kỹ thuật xử lí phương trình, hệ phương trình
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Các phương pháp cơ bản giải phương trình, hệ phương trìnhTRANSCRIPT
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1 | T H T H U T P H N G T R N H H P H N G T R N H
TI LIU N THI TRUNG HC PH THNG QUC GIA
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K THUT X L PHNG TRNH H PHNG TRNH V T
PHN I: PHN II: PHN III: PHN IV: PHN V: PHN VI: PHN VII:
PHNG PHP XT TNG V HIU D ON NHN T T NGHIM V T H S BT NH O HM MT BIN LNG GIC HA T 2 N PH PHN VII: PHNG PHP NH GI
Bin son: ON TR DNG
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2 | T H T H U T P H N G T R N H H P H N G T R N H
PHN I: PHNG PHP XT TNG V HIU
Phng php xt tng v hiu s dng cho cc phng trnh v t hoc mt phng trnh c trong mt h
phng trnh dng A B C . iu kin s dng ch ta nhn thy C l mt nhn t ca A B .
BI 1: 2 2 2 1 1x x x x
Nhn thy 2 22 2 1 1A B x x x x c mt nhn t l 1C x
2 22
2
2
2
2
2 2 1 12 2 1 1
12 2 1
2 2 1 12 2 2 0
2 2 1 1
x x x xx x x x
xx x x
x x x xx x x x
x x x x
BI 2: 3 2 2 21 2 1x x x x x
Nhn thy 3 2 2 31 2 1A B x x x x c mt nhn t l 2 1C x x
3 2 2 33 2 2
23 2 2
3 2 2 2
2 2 2
3 2 2
1 2 11 2 1
11 2
1 2 12 2 1 1 2 2
1 2 1
x x x xx x x x
x xx x x
x x x x xx x x x x x
x x x x
Th li nghim ta thy ch c 2x tha mn nn phng trnh c mt nghim duy nht l 2x
BI 3: 48 7 1 1x x x x x
Nhn thy 8 7 1 1A B x x x x x c mt nhn t l 4 1C x
4
4
4
4
8 7 1 18 7 1 1
18 7 1
8 7 1 12 7 1 2 7 0 0
8 7 1 1
x x x x xx x x x x
xx x x x
x x x x xx x x x x
x x x x x
BI 4: 3 4 5 4 4
5 3 7 2 2 1 4
y x y x
y x x y
Nhn thy phng trnh u c 3 4 5 4 8A B y x y x x c lin quan n gi tr 4
2
3 4 5 4 83 4 5 4 2
43 4 5 4
3 4 5 4 42 3 4 4 2 3 4 2 , 2.
3 4 5 4 2
y x y x xy x y x x
y x y x
y x y xy x x y x x y x x x
y x y x x
Thay vo phng trnh th 2 ta c
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3 | T H T H U T P H N G T R N H H P H N G T R N H
2 25 5 3 7 2 4 6 1 0x x x x x
2 2
2
2
5 5 3 1 2 7 2 4 7 2 0 *
1 14 7 2 1 0
2 7 25 5 3 1
x x x x x x x
x xx xx x x
V 2
2
2 1 1 7 17 5 4 171 0 4 7 2 0 ,
7 8 322 7 25 5 3 1x x x x y
x xx x x
Trong phn ny c chi tit trc cn thc bc * s c gii thch trong Phn II: H S BT NH
BI 5:
22
2 244 1 0
2 1
5 5 1 6
yx x y
y
x y x y
Phng trnh th 2 c 5 5 1 6 1A B x y x y x c lin quan n gi tr 6
2
5 5 1 6 15 5 1 1
65 5 1
75 5 1 12 1 7
4 5 20 55 5 1 6
x y x y xx y x y x
x y x y
xx y x y xx y x
y x yx y x y
phng trnh 1 c
2 22
2 2
2 24 5 2 2 94 1 0 1 2 0 5
2 1 2 1
y y y yx x y x y
y y
Vy h c nghim duy nht l 5x y
BI 6: 3 2 2
2 2 4 4
4 2 4 1 3 0
