
⇔
≥∧−=−+⇔
≥∧=−++⇔
≥∧−++
+
=+
x
Lx
x
xx
x
xxxx
xxx
xxx
x
x
Gi¶i2:
)2(62)22(3.2 ++=−+ xxx
§iÒu kiÖn: 2≥x ; Ph−¬ng tr×nh ® cho t−¬ng ®−¬ng víi:
62623 −=+−− xxx
Nh©n hai vÕ víi biÓu thøc liªn hîp 623 ++− xx
L m t−¬ng tù nh− phÇn 1) ta ®−îc tËp nghiÖm:





 −
=
2
5311
;3T
xxx ≥−−+ 11 (§H Ngo¹i th−¬ng HCM-2001).
Gi¶i1:
Nh©n hai vÕ víi biÓu thøc liªn hîp xx −++ 11 th× bÊt ph−¬ng
tr×nh ® cho t−¬ng ®−¬ng víi:











−++>
≤<



−++<
≤≤−
⇔



≥−−+−
≤≤−
⇔





≥
−++
≤≤−
xx
x
xx
x
xxx
x
x
xx
x
x
112
10
112
01
0)112(
11
11
2
11
10
10
0
01
≤≤⇔










∀
≤<



=
≤≤−
⇔ x
x
x
x
x
B i l m thªm: (Nh©n liªn hîp)
2 2 2 2
1. 1 4 9 0;
3
2. 4 1 3 2 ;
5
3. 3(2 2) 2 6;
4. 3 7 3 2 3 5 1 3 4;
5. 21 21 21;
6. 21 21 ;
2 2
7. 2 2;
2 2 2 2
8. 2 1 2 2
x x x x
x
x x
x x x
x x x x x x x
x x
x x x
x x
x x
x x x
− + − + + + =
+
+ − − =
+ − = + +
− + − − = − − − − +
+ + − =
+ − − =
+ −
+ =
+ + − +
− − + = −
--------------------------------------------------------------------------
5.Ph−¬ng ph¸p Ph©n chia miÒn x¸c ®Þnh5.Ph−¬ng ph¸p Ph©n chia miÒn x¸c ®Þnh5.Ph−¬ng ph¸p Ph©n chia miÒn x¸c ®Þnh5.Ph−¬ng ph¸p Ph©n chia miÒn x¸c ®Þnh
( ) ( ) ( ) ( ) ( )xfxhxfBxgxfA =+
XÐt ba tr−êng hîp :
Tr−êng hîp 1: ( ) ( )tmxf 0=
Tr−êng hîp 2: ( ) 0>xf Khi ®ã ph¶i cã
( )
( )


≥
≥
0
0
xh
xg
Ph−¬ng tr×nh ® cho trë th nh ( ) ( ) ( )xfxhBxgA =+ (Ph−¬ng tr×nh
c¬ b¶n)
Tr−êng hîp 3: ( ) 0<xf Khi ®ã ph¶i cã
( )
( )


≤
≤
0
0
xh
xg

Ph−¬ng tr×nh ® cho trë th nh ( ) ( ) ( )xfxhBxgA −=−+−
(Ph−¬ng tr×nh c¬ b¶n)
B i 5.1: Gi¶i ph−¬ng tr×nh sau
)1(221682.1 22
+=−+++ xxxx
(§H B¸ch khoa H Néi -2001).
Gi¶i1: 2 2
1. 2x 8x 6 x 1 2x 2 (1)+ + + − = ++ + + − = ++ + + − = ++ + + − = +
§iÒu kiÖn :



−=
≥
⇔





≥+
≥−
≥++
1
1
022
01
0682
2
2
x
x
x
x
xx
NhËn thÊy x=-1 l mét nghiÖm cña ph−¬ng tr×nh ® cho
Víi 1≥x : Ph−¬ng tr×nh t−¬ng ®−¬ng víi:
1
16422
1
121)3(2
1
)1(2)1)(1()3)(1(2
1
2
=⇔




