một số bt giải pt phổ biến
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7/23/2019 Mt S Bt Gii Pt Ph Bin
1/13
M T S B I TO N GI I PHNG TR NH
PH BIN NHT TRONG THI I HC
CM NANG CHO MA THI
NGUY N H U BI N
https://www.facebook.com/ng.huubienEmail: ng.huubien@gmail.com
(N THI THPT QUC GIA)
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MT S DNG PHNG TRNH THNG XUT HIN TRONG THI I HC
Trang 1NGUYN HU BIN -https://www.facebook.com/groups/nguyenhuubien
Bi 1:Gii phng trnh ( )2
1 1x
2 x x8 2
2 2
1 79 .log x x 2 3 .log 2 x 0
2 4
+
+ + =
Hng dn+ Khi gp phng trnh mhoc log, trc ht ta bin i theo csnhnht (y lcs3), bin i phng trnh cho, ta c:
( )
( )
( )
( )
2
2
2
2
2
1 1x
2 x x8 22 2
12x 1
242 2x x
1
42
2 22x 1 x x
1x x 2x 124
2 2
1x x
4
1 79 .log x x 2 3 .log 2 x 02 4
1 73 .log x x 2 .log 2x 1 0
43
3 1 7.log x x 2 .log 2x 1 0
43 3
73 .log x x 2 3 .log 2x 1 0
4
3 .log
+
+
+
+ + =
+ + =
+ + =
+ + =
2x 122 21 7 7x x 3 .log 2x 14 4 4
(1) + + = +
+ Xt hm s t 27
f (t) 3 .log t4
= +
, ta thy
2
2 1 1x x x 0t 04 2
2x 1 0
+ =
t t
2
7 1f '(t) 3 .ln 3.log t 3 . 0, t 0
4 t.ln 2
= + + >
f(t) l hm ng bin
+ T(1) ( )2 21 1
f x x f 2x 1 x x 2x 14 4
+ = + =
(2)
+ Xt 2 TH: 2x 1 0;2x 1 0 < bdu GTT(2), gii PT (2) ta c S:1 5
x ;2 2
=
Bi 2:Gii phng trnh 3 2 3x x 19x 16 3x x 1+ = + Hng dn+ K: x 1 + Nhn thy biu thc trong cn VP l: ( ) ( )3 2x 1 x 1 x x 1+ = + + nn rt c thy l cscho ta phn tch VT ca phng trnh, tht vy:
( ) ( ) ( )
( )( ) ( ) ( )
3 2 3 2
3 2 2 2
x x 19x 16 x 1 x x 1 18 x 1
x x 19x 16 x 1 x x 1 x x 1 18 x 1
+ = + + + +
+ = + + + + +
+ Phng trnh cho trthnh:
( ) ( ) ( ) ( ) ( ) ( )( )2 2 2x 1 x x 1 x x 1 18 x 1 3 x 1 1 x 1 x x 1 + + + + + = + + + (1)
+ t n ph2
a x 1 0
b x x 1 0
= +
= +
thay vo (1) ta c: ( )2 2 2 2 2a b b 18a 3 a 1 ab+ = (2)
+ Phng trnh (2) mun gii c th chcn cch phn tch a thc thnh nhn t, cngvic phn tch ny khng phi ddng, chng ta sdng mo sau phn tch:
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MT S DNG PHNG TRNH THNG XUT HIN TRONG THI I HC
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- Trc ht ta bin i a (2) vphng trnh bc 2 i vi bin l b:
( ) ( ) ( )2 2 2 2 2 2 2 3 2a b b 18a 3 a 1 ab a 1 b 3a 3a b 18a 0(3)+ = + =
- Phng trnh (3) tnh c ( ) ( )( ) ( )2 2
3 2 2 2 23a 3a 4 a 1 18a ... 9a a 3 = + + = = +
phng trnh (3) c 2 nghim
( )
( )
3 2
2
23 2
2 2
3a 3a 3a a 3b 3a
b 3a 0a 1
a b b 12a 03a 3a 3a a 3 12ab
a 1 a 1
+ + = =
=+
+ + = + = = + +
+ Vy (2) : ( ) ( )( )2 2 2 2 2 2a b b 18a 3 a 1 ab b 3a a b b 12a 0+ = + + =
+ Do 2a b b 12a 0 a b 0+ + = = = (do a, b 0 ) 2x 0
x 1 x x 1x 2
= + = +
=