x y x y
x x y y x
Nhn thy phng trnh u c 2 2 4 2 2A B x y x y y c lin quan n gi tr 4
2
2 2 4 2 2 22 2 4
4 22 2 4
2 2 4 426
2 2 1 122 322 2 4
2
x y x y y yx y x y
x y x y
x y x yyy
x y xyx y x y
Mt khc phng trnh th 2 bin i thnh:
3 2 2
3 2 2 2
2 23
4 2 4 1 3 0
1 4 4 4 4 0
1 2 2 0
x x y y x
x x xy y y y
x x y y
V 0 1VT x cho nn h phng trnh c nghim duy nht l 1, 2x y
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4 | T H T H U T P H N G T R N H H P H N G T R N H
BI 7: 2
2 1 1 2
( 1)
y x y y
x x y x x y
Nhn thy phng trnh u c 2 1 1 2 2 2A B y x y y x khng lin quan n 2C y
Cn phng trnh th 2 c 2( 1)A B y x x y x y x c th rt gn vi C x x
2
2
2
22
2
2 22 2 2 2 2 2
( 1)( 1)
( 1)
( 1)
2 ( 1)( 1)
2 4 0
y x x y x y x y xy x x y
x x xy x x y
y x x y x xy x x x y
y x x xy xx xy x x y
x
y x x x x y y x x x x y x x y y x x
Thay vo phng trnh th nht ta c: 2 2 21 1 2x x x x x x
n tnh hung ny ta dung k thut nhm nghim nhn ra phng trnh c nghim duy nht 1x (Hoc s
dng my tnh SHIFT SOLVE). Khi 1x th 2 1x x = 1, 2 1x x = 1. Do ta s dng bt ng thc
Cauchy nh gi:
2 22 2
2 22 2
2 2
2 2 2 2
1 11. 1 1
2 2
1 1 21. 1 1
2 2
21 1 1 1 1
2
x x x xx x x x
x x x xx x x x
x x x xx x x x x x x x x
V 2 2 2 2 21 1 1 1 2 1 1 2x x x x x x x x x x x x
Vy ng thc xy ra khi 1, 0x y
BI 8: 2 216 2 3 4 1 1x x x x
Bi ton ny nghim rt p 3, 0x x nhng gii ra nghim ny bng cch trc cn thc n thun th
gn nh s khng c nhiu im. gii quyt trit ta s dng k thut xt tng hiu:
2 2
2 2
2 2
2
2 2
16 2 3 4 1 1
16 4 3 4 1 1
1 116 2 3 4
3 12
1 116 2 3 4
x x x x
x x x x
xx x x
x x x
xx x x
Nh vy nghim u tin l 0x . Nu 0x th
2 216 2 3 4 3 4 1 1x x x x x
Do ta c h:
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5 | T H T H U T P H N G T R N H H P H N G T R N H
2 2
2
2 2
2
2
2
2
2
2
16 2 3 4 3 4 1 12 16 13 3 1 11 3
16 2 3 4 1 1
16 53 2 16 5 13 3 1 2 27 9
1 2
2 16 5 3 13 1 2 9 3 0
2 9 3 13 39 3 0
1 216 5
2 3 3 13
16 5
x x x x xx x x x
x x x x
xx x x x x
x
x x x x
x x xx
xx
x xx
x
39 0
1 2x
V 1 3 0x x . Ta xt 3 13 3 5 9 1
9 0 11 2 1 2
x x xx
x x
Vy phng trnh c 2 nghim duy nht l 3 0x x
BI TP P DNG:
BI 1:
2 21 1 1
1 1 2
x y y x
x y
BI 2:
2 2 2 2 2
5 3
x y x y y
x y
BI 3:
2
3
12 12 12
8 1 2 2
x y y x
x x y
BI 4:
2 2
2 2
1
1 2 1 1
2 1
2x x x
x y x y y
y
BI 5: 2 2 2 2 2x x x
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6 | T H T H U T P H N G T R N H H P H N G T R N H
PHN II: D ON NHN T T NGHIM V T
Phng php ny tn dng nghim v t m my tnh d c on trc nhn t ca phng trnh, h
phng trnh. s dng k thut ny, chng ta cn phi nm c tt quy tc d nghim SHIFT SOLVE.