−=−+
≥
⇔




+=−++
≥
⇔




+=+−+++
≥
⇔
x
xxx
x
xxx
x
xxxxx
x
VËy ph−¬ng tr×nh ® cho cã hai nghiÖm l x=1 v x=-1
113234.1 22
−≥+−−+− xxxxx (§H KÕ to¸n H Néi -2001)
4523423.2 222
+−≥+−++− xxxxxx (§H Y HCM -2001)
Gi¶i1: 113234.1 22
−≥+−−+− xxxxx
§iÒu kiÖn:







≤
≥
=
⇔



≥−−
≥−−
2
1
3
1
0)12)(1(
0)3)(1(
x
x
x
xx
xx
NhËn thÊy x=1 l mét nghiÖm cña bÊt ph−¬ng tr×nh
Víi 3≥x Ta t¸ch c¨n cña bÊt ph−¬ng tr×nh ® cho v ®−îc



−≥−−−
≥
1123
3
xxx
x
HÖ n y v« nghiÖm v× 13 −<− xx

Víi
2
1
≤x Ta t¸ch c¨n cña bÊt ph−¬ng tr×nh ® cho v ®−îc
2
1
3)1)(3(2
2
1
1213
2
1
≤⇔





−≥−−
≤
⇔





−−≥−−−
≤
x
xx
x
xxx
x
KÕt luËn: TËp nghiÖm {} 





∞−∪
2
1
;1
Gi¶i2: 4523423.2 222
+−≥+−++− xxxxxx
§iÒu kiÖn: 


≤
≥
4
1
x
x
NhËn thÊy x=1 l mét nghiÖm cña bÊt ph−¬ng tr×nh
Víi 4≥x Ta t¸ch c¨n cña bÊt ph−¬ng tr×nh ® cho v ®−îc bpt
4232 −≥−+− xxx
BPT tho¶ m n víi 4≥x v×: 432 −>−>− xxx
Víi 1≤x Ta t¸ch c¨n cña bÊt ph−¬ng tr×nh ® cho v ®−îc bpt
xxx −≥−+− 4232
BPT v« nghiÖm v× xxx −<−<− 432
KÕt luËn: TËp nghiÖm {} [ )+∞∪ ;41
B i tËp l m thªm:
B i 3: (PP ph©n chia MX§)
2
2
2
2 2
1. 1 1 1;
2. ( 3) (2 1);
3. ( 1)(2 7) 3( 1)( 6) ( 1)(7 1);
4. ( 1) ( 2) 2
5. 2 5 2 2) 3 6;
x x x
x x x x x
x x x x x x
x x x x x
x x x x x
− − + = +
+ − = −
− + + − − = − +
− + + =
+ + − + − = +
2
2 2
2
2 2
6. 1 1;
7. 2 8 6 1 2 2;
8. 4 1 4 1 1
9.( 3) 10 12
x x
x x x x
x x
x x x x
− = +
+ + + − = +
− + − =
+ − = − −
6.Ph−¬ng ph¸p Khai c¨n

( )( ) ( )( ) ( )xgBAxfAxf .
22
=−++
Khai c¨n v lÊy ®Êu gi¸ trÞ tuyÖt ®èi ta ®−îc ph−¬ng tr×nh
( ) ( ) ( )xgBAxgAxf .=−++
Ph¸ dÊu gi¸ trÞ tuyÖt ®èi b»ng c¸ch ph©n chia miÒn x¸c ®Þnh ta ®−îc mét tuyÓn
hai hÖ
( )
( ) ( )
( )
( )











