kỹ thuật xử lí phương trình, hệ phương trình

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


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


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


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:


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


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


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:


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


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


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


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 .


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

.


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


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


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


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

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