+ Thli vo PT cho thy x = 0; x = 2 khng l nghim (loi)
+ Vy chcn khnng b 3a= 2x x 1 3 x 1 x 5 33 + = + =
Bi 3:Gii phng trnh
( )( )4 3 2 24x 13x 9x 16 2x 3x x 3 x 1 8 + + + + + =
Hng dn
+ K4 3 24x 12x 9x 16 0
x 1
+ +
+ Ta bin i PT ( ) ( ) ( )2
2 22x 3x 16 2x 3x x 1 4 x 1 8 + + + =
+ t2a 2x 3x
b x 1
=
= thay vo phng trnh ta c:
( ) ( )
( )
( )
2 2
2
2
2
2
22
2 2
2 2
a 16 a b 4 b 88
b 4 ba 16 a
8 a 16 ab 4 b
16
a 16 ab 4 b
2
2 b 4 2b a 16 a
2b 16 2b a 16 a (1)
+ + + =
+ + =
+
+ +
+ + =
+ + + + =
+ + = + +
+ + = + +
+ Xt hm s 22
tf (t) t 16 t, t R f '(t) 1 0, t R f (t)
t 16(*)= + + = + >
+
l hm
ng bin.
Vy t(1) f (2b) f (a) 2b a = =( ) ( )
2
22
2
2x 3x 02 x 1 2x 3x (I)
4 x 1 2x 3x
=
=
+ Cc bn tgii h(I) x 2 =
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MT S DNG PHNG TRNH THNG XUT HIN TRONG THI I HC
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(Ch : Ta gii thch cho f '(t) 0> bng phng php phn chng nhsau:
Gis 2 22 2
t 0f '(t) 0 t t 16 0 t 16 t
t 16 t
< + + < + <
+
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MT S DNG PHNG TRNH THNG XUT HIN TRONG THI I HC
Trang 4NGUYN HU BIN -https://www.facebook.com/groups/nguyenhuubien
( )
( )2
3 3
4 27 9 8(6 2 )( 3)(4 27) 0
4 10 216 4 9 37 9 37
x xx x
xx x
+ + + + + =
+ +
- TH1 3 0 3x x+ = = (TMPT)
- TH 2. 3x
( )2
3 3
36 164 27 0
4 10 216 4 9 37 9 37
x
xx x
+ + =
+ +
( )2
3
36 164 27 0
4 10 212 9 37 2x
xx
+ + =+ +
+ Do 5x nn36 16
4.5 27 012 4
VT + + = . ng thc xy ra 5x =
Vy phng trnh c 2 nghim l 3 v 5
Bi 7:Gii phng trnh ( )24 2 1
log 2 2.8 3.2 1 (*)
2.16 2.4 1
+= +
+
x xx x x
x x.
Hng dn:
( )
2 2
2 2
'
2
log (4 2 1) log (2.16 2.4 1) 2.16 3.4 2
log (4 2 1) 4 2 1 log (2.16 2.4 1) 2.16 2.4 1
1Xt ( ) log , 0, ( ) 1 0, 0 ng bin trn 0;
ln 2
+ + = +
+ + + = + + +
= + > = + > > +
x x x x x x x
x x x x x x x x
f t t t t f t t ft
( )2 4 2
4 2 3
2
* (4 2 1) (2.16 2.4 1)
4 2 1 2.16 2.4 1 2 2 2.2 2.2
2.2 3.2 2 0 2.2 3.2 1 0
1 3 1 3 (2 1)(2 )(2 ) 0
2 2
2 10
1 3 2 1 3
log22
1 32 (loi)
2
+ = +
+ = + =
+ = + =
+ + + =
=
= + = + =
+=
x x x x
x x x x x x x x
x x x x x
x x x
x
x
x
f f
x
x
Bi 8:Gii phng trnh: .16212244 2 +=++ xxxx
Hng dn: iu kin xc nh: .4x Vi iu kin , phng trnh cho tng
ng vi:
( )2
2x 4 x 4 (x 4) (x 4) 12 2 x 16 x 4 x 4 x 4 x 4 12+ + = + + + + + = + +
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MT S DNG PHNG TRNH THNG XUT HIN TRONG THI I HC
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+ t t = 44 ++ xx , t > 0 ta c
=
==
4
)(3012
2
t
loaittt
+ Vi t = 4 , ta c
+=
==++
22
2
166416
84816444
xxx
xxxxx
.55
84
=
=
xx
x
Vy nghim ca phng trnh l x = 5.