=
≤




=
≥
xgBA
Axf
xgBxf
Axf
.2
.2
Gi¶i hai hÖ n y ta sÏ t×m ®−îc nghiÖm cña ph−¬ng tr×nh
® cho.
294444.1 2
+−=−−+−+ xxxxxx
2
5
2122122
+
=++−++++
x
xxxx
Gi¶i 1:
294444.1 2
+−=−−+−+ xxxxxx
2492424 2
+−=−−++−⇔ xxxx
NÕu 8≥x pt trë th nh:
( )( )
( )
42
4
2
54
1
45442
42094224942 22
−
+
−−
=⇔
+−−=−⇔
++−=−⇔+−=−
x
xx
xxx
xxxxxx
V× 8≥x Nªn
( ) 3
42
4
2
54
≥
−
+
−−
x
xx
vËy ph−¬ng tr×nh n y v« nghiÖm
NÕu 84 <≤ x pt trë th nh:
542494 2
=∨=⇔+−= xxxx
VËy pt ® cho cã nghiÖm l x=4 v x=5.
Gi¶i 2:
2
5
2122122
+
=++−++++
x
xxxx

2
5
1111
+
=−++++⇔
x
xx
Gi¶i t−¬ng tù ta ®−îc nghiÖm l x=-1 v x=3.
21212 =−−−−+ xxxx
Gi¶i:
21111 =−−−+−⇔ xx
( ) 2
2
1111
21
21111
2
≥⇔







=



−−+−
<≤



=−−−+−
≥
⇔ x
xx
x
xx
x
TËp nghiÖm:[ )+∞;2
7.Ph−¬ng ph¸p §¹o hµm
D¹ng : B i to¸n t×m m ®Ó ph−¬ng tr×nh f(x)=m cã nghiÖm,
B i to¸n chøng minh ph−¬ng tr×nh f(x)=A cã nghiÖm duy nhÊt,
B i to¸n biÖn luËn sè nghiÖm cña ph−¬ng tr×nh f(x)=m theo tham sè m.
* T×m tËp x¸c ®Þnh D cña h m sè y=f(x)
* TÝnh ®¹o h m f’
(x) ,lËp b¶ng biÕn thiªn .
* Dùa v o b¶ng biÕn thiªn ®Ó biÖn luËn sè nghiÖm cña ph−¬ng tr×nh .
B i 7.1:T×m m ®Ó ph−¬ng tr×nh sau cã nghiÖm
)45(12 xxmxxx −+−=++
Gi¶i:
Nh©n hai vÕ víi biÓu thøc liªn hîp: xx −−− 45 ta ®−îc:
mxxxxx =−−−++ )45)(12(
XÐt )(xfVT = TX§ [ ]4;0=D
12)( ++= xxxxg ; 0
122
1
2
3
)( >
+
+=′
x
x
xg
)(xg⇒ ®ång biÕn v lu«n d−¬ng trªn D.
xxxh −−−= 45)( ; 0
452
45
)( >
−−
−−−
=′
xx
xx
xh
( )xh⇒ ®ång biÕn v lu«n d−¬ng trªn D.
Suy ra h m sè )()()( xhxgxf = còng sÏ l h m sè ®ång biÕn trªn D.
Tõ ®ã ( ) 44512)4()0( ≤≤−⇔≤≤ VTfVTf

VËy ®Ó ph−¬ng tr×nh ® cho cã nghiÖm th×:
( ) 44512 ≤≤− m
8.Ph−¬ng ph¸p ®¸nh gi¸ hai vÕ
Ph−¬ng ph¸p:
Sö dông bÊt ®¼ng thøc ®Ó chøng minh VPVTVPVT ≤∨≥ v t×m ®iÒu kiÖn ®Ó dÊu
b»ng x¶y ra
2152.1 2
=−++− xxx
11414.2 2
=−+− xx (§HQG H Néi-2001)
Gi¶i1: )1(2152.1 2
=−++− xxx
§iÒu kiÖn: 1
01
0522
≥⇔



≥−
≥+−
x
x
xx
Ta cã: ( ) xxxx ∀≥+−=+− 44152
22
VPxxxVT =≥−++−=⇒ 21522
DÊu b»ng x¶y ra khi x=1.
VËy pt ® cho cã nghiÖm duy nhÊt x=1
Gi¶i 2: 11414.2 2
=−+− xx
§iÒu kiÖn:
2
1
2
1
4
1
≥⇔