BI 1: 2 25 5 3 7 2 4 6 1 0x x x x x
iu kin: 2
7x . S dng my tnh SHIFT SOLVE vi 1x ta c 1,390388203x .
Khi thay vo gi tr cn thc: 25 5 3 2,390388203 1
7 2 2,780776406 2
x x x
x x
. Do 25 5 3x x cn phi to
thnh nhm biu thc 25 5 3 1x x x cn 7 2x cn phi to thnh nhm biu thc 2 7 2x x .
2 25 5 3 2,390388203 5 5 3 1x x x x x . Nh vy ta thy rng.
Vit li phng trnh ban u ta c: 2 25 5 3 7 2 4 6 1 0x x x x x
2 2
2
2
5 5 3 1 2 7 2 4 7 2 0 *
1 14 7 2 1 0
2 7 25 5 3 1
x x x x x x x
x xx xx x x
V 2
2
2 1 1 7 17 5 4 171 0 4 7 2 0 ,
7 8 322 7 25 5 3 1x x x x y
x xx x x
BI 2: 2 2 2 3 1 2 3 1x x x x x
iu kin: 2 2 0x x . S dng SHIFT vi 0x ta c 4,236067977x
Thay vo cc cn thc ca bi ton:
2 2 1
2 3 1 5,236067977
x x
x
. Nh vy 2 2x x s tr i 1 cn
2 3 1x s tr i 1x . Vit li phng trnh:
2 23 1 2 2 2 3 1 2 2 1 1 2 3 1 4 1 0x x x x x x x x x x x
2
2
22
2 22
2 1 2 3 12 2 14 1 0
1 2 3 12 2 1
1 2 2 4 14 1 0
1 2 3 12 2 1
2 1 2 2 4 14 1 0
1 2 3 12 2 1 1 2 2
x x xx xx x
x xx x
x x x xx x
x xx x
x x x x xx x
x xx x x x
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7 | T H T H U T P H N G T R N H H P H N G T R N H
2 1 14 1 1 01 2 3 12 2 1 1 2 2
x xx xx x x x
V 2 2 0x x nn
2 2 1 1 2 2 02 2 2.0 2 2 1
1 2 3 1 0
x x x xx x x
x x
Vy 2 4 1 0, 2 2 5x x x x .
BI 3: 3 2 24 3 2 5 2 13x x x x x x
SHIFT SOLVE vi 0x ta c 0,828427124x . Thay vo cc gi tr cn thc:
2 5 4,828427125 4
2 13 3,828427125 3
x x
x x
. Do ta vit li phng trnh ban u:
2
2 2 2
22
4 2 5 3 2 13 0
8 16 4 5 6 9 2 130
4 2 5 3 2 13
14 4 0
4 2 5 3 2 13
x x x x x
x x x x x x x
x x x x
xx x
x x x x
n y ta s chng minh 4 2 5x x v 3 2 13x x u dng. nh gi c iu ny ta phi
xut pht t phng trnh ban u v nh gi iu kin ngoi cn: 3 24 3 0x x x . Tuy nhin phng
trnh bc 3 ny nghim rt xu, v trong chng trnh THPT th khng nn s dng phng php Cardano
gii bt phng trnh ny m ta s thm bt mt vi hng t nh bt phng trnh d gii hn:
3 2 3 2 24 3 0 4 4 0 4 1 0 4x x x x x x x x x .
Do :
4 2 5 0
3 2 13 4 3 2. 4 13 0
x x
x x
. Vy ta c 2 4 4 0
2 2 24
x xx
x
BI 4: 2 3 33 4 2x x x
Phng trnh ny nu gii bng phng php o hm s p hn rt nhiu nhng chng ta s th ph cn 2
v v s dng k thut h s bt nh thng qua SHIFT SOLVE thy rng bi ton c th c nhng cch
gii rt ph thng. Lp phng hai v ta c: 6 327 4 2x x x .
