Chuyªn ®Ò:
ph¬ng tr×nh,bÊt ph¬ng tr×nh v« tØ,hÖ ph¬ng tr×nh
vµ hÖ bÊt ph¬ng tr×nh
Biªn so¹n :trÞnh xu©n t×nh
PhÇn I: Ph¬ng tr×nh v« tØ
Ph ¬ng ph¸p 1:Ph ¬ng ph¸p gi¶i d¹ng c¬ b¶n:
1/
( ) ( )
f x g x=
( )
( ) ( )
2
g x 0
f x g x
=
2/
( ) ( ) ( )
f x g x h x+ =
B×nh ph¬ng hai vÕ
1-(§HQGHN KD-1997)
16x 17 8x 23+ = −
2-(§H C¶nh s¸t -1999)
2 2
x x 11 31+ + =
3-(HVNHHCM-1999)
2
x 4x 2 2x
− + + =
4-(§H Th¬ng m¹i-1999) Gi¶i vµ biÖn luËn pt:
2
m x 3x 2 x− − + =
5-(§HC§ KB-2006) T×m m ®Ó pt sau cã hai nghiÖm thùc ph©n biÖt:
2
x mx 2 2x 1+ + = +
6-(§GKTQD-2000)
7-(§HSP 2 HN)
( ) ( )
2
x x 1 x x 2 2 x− + + =
8-(HVHCQ-1999)
x 3 2x 1 3x 2+ − − = −
9-(HVNH-1998)
3x 4 2x 1 x 3+ − + = +
10-(§H Ngo¹i th¬ng-1999)
2 2
3 x x 2 x x 1− + − + − =
Ph ¬ng ph¸p 2: ph ¬ng ph¸p ®Æt Èn phô:
I-§Æt Èn phô ® a pt vÒ pt theo Çn phô:
D¹ng 1: Pt d¹ng:
2 2
ax bx c px qx r+ + = + +
trong ®ã
a b
p q
=
C¸ch gi¶i: §Æt
2
t px qx r= + +
§K
t 0
1-(§H Ngo¹i th¬ng-2000)
( ) ( )
2
x 5 2 x 3 x 3x+ − = +
2-(§H Ngo¹i ng÷ -1998)
( ) ( )
2
x 4 x 1 3 x 5x 2 6+ + − + + =
1
3-(§H CÇn th¬-1999)
2
(x 1)(2 x) 1 2x 2x+ − = + −
4-
2 2
4x 10x 9 5 2x 5x 3+ + = + +
5-
3
2 2
18x 18x 5 3 9x 9x 2− + = − +
6-
2 2
3x 21x 18 2 x 7x 7 2+ + + + + =
D¹ng 2: Pt D¹ng:
P(x) Q(x) P(x).Q(x) 0α +β + γ =
( )
0
αβγ
C¸ch gi¶i: * NÕu
( )
P x 0
=
( )
( )
P x 0
pt Q x 0
=
� � =
* NÕu
( )
P x 0
chia hai vÕ cho
( )
P x
sau ®ã ®Æt
( )
( )
Q x
tP x
=
t 0
1-(§HC§ KA-2007) T×m m ®Ó pt sau cã nghiÖm:
42
3 x 1 m x 1 2 x 1− + + = −
2-
( )
2 3
2 x 3x 2 3 x 8− + = +
3-
( )
2 3
2 x 2 5 x 1+ = +
D¹ng 3: Pt D¹ng :
( ) ( )
( )
( ) ( )
( )
( ) ( )
( )
2 2
P x Q x P x Q x
2 P x .Q x 0 0
α + +β
α + γ = α +β
C¸ch gi¶i: §Æt
( ) ( ) ( ) ( ) ( ) ( )
2
t P x Q x t P x Q x 2 P x .Q x= = +�� �
1-(§HQGHN-2000)
2
2
1 x x x 1 x
3
+ − = + −
2-(HVKTQS-1999)
2
3x 2 x 1 4x 9 2 3x 5x 2− + − = − + − +
3-(Bé quèc phßng-2002)
2
2x 3 x 1 3x 2 2x 5x 3 16+ + + = + + + −
4-
2
4x 3 2x 1 6x 8x 10x 3 16+ + + = + + + −