≥
≥
x
x
x
VËy VPxxVT =≥−+−= 11414 2
DÊu b»ng x¶y ra khi
2
1
014
114
2
=⇔



=−
=−
x
x
x
VËy pt ® cho cã nghiÖm: 2
1=x
xxxxxxx 32 +++=++
Gi¶i:
§iÒu kiÖn: 0≥x
NhËn thÊy x=0 l mét nghiÖm cña ph−¬ng tr×nh

Víi x>0
xxxxxxx
xxxx
xxx
32
32
+++<++⇒




+<+
+<
DÊu b»ng kh«ng x¶y ra nªn pt v« nghiÖm víi x>0
KÕt luËn:nghiÖm x=0
0321 333
=+++++ xxx
Gi¶i:
NhËn thÊy x=-2 l mét nghiÖm
Víi x>-2 th× x+1>-1
0
13
02
11
3
3
3
>⇒






>+
>+
−>+
⇒ VT
x
x
x
DÊu b»ng kh«ng x¶y ra nªn pt v« nghiÖm víi x>-2
T−¬ng tù víi x<-2
0
13
02
11
3
3
3
<⇒






<+
<+
−<+
⇒ VT
x
x
x
DÊu b»ng kh«ng x¶y ra nªn pt v« nghiÖm víi x<-2
KÕt luËn : nghiÖm x=0
B i tËp l m thªm : C¨n bËc ba.
3 3 3
3 3 3
3 3 3
3 3
3 3
1. 1 2 2 3;
2. 5 6 2 11;
3. 1 3 1 1;
4. 1 1 2
5. 2 1 2 1 2;
x x x
x x x
x x x
x x
x x x x
− + − = −
+ + + = +
+ + + = −
+ + − =
+ − + − − =
2
3 2 2
2
2
1. 2 5 1 2;
2. 2 7 11 25 12 6 1;
1 1
3. 2 2 4 ;
4. 2 1 3 4 1 1;
x x x
x x x x x
x x
x x
x x x x
− + + − =
− + − = + −
 
− + − = − + 
 
− − + + − − =

( )
2 2
3 2 2
5. 1 1 2;
6. 1 2 2 1 2 2 1;
7. 2 2 1 2 1 3;
8. 2 5 3 3 2 6 1;
x x x x
x x x x
x x x x x
x x x x x
− − + + − =
− + − − − − − =
+ − − − − = +
+ + − = + −
2
6
9. 2 1 19 2
10 24
x x
x x
− + − =
− + −
2 2 3 3 4 43 3 4 4
10. 1 1 1 1 1 1 6;x x x x x x+ + − + + + − + + + − =
4 4 4
11. 1 1 2 8;x x x x+ − + + − = +
4 24
2 4 4 34
2 44 4
12. 2 3 4;
13. 2 1;
14. 2 2 4;
5
15. 2 2 1 2 2 1 ;
2
x x x
x x x x
x x x x
x
x x x x
− = − +
− = − +
+ + − + − =
+
+ + + + + − + =
16. 3 4 1 15 8 1 6;
17. 6 9 6 9 6;
18. 5 4 1 2 2 1 1;
19. 2 2 2 1 2 2 3 4 2 1 3 2 8 6 2 1 4;
x x x x
x x x x
x x x x
x x x x x x
+ + − + + − − =
+ − + − − =
+ − + + + − + =
− − − + − − + + − − =
9.Ph−¬ng ph¸p Tam thøc bËc hai
D¹ng : B i to¸n biÖn luËn sè nghiÖm cña ph−¬ng tr×nh f(x)=m theo tham sè m.
Trong ®ã ta ®Æt ®−îc: ( ) ( )0≥= ttxu ;
B i to¸n khi ®ã trë th nh :BiÖn luËn theo m sè nghiÖm cña ph−¬ng tr×nh bËc
hai
02
=++ cbtat
B¶y b i to¸n so s¸nh nghiÖm cña tam thøc bËc hai víi mét sè, hai sè:
21
21
21
,3
,2
,1
xx
xx
xx
<<
<<
<<
α
α
α
βα
βα
βα
βα
βα
<<<