S dng SHIFT SOLVE lin tc vi cc gi tr khc nhau ta thu c ch c duy nht 2 nghim l:
1 0,434258545x (S dng tip SHIFT RCL A gn vo bin A) v 2 0,767591879x (S dng tip
SHIFT RCL B gn vo bin B). Khi ta s dng nh l Viet o: 1 2
1 2
1
3
1
3
x x A B
x x AB
. Nh vy ta s
nhn ra nhn t nu c s l 21 1
03 3
x x hay 23 1 0x x . Thc hin php chia a thc ta thu c:
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8 | T H T H U T P H N G T R N H H P H N G T R N H
6 3 2 4 3 227 4 2 3 1 9 3 4 2 2 0x x x x x x x x x
V 4 3 2 4 2 2 29 3 4 2 2 6 3 1 2 1 1 0x x x x x x x x x x nn 21 13
3 1 06
x x x
BI 5: 2 215 2 1 5x x x x
SHIFT SOLVE ta c 20,767591879 1 1,535183758 2x x x x . Nhn t l 2 1 2x x x .
2 2 2 2
2
2 2
2 2
15 2 1 5 0 2 2 1 15 5 5 0
2 3 1 25 3 1 0 3 1 5 0
2 1 2 1
x x x x x x x x x
x xx x x x
x x x x x x
Xt2
2
25 0 10 5 1 2 0
2 1x x x
x x x
(Phng trnh bc 2).
Kt hp 23 1 0x x v 215 5 0x x ta c 1 13
6x
BI 6: 3 2 2 1 1 1x x x x
SHIFL SOLVE ta c 1,618033989 1 1,618033989x x x . Do c nhn t 1x x .
3 2 2 2 2
2 2 22 2
1 1 1 0 1 1 1 0
1 1 11 0 1 1 0
1 1
x x x x x x x x x
x x x xx x x x
x x x x
Xt 2
21 1 0 1 1 01
xx x x
x x
(V nghim). Vy 2
1 51 0
2x x x
.
3 2 3 2 23
1 8 8 3 0 2 1 4 6 3 08
x x x x x x x . 21 1 5
4 6 3 02 2
x x x x x
BI TP P DNG
BI 1: 2 3 2 1 2 1x x x x
BI 2: 2 2 3x x x x
BI 3: 3 2 23 2 2 4 2 11x x x x x x
BI 4: 2 21 2 2 2x x x x x
BI 5: 2 2
2
1 1
4 2 1
x x x
x x
BI 6: 2 6 2 8x x x
BI 7: 3 1 3 2x x x
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9 | T H T H U T P H N G T R N H H P H N G T R N H
PHN III: H S BT NH
Mc ch ca phng php h s bt nh l to ra cc thm bt gi nh sao cho c nhn t chung ri ng
nht h s tm ra cc gi nh . H s bt nh c bn cht l phn tch nhn t v c tc dng mnh trong
cc bi ton c nhiu hn 1 nghim.
BI 1: 4 2 4 24 20 4 7x x x x x
iu kin: 0x . Ta nhn thy cn phi khai trin 7x ax bx vi ,a b l hai s gi nh no sao cho khi
chuyn sang bn tri, nhn lin hp ta s tm c hai nhn t chung. Do ta s trin khai trin gi nh:
4 2 2 4 2 24 2 4 2
4 2 4 2
1 4 20 44 20 4 0 0
4 20 4
x a x x b xx x ax x x bx
x x ax x x bx
Mc ch ca ta l hai t s c cng nhn t chung do ta c
22
11 42, 51 20 4
7
a
a bb
a b
Nh vy ta khai trin li bi ton nh sau: 4 2 4 24 2 20 4 5 0x x x x x x
4 2 4 2
4 2
4 2 4 2 4 2 4 2
5 4 5 4 1 10 5 4 0
4 2 20 4 5 4 2 20 4 5
x x x xx x
x x x x x x x x x x x x
V 0x nn phng trnh c 2 nghim duy nht l 1 2x x .
BI 2: 2 26 1 2 1 2 3x x x x x
iu kin: 2 6 1 2 1 0x x x . Do phng trnh tng ng vi 2
26 1 2 32 1
x xx x
x
nn ta s i
tm mt nhm ax b gi nh sao cho phng trnh 2
26 1 2 32 1
x xax b x x ax b
x
c v tri
sau khi quy ng v v phi sau khi trc cn thc c cc nhn t ging nhau.