Bi 9:Gii phng trnh 3 2 2 332 10 17 8 2 5 + + = x x x x x x
Hng dn: Nhn xt: x = 0 khng tha phng trnh cho
+ Chia hai vca phng trnh cho x3, ta c: 32 3 210 17 8 5
2 2 1x x x x
+ + =
+ t ( )1
0t tx
= , phng trnh trthnh: 2 3 232 10 17 8 2 5 1t t t t + + =
( ) ( )
( )
33 2 23 32 1 2 2 1 5 1 2 5 1t t t t + = + ( )
( )
232 1 5 1f t f t =
+ Vi ( ) 3 2 ,= + f t t t t R . Ta c: ( ) 2' 3 2 0,= + > f t t t R nn f ng bin trn R , v vy:
( ) ( )2 23 32 1 5 1 2 1 5 1f t f t t t = =
( )3 2 3 2
0
17 972 1 5 1 8 17 6 0
16
17 97
16
t
t t t t t t
t
=
+ = + = =
=
(loai)
(nhan)
(nhan)
+17 97 17 97
16 12t x
+ = =
+17 97 17 97
16 12t x
+ += =
Vy phng trnh cho c 2 nghim:17 97
12x
=
Bi 10:Gii phng trnh: 3
1 1 3x 2x (x R). (1)2+ + =
Hng dn:iu kin:
1
3x (*).
+ t 1 1 3x2
y = + 1 3x 2y + = (2), (1) trthnh 1 3 2y x+ = (3)
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MT S DNG PHNG TRNH THNG XUT HIN TRONG THI I HC
Trang 6NGUYN HU BIN -https://www.facebook.com/groups/nguyenhuubien
+ T(2) v (3) ta c h1 3x 2
1 3 2x
y
y
+ =
+ =
2
2
0, 0
1 3x 4 (4)
1 3 4x (5)
x y
y
y
+ =
+ =
.
+ Trvvi v (4) v (5) ta c 2 23( ) 4( )x y x y = ( )(3 4 4 ) 0x y x y y x + + = = (v
0, 0x y ). Thy = x vo (5) ta c 21
4x 3x 1 0 1
4
x
x
= = =
.
+ Kt hp vi 0x , suy ra phng trnh (1) c nghim duy nht x = 1.
Bi 11:Gii phng trnh 41292 22 +=++++ xxxxx
Hng dn:
Ta thy: )4(2)12()92( 22 +=+++ xxxxx + x = -4 khng phi l nghim+ Xt 4x , trc cn thc ta c:
2129241292
8222
22 =++++=+++
+
xxxxxxxxx
x
+ Vy ta c h
=
=
+=++
+=++++
=+++
7
8
0
692241292
212922
22
22
x
x
xxxxxxxx
xxxx
+ Thli vo phng trnh ta thy tha mn.
Vy phng trnh c 2 nghim: x = 0, v x =7
8
Bi 12:Gii phng trnh 0123)13(1 32 =++++ xxxxx
H
ng dn: iu kin: 1
x
=++
=+
=+++
=+++++
=+++++
0131
01
0)131)(1(
0)131()131(1
0311131
2
2
22
32
xx
xx
xxxx
xxxxxx
xxxxxxxx
Kt hp vi iu kin ta thy phng trnh v nghim.