<<<
<<<
<<<
<<<
21
21
21
21
21
,7
,6
,5
,4
xx
xx
xx
xx
xx
Ba b i to¸n c¬ b¶n cña tam thøc bËc hai:
1, T×m ®iÒu kiÖn ®Ó f(x)>0 víi mäi x thuéc R

2, T×m ®iÒu kiÖn ®Ó f(x)>0 víi mäi x thuéc kho¶ng (α;+∞);
3, T×m ®iÒu kiÖn ®Ó f(x)>0 víi mäi x thuéc kho¶ng (α;β);
--------------------------------------------------------------------------
( )( ) 01562
=−−++− xxmxx (C§ SP HCM-2001).
--------------------------------------------------------------------------
Gi¶i: §iÒu kiÖn: 51 ≤≤ x
§Æt ( )( ) ( ) 2043415
22
≤≤⇒≤−−=⇒=−− txttxx
B i to¸n ® cho trë th nh:
T×m m ®Ó ph−¬ng tr×nh t2
-t+5-m=0
cã nghiÖm [ ]2;0∈t ,nghÜa l





<≤<
≤≤≤
≤≤≤
20
20
20
21
21
21
tt
tt
tt
HÖ ®iÒu kiÖn trªn t−¬ng ®−¬ng víi:
( ) ( )
( )
( )

















<<
>
>
≥∆
≤
2
2
0
02
00
0
02.0
s
f
f
ff
( )( )
7
4
19
2
2
1
0
7
5
4
19
075
≤≤⇔




















<<
<
<
≥
≤−−
⇔ m
m
m
m
mm
--------------------------------------------------------------------------
mxxxx ++−=−+ 99 2
(C§ Y HCM-1997).
--------------------------------------------------------------------------
Gi¶i: §iÒu kiÖn: 90 ≤≤ x
§Æt : ( ) ( )
4
81
2
9
4
1
09
2
2
≤





−−=⇒≥=− xtttxx
2
9
0 ≤≤⇒ t
B i to¸n ® cho trë th nh:
T×m m ®Ó ph−¬ng tr×nh t2
-2t+m-9=0

cã nghiÖm 



∈
2
9
;0t ,nghÜa l









<≤<
≤≤≤
≤≤≤
2
9
0
2
9
0
2
9
0
21
21
21
tt
tt
tt
HÖ ®iÒu kiÖn trªn t−¬ng ®−¬ng
víi:
( )
( )
( )
10
4
9
109
9
4
9
0
4
9
09
010
0
4
9
9
2
2
0
0
2
9
00
0
0
2
9
.0
'
≤≤−⇔




<<
≤≤−
⇔

















>+
>−
≥+−
<





+−
⇔





















<<
>





>
≥∆ ′
≤





m
m
m
m
m
m
mm
s
f
f
ff
10.HÖ ph−¬ng tr×nh
HÖ ®èi xøng lo¹i 1:
L hÖ ph−¬ng tr×nh m khi thay ®æi vai trß cña x v y th× mçi ph−¬ng tr×nh cña hÖ
kh«ng thay ®æi.
C¸ch gi¶i: + §Æt ( )PS
Pxy
Syx
42
≥



=
=+
+ Gi¶i hÖ víi hai Èn S,P
+ Thö ®k v lÊy x,y l hai nghiÖm pt X2
-SX+P=0
B i 10.1:
Gi¶i hÖ:





=+
+=+
78
1
7
xyyxyx
xyx
y
y
x
(§H H ng H¶i 1999).
Gi¶i:HÖ ® cho t−¬ng ®−¬ng víi: ( )