5-(C§SPHN-2001)
2
x 2 x 2 2 x 4 2x 2
− − + = − − +
D¹ng 4: Pt D¹ng:
( ) ( )
a cx b cx d a cx b cx n+ + − + + − =
Trong ®ã
a, b,c,d, n
lµ c¸c h»ng sè ,
c 0,d 0>
C¸ch gi¶i: §Æt
( )
t a cx b cx( a b t 2 a b= + + − + +
1-(§H Má-2001)
2 2
x 4 x 2 3x 4 x+ − = + −
2
2-
( ) ( )
3 x 6 x 3 x 6 x 3+ + − − + − =
3-(§HSP Vinh-2000) Cho pt:
( ) ( )
x 1 3 x x 1 3 x m+ + − − + − =
a/ Gi¶i pt khi
m 2=
b/T×m c¸c gt cña m ®Ó pt cã nghiÖm
4-(§HKTQD-1998) Cho pt
1 x 8 x (1 x)(8 x) a+ + − + + − =
a/Gpt khi
a 3=
b/T×m c¸c gt cña a ®Ó pt cã nghiÖm
5-TT §T Y tÕ tphcm-1999) T×m c¸c gt cña m ®Ó pt cã nghiÖm
x 1 3 x (x 1)(3 x) m− + − + − − =
6-(§H Ngo¹i ng÷-2001)
x 1 4 x (x 1)(4 x) 5+ + − + + − =
D¹ng 5: Pt d¹ng:
2 2
x a b 2a x b x a b 2a x b cx m+ − + − + + − − − = +
Trong ®ã
a, b,c,m
lµ h»ng sè
a 0
C¸ch gi¶i : §Æt
t x b= −
§K:
t 0
®a pt vÒ d¹ng:
2
t a t a c(t b) m+ + − = + +
1-(§HSP Vinh-2000)
x 1 2 x 2 x 1 2 x 2 1− + − − − − − =
2-(HV BCVT-2000)
x 2 x 1 x 2 x 1 2+ − − − − =
3-(§HC§ KD-2005)
2 x 2 2 x 1 x 1 4+ + + − + =
4-(§H Thuû s¶n -2001)
x 5
x 2 2 x 1 x 2 2 x 1 2
+
+ + + + + − + =
5-
x 3
x 2 x 1 x 2 x 1 2
+
+ − + − − =
6- XÐt pt:
x m
x 6 x 9 x 6 x 9 6
+
+ − + − − =
a/ Gi¶i pt khi
m 23=
b/ T×m c¸c gt cña m ®Ó pt cã nghiÖm
II-Sö dông Èn phô ® a pt vÒ Èn phô ®ã ,cßn Èn ban ®Çu coi lµ tham sè :
1-
( )
2 2
6x 10x 5 4x 1 6x 6x 5 0− + − − − + =
2-(§H Dîc-1999)
( )
2 2
x 3 10 x x x 12+ − = − −
3
3-(§H Dîc-1997)
( )
2 2
2 1 x x 2x 1 x 2x 1− + − = − −
4-
5-
( )
2 2
2 1 x x x 1 x 3x 1− + + = − −
6-(§HQG-HVNH KA-2001)
2 2
x 3x 1 (x 3) x 1+ + = + +
III-Sö dông Èn phô ® a vÒ hÖ pt:
D¹ng 1: Pt D¹ng:
nn
x a b bx a+ = −
C¸ch gi¶i: §Æt
n
y bx a
= −
khi ®ã ta cã hÖ:
n
n
x by a 0
y bx a 0
− + =
− + =
1-(§HXD-DH HuÕ-1998)
2
x 1 x 1− = +
2-
2
x x 5 5+ + =
3-
2
x 2002 2002x 2001 2001 0− − + =
4- (§H Dîc-1996)
3 3
x 1 2 2x 1+ = −
D¹ng 2: Pt D¹ng:
( )
2
ax b r ux v dx e+ = + + +
trong ®ã
a, u, r 0
Vµ
u ar d, v br e= + = +
C¸ch gi¶i: §Æt