V vy ta s khai trin gi nh nh sau:
2
26 1 2 32 1
x xax b x x ax b
x
2 2 222
1 2 2 31 2 6 2 1
2 1 2 3
a x ab x ba x a b x b
x x x ax b
Do ta cn 2 t s c nhn t ging nhau nn ta c 2 2
1 2 6 2 10, 2
1 2 2 3
a a b ba b
a ab b
.
Khi ta khai trin li bi ton nh sau:
22 2 22
2 2
2 1 06 1 2 1 2 12 2 3 2
2 1 2 1 2 3 1 2 3 2 1
x xx x x x x xx x
x x x x x x x
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10 | T H T H U T P H N G T R N H H P H N G T R N H
BI 3: 2 22 1 1 2 2 6x x x x x x x
iu kin: 0x . Vit lai bi ton di dng: 3 2
3 22 2 6 2 2 41
x x xx x x
x
. nn ta s i tm mt nhm
ax b gi nh sao cho phng trnh 3 2
3 22 2 6 2 2 4 *1
x x xax b x x x ax b
x
c v tri
sau khi quy ng v v phi sau khi trc cn thc c cc nhn t ging nhau. Ta khai trin gi nh nh sau:
3 2 2 23 2
3 2
2 2 4 22 2 1 6*
1 2 2 4
x a x ab x bx a x a b x b
x x x x ax b
Do ta cn 2 t s c nhn t ging nhau nn ta c: 2 2
2 2 1 61, 2
2 2 4 2
a a b ba b
a ab b
Khi khai trin li bi ton vi 1, 2a b ta c: 3 2
3 22 2 6 2 2 2 4 21
x x xx x x x x
x
3 23 2 3 2
3 23 2
2 3 4 02 3 4 2 3 4
1 2 2 4 3 02 2 4 2
x xx x x x
x x x x VNx x x x
BI 4: 2 22 3 21 17 0x x x x x
SHIFT SOLVE 1 2x x . lm xut hin nhn t ny, ta cn khai trin gi nh bi ton thnh:
2 22 3 21 17 0x x mx n px q x x x m p x n q
Xt 22 3x x mx n ta c: 1 2
12 2 3
x m nm n
x m n
Xt 21 17px q x ta c: 1 2 3
2 2 5 1
x p q p
x p q q
Vy ta khai trin li bi ton nh sau: 2 22 3 1 3 1 21 17 3 2 0x x x x x x x
22
1 93 2 1 0
3 1 21 172 3 1x x
x xx x x
. V
1 017
3.1721 3 1 1 0
21
x
xx
Do phng trnh c 2 nghim duy nht l 1 2x x .
BI TP P DNG
BI 1: 2 36 5 1 3 2 3x x x x x
BI 2: 2 2 3 23 1 3 4 1x x x x x x BI 3: 2 23 4 1 4 2x x x x x BI 4: 32 1 3 2 2x x x
BI 5: 3 2 3 25 4 5 1 2 6 2 7x x x x x x
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11 | T H T H U T P H N G T R N H H P H N G T R N H
PHN IV: O HM MT BIN
K thut 1: Coi x l n, y l tham s, tnh o hm ' ,xf x y v chng minh hm s n iu v lin
tc theo x.
K thut 2: Phng trnh 0f x c ti a 1 nghim nu f x n iu v lin tc theo x.
K thut 3: f x f y x y nu f x n iu v lin tc theo x.
BI 1:
2 2
2 2
1
1 2 1 1
2 1
2x x x
x y x y y
y
Nu 22x y th phng trnh u tr thnh 20 0
1 11 2
y xy y
y x
. Thay cc cp nghim trn
vo phng trnh 2 ta thy khng tha mn.