=+
=−+
>
78
7
0,
xyyx
xyyx
yx

§Æt



>
>




=
+=
0
0
;
v
u
xyv
yxu
HÖ ® cho trë th nh



=
=
⇔



=
=−
6
13
78
7
v
u
uv
vu
Gi¶i ra ta ®−îc 2 nghiÖm ( ) ( )4;9;9;4
HÖ ®èi xøng lo¹i 2:
- L hÖ ph−¬ng tr×nh m khi thay ®æi vai trß cña x v y th× hai ph−¬ng tr×nh cña hÖ
®æi chç cho nhau.
C¸ch gi¶i: -Trõ vÕ víi vÕ cña hai ph−¬ng tr×nh ®Ó ®−îc mét ph−¬ng tr×nh cã d¹ng
tÝch.
- HÖ ® cho sÏ t−¬ng ®−¬ng víi tuyÓn hai hÖ ph−¬ng tr×nh.
- Gi¶i hai hÖ n y ®Ó t×m nghiÖm x v y.
B i 10.2: Cho hÖ:




=−++
=−++
mxy
myx
21
21
1,Gi¶i hÖ khi m=9;
2,T×m m ®Ó hÖ cã nghiÖm (§H SP HCM 2001).
Gi¶i:
§iÒu kiÖn: 0;2;1 ≥≥≥ myx
B×nh ph−¬ng hai vÕ ta ®−îc hÖ:
( )( )
( )( )



=−++−+
=−++−+
mxyyx
myxyx
211
211
Trõ vÕ víi vÕ cña hai ph−¬ng tr×nh trªn ta ®−îc hÖ:
( )( ) ( )( )
( )( ) ( )( )



−+=−+
=
⇔




=−++−+
−+=−+
xmxx
yx
mxyyx
xyyx
21212211
2121
1, Víi m=9 ta cã hÖ:
( )( )
( )( ) ( )
3
521
5
521 2
==⇔





−=−+
=
≤
⇔




−=−+
=
yx
xxx
yx
x
xxx
yx
2,T×m m ®Ó hÖ cã nghiÖm :
HÖ ( )( )









++
=
+
≤=≤
≥
⇔




−+=−+
=
m
mm
x
m
yx
m
xmxx
yx
4
82
2
1
2
0
21212
2

§iÒu kiÖn mmmmm
m
x 22928
2
1
2 22
+≤++≤⇔
+
≤≤
( ) 3
3
03
09
096 2
2
2
≥⇔



≥
≥−
⇔




≥−
≥+−
⇔ m
m
m
m
mm
KÕt luËn: 3≥m .
11.Ph−¬ng ph¸p ®Æc biÖt
1.Ph−¬ng tr×nh chøa c¨n bËc hai vµ luü thõa bËc hai
B i to¸n tæng qu¸t:
Gi¶i ph−¬ng tr×nh: ( ) ( )Iedxvuxrbax +++=+
2
Víi a ≠ 0, u ≠ 0 , r ≠ 0 ;
Ph−¬ng ph¸p gi¶i:
§iÒu kiÖn dÓ ph−¬ng tr×nh cã nghÜa: 0≥+ bax
§Æt Èn phô : ( )1)( 2
baxvuybaxvuy +=+⇔+=+
Víi ®iÒu kiÖn 0≥+ vuy
Lóc ®ã (I) trë th nh : evdxuyvuyr −+−=+ 2
)(
Gi¶ sö çc ®iÒu kiÖn sau ®−îc tho¶ m n: u=ar +d v v=br+e
Lóc ®ã ph−¬ng tr×nh ® cho trë th nh hÖ
( )
( ) ( )



+−+=+
+=+
brxuaruyvuxr
brarxvuyr
2
2
Gi¶i hÖ trªn b»ng çch trõ vÕ víi vÕ cña hai ph−¬ng tr×nh , ®−îc mét tuyÓn hai hÖ
ph−¬ng tr×nh trong ®ã cã mét nghiÖm x=y
--------------------------------------------------------------------------
B i 11.1:
Gi¶i ph−¬ng tr×nh: ( )1203232152 2
−+=+ xxx
(T¹p chÝ To¸n Häc Tuæi TrÎ – Sè 303)
--------------------------------------------------------------------------
Lêi gi¶i: §iÒu kiÖn 0152 ≥+x
BiÕn ®æi ph−¬ng tr×nh (1) th nh: ( ) 28242152
2
−+=+ xx
§Æt Èn phô : ( )024152)24(15224 2
≥++=+⇔+=+ yxyxy .
Ph−¬ng tr×nh (1) trë th nh : 152)24( 2
+=+ yx
VËy ta cã hÖ:




+=+
+=+
152)24(
152)24(
2
2
xy
yx
HÖ n y l hÖ ®èi xøng lo¹i hai
Gi¶i hÖ trªn b»ng çch trõ vÕ víi vÕ cña hai ph−¬ng tr×nh ,
Ta ®−îc 2 nghiÖm l
16
2219
2
1
21
−−
=∧= xx
2.Ph−¬ng tr×nh chøa c¨n bËc ba vµ luü thõa bËc ba

B i to¸n tæng qu¸t:
Gi¶i ph−¬ng tr×nh: ( ) ( )IIedxvuxrbax +++=+
33
Víi a ≠ 0, u ≠ 0 ,
r ≠ 0 ;
Ph−¬ng ph¸p gi¶i:
§Æt Èn phô : ( )1)( 33
baxvuybaxvuy +=+⇔+=+
Lóc ®ã (II) trë th nh : evdxuyvuxr −+−=+ 3
)(
Gi¶ sö çc ®iÒu kiÖn sau ®−îc tho¶ m n: u=ar +d v v=br+e
( )
( ) ( )



+−+=+
+=+
brxuaruyvuxr
brarxvuyr
3
3
Gi¶i hÖ trªn b»ng çch trõ vÕ víi vÕ cña hai ph−¬ng tr×nh , ®−îc mét tuyÓn hai
hÖ ph−¬ng tr×nh trong ®ã cã mét nghiÖm x=y.
B i 11.2:
Gi¶i ph−¬ng tr×nh: ( )2255336853 233
−+−=− xxxx
(T¹p chÝ To¸n Häc Tuæi TrÎ – Sè 303)
--------------------------------------------------------------------------
Lêi gi¶i: ( ) ( ) ( )2232532
33
+−−=−⇔ xxxPT
§Æt Èn phô : ( ) 53325332
33
−=−⇔−=− xyxy
Lóc ®ã (2) trë th nh ( ) 5232
3
−+=− xyx
( )
( )



−=−
−+=−
5332
5232
3
3
xy
xyx
Gi¶i hÖ trªn b»ng çch trõ vÕ víi vÕ cña hai ph−¬ng tr×nh ,
Ta ®−îc 3 nghiÖm:
4
35
;
4
35
;2 321
−
=
+
== xxx
3 3
1. 1 2 2 1;x x+ = −
( )3 33 3
2. 35 35 30;x x x x− + − =
3 3
2
2
2 2
3. 3 3 2 2;
4. 1 1;
5. 5 5;
6. 5 (5 ) ;
7. 3 3 ;
x x
x x
x x
x x
x x
− + =
+ + =
+ + =
= − −
+ + =

2 2
8. ( ) ;x a b a bx= − −
3.Sö dông tÝnh chÊt vÐc t¬:
baba
ϖϖϖϖ
+≤+
DÊu b»ng x¶y ra khi hai vÐc t¬ a
ϖ
v b
ϖ
cïng h−íng , t−¬ng ®−¬ng víi:
( )0>= kbka
ϖϖ
;
D¹ng :Gi¶i ph−¬ng tr×nh
( ) ( ) ( ) ( )222222
BAxhBxgAxf ++=+++
Víi
( ) ( ) ( )



=+
=+
CBA
xhxgxf
DÆt :
( )( )
( )( )
( ) ( )( ) ( )( )BAxhBAxgxfba
Bxgb
Axfa
+=++=+⇒



=
=
;;
;
; ϖϖ
ϖ
ϖ
;
Ph−¬ng tr×nh ® cho trë th nh baba
ϖϖϖϖ
+=+ DÊu ®¼ng thøc x¶y ra khi v chØ khi
hai vÐc t¬ a
ϖ
v b
ϖ
cïng h−íng , t−¬ng ®−¬ng víi: ( )0>= kbka
ϖϖ
;
--------------------------------------------------------------------------
B i 11.3: Gi¶i ph−¬ng tr×nh:
2003267108168 22
=++++− xxxx
(TuyÓn tËp ®Ò thi Olimpic 30-4 -2003 )
--------------------------------------------------------------------------
Gi¶i:
§Æt
( )
( )
( )231;9
211;5
220;4
=+⇒