Bi 13:Gii phng trnh 11 33 2 =+ xxx
Hng dn: iu kin: 3 2x + Nhn thy x = 3 l nghim ca phng trnh , nn ta bin i phng trnh
52
)93)(3(
412)1(
31)3(52321
3
2
3 23 22
33 2
+
++=
++
++=+
x
xxx
xx
xxxxx
+ Ta chng minh :52
932
3)11(
31
412)1(
31
3
2
23 23 3 222 +
++
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MT S DNG PHNG TRNH THNG XUT HIN TRONG THI I HC
Trang 7NGUYN HU BIN -https://www.facebook.com/groups/nguyenhuubien
Bi 14:Gii phng trnh: 31
21
=+
+ x
x
x
x
Hng dn:iu kin: 01 >< xx , t
2
1
10,
1
tx
xt
x
xt =
+>
+=
3 2
2
1t1
2t 3 2t 3t 1 0 2tt 1
= = + =
=
+ Ly3
4
2
11
2
1==
+= x
x
xt
Bi 15:Gii phng trnh 82315 22 ++=+ xxx
Hng dn:
1
0)338
1
415
1)(1(
38
1)1(3
415
1
3833415
222
2
2
2
22
=
=
++
+
++
+
++
+=
++
++=+
x
x
x
x
xx
x
xx
x
x
xxx
Bi 16:Gii phng trnh : 1221)14( 22 ++=+ xxxx
Hng dn:t 1)0(1 222 =>+= txtxt
Phng trnh trthnh 12)1(2)14( 2 ++= xttx 012)14(2 2 =+ xtxt
Ta c: 2)34( = x
3
4
3
4
0
1)12(
)(4
3
12
2
1
22
2
=
=
=
=
=
=
= x
x
x
xx
Lx
xt
t
Bi 17: )(01713264 2 Rxxxxx =++
Hng dn: iu kin: 64 x
0)15132()16(
)16)(16(
)14(
)14)(14(
015132)16()14(1713264
2
22
=++
++
+
+
=+++=+
xxx
xx
x
xx
xxxxxxxx
x 5 5 x(x 5)(2x 3) 0
x 4 1 6 x 1
1 1(x 5)( (2x 3)) 0x 4 1 6 x 1
x 5
1 1(2x 3) 0
x 4 1 6 x 1
+ =
+ +
= + +
=
= + +
+ Ta c: )1(3216
1
14
10)32(
16
1
14
1=
+
+=
+
+x
xxx
xx
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Trang 8NGUYN HU BIN -https://www.facebook.com/groups/nguyenhuubien
114
1
16
1
14
1
++++=++++=+++ 03
4
3)
2
1(3
4
3
4
1
2
1.23 2222222 yyxyyxyxxyyx
Bi 19: 4384212)47(23)14( 2 ++=+ xxxxxx (vi x )R
Hng dn: iu kin:2
3
2
1 x
t )1(2)0(12
)0(23 22=+
=
= babxb
axa
+ V baxxxxaxbx .23.12384;1247;214 222 ==++==
+ Tphng trnh cho ta c )2(42)12()12( 22 +=+++ abbaab
+ Kt hp (1) v (2) ta c hphng trnh
+=+++
=+
42)12()12(
2
22
22
abbaab
ba
+ y l hphng trnh i xng loi I vi n a, b.+ Gii hphng trnh trn ta c nghim l : a = b =1
+ Vi a = b = 1 khi ta c: 1112
123 =
=
= xx
x (tha mn iu kin)
Vy nghim ca phng trnh cho l x = 1
Bi 20:Gii phng trnh: 83)23(2 32 +=+ xxx
Hng dn: iu kin 2x
+ Bin i phng trnh vdng: )42)(2(3)2(2)422(2 22 ++=++ xxxxxx
+ Do 2=x khng phi l nghim ca phng trnh nn chia chai vcho (x+2) ta c
02
2
423
2
)42(2 22=
+
+
+
+
x
xx
x
xx
+ t )0(2
422=
+
+tt
x
xxta c
=
==
2
2
1
0232 2
t
ttt (loi)
Vi t = 2 133046844242
42 222
==+=+=+
+ xxxxxx
x
xx
Bi 21:Gii phng trnh 3)6)(3(63 =+++ xxxx
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Hng dn: t 2