Nu 2 1x y th phng trnh u tr thnh 2 1 1 1 2y y y x . Thay cp nghim trn vo
phng trnh 2 ta thy cng khng tha mn. Vy 2 22 , 1x y x y . Khi ta xt hm s:
2 22 2
11
1 12 1 ' 0
2 2 2f x x y x y y f x
x y x y
. Do hm s n iu v lin
tc vi mi x thuc tp xc nh. M 2 20f y x y . Thay vo phng trnh 2 ta c:
2 2 21 2 1 1 4 0x x x y
2 2 2 2
2 2 2 2 2
24 1 2 1 1 1 1
1 1 1 1 1
0 1 2 2 4 0
2 2 2 0 2 2 0
x x x x x x
x x x x x
x x x
x x x
Do 2 2
2 2
41 1
1 4 1 12 0 2
0 3 3x x
x xx x x y
x
CH : tm ra nhn t 2x y ta c th lm nh sau:
t 2100 20000 10001 101 10000y x x x y
BI 2: 21 3 2 1 1x x x x
iu kin: x > 0. Ta vit li phng trnh thnh: 21 3 1 1
2
x x x
x
2
2
2
1 3 1 1 1 1 1 11 1 1
2 2 2 2
x x x xx
x x x x
Xt hm 2
2
2 2
11 ' 0
1 1
t tt tf t t t f t
t t
do f t lin tc v ng bin trn .
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12 | T H T H U T P H N G T R N H H P H N G T R N H
Do 1 1 1 1
12 2
x xf f x
x x
BI 3:
3
2
2 2 1 3 1
2 1 2
y y x x x
y y x
Xt hm s 32 2 1 3 1f y y y x x x vi y l n, x l tham s. Ta c hm f y lin tc trn v
c 2' 6 1 0f y y nn f y l hm ng bin trn .
Mt khc ta c 1 2 1 1 1 2 1 3 1 0f x x x x x x x do phng trnh c mt nghim duy nht l 1y x . Thay vo phng trnh 2 ta c:
23 2 13 2 1 2 2
3 2 1 3 2 1 1
xx xx x x x
x x x x
tm ra nhn t 1y x , ta x l nh sau:
t 399 198 99 99 1 99
2 1 3 1 0 1 1100 100 100 100 10 100
x y y y x
BI 4: 2 2 2 2
2 2 2 2
1 1 4
1 1
x y y x xy
x y y x x y x
T phng trnh 2 ta c c 2 2 21 1 1x x x y y . Do 2
2
1 00
1 1 0
x x x xy
y
. M x v y
cng du nn ta suy ra 0, 0x y . Khi phng trnh 2 vit li thnh: 22
1 1 11 1y y y
x x x .
Xt hm s 2
2 2
21 , 0; ' 1 1 0
1
tf t t t t t f t t
t
. Do f t l hm s lin tc
v ng bin trn 0; . V vy ta c 1 1
f f y yx x
. Thay vo phng trnh u ta c 1x .
BI 5: 2 2
2 2
2 5 3 4
3 3 1 0
x x x y y
x y x y
thy phng trnh th 2 l mt phn khuyt ca phng trnh u. Nu ta kt hp hai phng trnh
th c th xy dng hm c trng. V vy ta bin i phng trnh 2 tr thnh 2 23 1 3x x y y v cng
vo 2 v ca phng trnh u ta c:
2 22 2 2 2 2 22 1 2 5 4 1 1 4 4x x x x y y x x y y
Xt hm c trng 1
4, 0; ' 1 02 4
f t t t t f tt
. Do 2 21f x f y khi
v ch khi 2 2
11
1
y xx y
y x
.