uy v ax b
+ = +
khi ®ã ta cã hÖ:
( )
( )
2
2
uy v r ux v dx e
ax b uy v
+ = + + +
+ = +
1-(§HC§ KD-2006)
2
2x 1 x 3x 1 0− + − + =
2-
2
2x 15 32x 32x 20+ = + −
3-
2
3x 1 4x 13x 5+ = − + −
4-
2
x 5 x 4x 3+ = − −
5-
2
x 2 x 2= − +
6-
2
x 1 3 x x− = + −
D¹ng 3: PT D¹ng:
( ) ( )
n m
a f x b f x c− + + =
C¸ch gi¶i: §Æt
( ) ( )
n m
u a f x , v b f x= − = +
khi ®ã ta cã hÖ:
n m
u v c
u v a b
+ =
+ = +
1-(§HTCKT-2000)
3
2 x 1 x 1− = − −
2-
3 3
x 34 x 3 1+ − − =
3-
3
x 2 x 1 3− + + =
4-
4
4
97 x x 5− + =
5-
4
4
18 x x 1 3− + − =
Ph ¬ng ph¸p 3: Nh©n l îng liªn hîp:
D¹ng 1: Pt D¹ng:
( ) ( )
f x a f x b+ =
4
C¸ch gi¶i: Nh©n lîng liªn hîp cña vÕ tr¸i khi ®ã ta cã hÖ:
( ) ( )
( ) ( )
f x a f x b
f x a f x a b
+ =
+ =
m
1-
2 2
4x 5x 1 4x 5x 7 3+ + + + + =
2-
2 2
3x 5x 1 3x 5x 7 2+ + − + − =
3- 3- (§H Ngo¹i th¬ng-1999 )
2 2
3 x x 2 x x 1− + − + − =
4-(§H Th¬ng m¹i-1998)
2 2
x 3x 3 x 3x 6 3− + + − + =
5-(HVKTQS-2001)
1 1 1
x 4 x 2 x 2 x
+ =
+ + + + +
D¹ng 2: Pt D¹ng:
( ) ( ) ( ) ( )
( )
f x g x m f x g x = −
1-(HVBCVT-2001)
x 3
4x 1 3x 2 5
+
+ − − =
2-(HVKTQS-2001)
3(2 x 2) 2x x 6+ − = + +
Ph ¬ng ph¸p 4:Ph ¬ng ph¸p ®¸nh gi¸:
1-
2
x 2 4 x x 6x 11− + − = − +
2-
2 2 2
x x 1 x x 1 x x 2
+ − + − + = − +
3-(§HQGHN-Ng©n hµng KD-2000)
2
4x 1 4x 1 1− + − =
4-(§H N«ng nghiÖp-1999)
2
x 2x 5 x 1 2− + + − =
Ph ¬ng ph¸p 5:Ph ¬ng ph¸p ®k cÇn vµ ®ñ:
1-T×m m ®Ó pt sau cã nghiÖm duy nhÊt:
x 2 x m+ − =
2- T×m m ®Ó pt sau cã nghiÖm duy nhÊt
x 5 9 x m− + − =
3- T×m m ®Ó pt sau cã nghiÖm duy nhÊt
4 4
x 1 x x 1 x m+ − + + − =
Ph ¬ng ph¸p 6: Ph ¬ng ph¸p hµm sè (Sö dông ®¹o hµm)
1-(§HC§ KB-2004) - T×m m ®Ó pt sau cã nghiÖm :
()
2 2 4 2 2
m 1 x 1 x 2 2 1 x 1 x 1 x+ − − + = − + + − −
2- - T×m m ®Ó c¸c pt sau cã nghiÖm :
1*/
2
4 x mx m 2
− = − +
2*/
x 1 x 1 5 x 18 3x 2m 1+ + − − − − − = +
3--(§HC§ KA-2007) T×m m ®Ó pt sau cã nghiÖm:
42
3 x 1 m x 1 2 x 1− + + = −
4-(§HC§KB-2007) CMR
m 0∀ >
pt sau cã 2nghiÖm pb:
2
x 2x 8 m(x 2)+ − = −
5