9)6)(3(63
2
=+++= t
xxxxt
+ Phng trnh trthnh
=
==
3
13
2
92
t
ttt
+ Vi t = -1 163 =++ xx (v nghim)
+ Vi t = 3
=
=
=++ 6
3363
x
xxx
Cch khc:
Bi ton c thgii bng cch a vhphng trnh vi 3+= xu v xv = 6
Bi 22:Gii phng trnh: )1(055)3(66138 223 =++++ xxxxxx
Hng dn:iu kin: 0552 + xx
+ T(1) 055)3(6)25)(3( 22 =++ xxxxxx
=++
=
)2(055625
)(3
22 xxxx
loaix
+ Gii (2): t ,552 txx =+ iu kin 0t
(2)
=
==+
7
10762
t
ttt , t = -7 (loi), t = 1 (tha mn)
+ Vi t =1
=
==+
4
11552
x
xxx (tha mn iu kin)
Vy phng trnh c hai nghim x = 1 v x = 4
Bi 23: 16924422 2 +=++ xxx
Hng dn:
Bnh phng hai v, lc phng trnh :
169)2(16)4(216)42(4 22 +=+++ xxxx
xxxx 8)4(216)4(8 222 +=+
+ t t = )0()4(2 2 tx
+ Phng trnh trthnh : 08164 22 =+ xxtt
+ Gii phng trnh trn vi n t ta tm c
=
=
42
2
2
1
xt
xt
+ Do 2x nn 02
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+ Khi phng trnh cho trthnh )1(03638 32 =+++ xxxx
+ Nhn thy x = 0 khng tha mn nn (1) 03
63
8 =+++x
xx
x
+ t )12(3 4
=+ ttx
x , ta c
=
==+
4
20862
t
ttt (tha mn)
+ Vi t = 2
=
=31
xx
+ Vi t = 4
=
+=
618
618
x
x
Bi 25: 1635223132 2 +++=+++ xxxxx
Hng dn:
Bin i phng trnh vdng:20)1)(32(2132132 ++++++=+++ xxxxxx
+ t )0(132 +++= txxt
+ Khi )1)(32(21322 ++++++= xxxxt + Ta c phng trnh: 5;40202 === tttt + Ly t = 5 tha mn, thay t = 5 c x = 3
Bi 26:Gii phng trnh 0192
29
22 =
+
+
x
x
x
Hng dn: iu kin: 0x
+ Phng trnh cho tng ng 0392
292
22
2
=
+
++
x
x
x
x
+ t 092 2
+
=
x
xt . Phng trnh trthnh
=
=
=+
2
1
1
03212 t
t
tt
+ Vi2
23
924
0922
2
122
2=
+=
+= ,053)(' 2 , suy ra hm strn ng bin trn R
+ M phng trnh (*) c dng: )92()5( 3 = xfxf
=
=
=+
=+=
2
511
4
0)2911)(4(
0116731592)5(
2
233
x
x
xxx
xxxxx
Bi 32:Gii phng trnh 612243 =++ xx
Hng dn:iu kin: 12x
+ t xvxu =+= 12;243 vi 0v
+ Ta c
=+
=+
=+
=+
=+
=+
=+
=+
0)12(
6
012
6
36)6(
6
36
6
2232323uuu
vu
uuu
vu
uu
vu
vu
vu
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MT S DNG PHNG TRNH THNG XUT HIN TRONG THI I HC
=
=
=
=
=
=
=
=
=
3
88
24
3
3
10
4
6
0
x
x
x
v
u
v
u
v
u
(tha mn)
Bi 33:Gii phng trnh 2132727122 233 += xxxx
Hng dn:
)13(2)13(122)12( 33 +=+ xxxx
+ Xt hm 32)( aaaf += ta suy ra 0072727)13(12 233 ==+= xxxxxx
p s: x = 0
Bi 34: Gii phng trnh 3 233 23 1 xxxxx ++=++
Hng dn:
+ Xt x = 0 khng phi l nghim ca phng trnh+ Vi 0x , ta chia hai vcho x:
10)1(11
111 33333 ==
+++=+
+xx
x
xxx
x
x
(lm bi tp kt hp vi hc trong vghi)
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