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13 | T H T H U T P H N G T R N H H P H N G T R N H
PHN V: LNG GIC HA
BI 1: 2 21 2 1 2 1x x x x
2cos 0; 2 sin 2cos 1 2cos sin 2 sin cos 2 sin 2 sin sin 22 2 2 4
t t tx t t t t t t t t
BI 2: 3 2 24 12 9 1 2x x x x x
Phng trnh 3 2
4 1 3 1 1 1x x x . t 1 cos 0;x t t
34cos 3cos sin cos3 sint t t t t
BI 3: 2 4 2 31 16 12 1 4 3x x x x x
4 2 3
22 2
2
cos 0; sin 16cos 12cos 1 4cos 3cos
sin 4 2cos 1 2 2cos 1 1 cos3
sin 4cos 2 2 2cos 2 1 cos3
sin 2cos 4 2cos 2 1 cos3
sin 4cos3 cos 1 cos3
2cos3 sin 2 sin cos3
sin 5 sin
x t t t t t t t
t t t t
t t t t
t t t t
t t t t
t t t t
t
sin cos3t t t
BI 4:
222
2
2
111
2 2 1
xxx
x x x
2
2
222
tan , ; \ 0; ;2 2 4 4
2 11 21 , sin 2 , sin 2 cos 2
cos 1 1
x t t
x xxx t t t
t x x
222
2
2
2
11 1 1 21
2 cos sin 2 sin 42 1
2cos 2 2sin 12sin 1 2 2
sin 2 sin 4 sin 4 sin 4
1cos 2 2sin 1 1 1 2sin 2sin 1 1 sin
2
xxx
x t t tx x
t tt
t t t t
t t t t t
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14 | T H T H U T P H N G T R N H H P H N G T R N H
PHN VI: T 2 N PH
K thut 1: t 2 n ph a v h phng trnh c bn.
K thut 2: t 2 n ph phn tch a thc thnh nhn t.
BI 1: 3 1 2 2 1 8
5 2 9
x y x y x y
x x y y
t 2 2 22 1 1
0, 2 1 0 ,2 2
a b ba x y b y x y
. Thay vo h phng trnh ta c:
2 2 2 22 2
2 1 2 1 81 2, 1
2 1 4
a b a b a ba b x y
a a b
BI 2:
2 2
1 1 2
8 8 8
y x y x y y x
x y y x
t 2 2 20, 0 ,a x y b y x a b y b . V phng trnh 2 kh ln nn ta tp trung vo phng
trnh u phn tch nhn t: 2 2 2 21 1 2 1 1 2 0b a a b a b a b a b (1)
n y l ta c th s dng phng php th c ri. Tuy nhin nu k th phng trnh 2 c th x
l c mt cch c lp: 2 2 2 2 2 2 28 8 8 8 16 8 64 8x y y x x y x y y x
2
2 2 2 2 28 16 8 8 8 0 8 0 8x x y y x y x y (2).
Kt hp (1) v (2)
2
2
9 71, 8 ,
2 2
1, 8 3, 1
a x y x y
b x y x y
BI 3: 2
2 1 5
2
x y y x y
y xy y
t 2 20, 2 1 0 1a x y b y a b x y . Thay vo phng trnh u: 2 2 4a b a b
V phng trnh ny khng phn tch c thnh nhn t nn ta phi tm cch bin i phng trnh 2.
ta thy rng 2 2 22 1 2a b x y y x y xy y trong c 2 2xy y y xut hin trong phng trnh 2. Do : 2 2 2 2 2 2 2 22 2 4 1 3 3a b x y y x y a b x y a b a b .
Vy ta c h i xng loi 1: 2 2 2 2 2 24, 3 1 2, 1a b a b a b a b a b x y .
BI 4: 2441 1 1x x x x
thy 21 1 1x x x nn ta s t n ph da trn yu t ny. t 4 41 0, 1 0a x b x . Khi
ta c: 4 42x a b . Nhn 2 c 2 v ca phng trnh ta c:
2 4 4 2 22 2 2 0 0a a b b ab a b a b a b a b a b x
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15 | T H T H U T P H N G T R N H H P H N G T R N H
PHN VII: PHNG PHP NH GI
K thut 1: a phng trnh, h phng trnh v dng 2 2 0A B
K thut 2: S dng Cauchy vi nhng bi c cn bc ln.
K thut 3: S dng Bunyakovsky: 2 2 2 2ax by a b x y . Du bng: a b
x y
K thut 4: S dng Minkowski: 2 22 2 2 2a b x y a x b y . Du bng:
a b
x y
K thut 5: S dng Schwartz:
22 2 a ba b
x y x y
. Du bng:
a b
x y
K thut 6: S dng bt ng thc Jensen dnh cho hm li, hm lm:
" 0 22
" 0 22
a bf x f a f b f
a bf x f a f b f
. Du bng xy ra khi a b
BI 1: 3 2 44 4 5 9 4 16 8x x x x
SHIFT SOLVE ta tm c nghim duy nht 1
2x v xut hin cn bc 4 nn ta ngh ti vic s dng bt
ng thc Cauchy gii quyt bi ton gn nh hn. Tuy nhin Cauchy th cc ng thc phi bng nhau.