+=
−=
ba
xb
xa ϖϖ
ϖ
ϖ
VËy ta cã:
2003
;26710;8168 22
=+
++=+−=
ba
xxbxxa
ϖϖ
ϖϖ
Ph−¬ng tr×nh ® cho trë th nh baba
ϖϖϖϖ
+=+
DÊu ®¼ng thøc x¶y ra khi v chØ khi hai vÐc t¬ a
ϖ
v b
ϖ
cïng h−íng , t−¬ng
®−¬ng víi: ( )0>= kbka
ϖϖ
; Gi¶i ra ®−îc
31
56−
=x
----------------------------------------------------
a
ρ b
ϖ
ba
ϖϖ
+

3.Sö dông phÐp ®Æt l−îng gi¸c:
D¹ng 1: B i to¸n cã chøa 2
1 x− .
Ph−¬ng ph¸p gi¶i : §iÒu kiÖn 1≤x .Dùa v o ®iÒu kiÖn n y ta ®Æt x=sint
víi 




 ΠΠ
−∈
2
;
2
t ; hoÆc x=cost víi [ ]Π∈ ;0t ; v gi¶i ph−¬ng tr×nh l−îng gi¸c.
D¹ng 2: B i to¸n cã chøa 12
−x .
Ph−¬ng ph¸p gi¶i : §iÒu kiÖn 1≥x .Dùa v o ®iÒu kiÖn n y ta ®Æt t
x
sin
1
=
víi 




 ΠΠ
−∈
2
;
2
t ; hoÆc t
x
cos
1
= víi [ ]Π∈ ;0t ; v gi¶i ph−¬ng tr×nh l−îng
gi¸c.
--------------------------------------------------------------------------
B i 11.4: Gi¶i ph−¬ng tr×nh: :
xxx 341 22
−=− ;
--------------------------------------------------------------------------
Gi¶i: §iÒu kiÖn 1≤x .Dùa v o ®iÒu kiÖn n y ta ®Æt x=cost víi [ ]Π∈ ;0t ; v gi¶i
ph−¬ng tr×nh l−îng gi¸c:
( )






Π
+
Π
−=
Π
+
Π
=
⇔





−
Π
=⇔
>=⇔=−
24
28
2
cos3cos
0sinsin3cossincos3cos4 23
kt
kt
tt
tttttt
Do [ ]Π∈ ;0t nªn ta chän:










+−
=
+
=
−
=
⇔









Π
=
Π
=
Π
=
4
22
4
22
2
2
8
5
8
4
3
x
x
x
t
t
t
--------------------------------------------------------------------------
B i 11.5: Gi¶i ph−¬ng tr×nh: :
12
35
1 2
>
−
+
x
x
x ;

--------------------------------------------------------------------------
Gi¶i : §iÒu kiÖn 1>x .V× vÕ tr¸i lu«n d−¬ng nªn yªu cÇu x > 0 , do ®ã x>1
Dùa v o ®iÒu kiÖn n y ta ®Æt :
t
x
cos
1
= víi 




 Π
∈
2
;0t ; v gi¶i bÊt ph−¬ng tr×nh l−îng gi¸c
( )
( )
( )
( )






<<
>
⇔






<<
<<
⇔






<<
<<
⇔
<−<⇔<<⇔<<⇔
=<−−⇔
>+⇔
>+⇔>+
4
5
1
3
5
1cos
5
4
5
3
cos0
1cos
25
16
25
9
cos0
625
144
cos1cos0
25
12
cossin0
25
12
0
cossin0144144.21225
cossin1225cossin21144
cossin35cossin12
12
35
sin
1
cos
1
2
2
22
2
22
x
x
t
t
t
t
tttty
ttyyy
tttt
tttt
tt
---------------------------------------------------------------------------

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