Ta thy rng 16 8 8 2 1x x trong 2 1 2x nn ta s tch:
4 4 44 4 2 2 2 2 1 2 716 8 2 2 2 2 14 4
x xx x
. Nh vy ta c 3 24 4 5 9 2 7x x x x . Do :
23 24 4 7 2 0 2 2 1 0x x x x x . V
21 12 0 2 1 0
2 2x x x x .
BI 2: 4 244 1 8 3 4 3 5x x x x x
SHIFL SOLVE ta tm c nghim duy nht 1
2x v xut hin cn bc 4 nn ta ngh ti vic s dng bt
ng thc Cauchy. Ta thy vi 1
2x th 44 1 1, 8 3 1x x nn ta ln lt s dng Cauchy bc 2 v
Cauchy bc 4 ta c: 4 41 4 1 1 1 1 8 3
1. 4 1 4 1 2 ,1.1.1. 8 3 8 3 22 4
x xx x x x x x
Vy 24 2 4 2 14 3 5 4 4 3 0 1 2 1 0
2x x x x x x x x x x x do
3
8x .
BI 3: 3 2 3 2 23 2 2 3 2 1 2 1x x x x x x x
SHIFL SOLVE ta tm c nghim duy nht 1x . Khi 3 2 3 23 2 2 3 2 1 1x x x x x m ta
thy c 2 biu thc lp phng i nhau trong 2 cn, nu 2 cn bnh phng th s trit tiu c nn ta
ngh n s dng Bunyakovsky:
3 2 3 2 2 2 3 2 3 21. 3 2 2 1. 3 2 1 1 1 3 2 2 3 2 1x x x x x x x x x x
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16 | T H T H U T P H N G T R N H H P H N G T R N H
Do ta c 2 22 1 2 3 2 1x x x x . Bnh phng 2 v 2 21 2 1 0 1x x x
BI 4:
22 3
2
4 1 4 8 1
40 14 1
y x x x
x x y x
S dng php th 240
14 1
x xy
x
vo phng trnh u v s dng SHIFT SOLVE ta c
1 3,
8 2x y .
Ch rng1
4 ,8 1 22
x x nn ta c: 3 33
8 18 1
8 1 8 124 8 1 8 .1 4 8 12 3 2
xx
x xx x x x x
V 23 14 1
14 1 14 12 2
y xx y y x
. Do h phng trnh tr thnh:
22
22
8 14 1 1
2
1440 2
2
xy x
y xx x
Ly 1 +2. 2 22 2 28 1
4 1 2 40 14 12
xy x x x y x
2
2 3 1 1 396 24 0 96 0 ,2 8 8 2
x x x x y
BI 5:
2 2
2
2 2 4 2
6 11 10 4 2 0
x x y y
x y x x
S dng php th 26 11 10 4 2y x x x vo phng trnh u v SHIFT SOLVE ta c 1; 3x y .
Khi 2 24 2 1, 10 4 2 2y y x x . V vy ta iu chnh cc s cho hp l v p dng Cauchy:
2 22 2 2
22 2
1 4 2 4 12 2 1. 4 2 2 2
2 2
7 22. 10 4 2 4 10 4 26 116 11
22 4
y y y yx x y y x x
x xx x x xx yx y
Do ta c:
2 2
2
2 4 4 3 0
10 2 15 0
x x y y
x x y
. Cng hai v ca hai phng trnh ta c:
2 22 23 6 6 12 0 3 1 3 0 1, 3x x y y x y x y
BI TP P DNG:
BI 1:
2 2 3
2 2
1 2 1
13
2
x y x x
x x y x x
BI 2: 2
3
12 12 12
8 1 2 1
x y y x
x x y