Lý thuyết sử dụng ẩn phụ căn thức

TTII LLIIUU TTHHAAMM KKHHOO TTOONN HHCC PPHH TTHHNNGG ____________________________________________________________________________________________________________________________

xyz

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CCHHUUYYNN PPHHNNGG TTRRNNHH VV BBTT PPHHNNGG TTRRNNHH

LL TTHHUUYYTT SS DDNNGG NN PPHH CCNN TTHHCC ((PPHHNN 44))

4 3 6D E F QQUUNN OONN BB BBIINNHH

CCHH OO:: SS DDNNGG HHAAII NN PPHH AA VV PPHHNNGG TTRRNNHH NNGG BBCC NNGG CCPP

TT HHAAII NN PPHH PPHHNNGG TTRRNNHH NNGG BBCC BBCC HHAAII.. TT HHAAII NN PPHH PPHHNN TTCCHH NNHHNN TT.. BBII TTOONN NNHHIIUU CCCCHH GGIIII..

CCRREEAATTEEDD BBYY GGIIAANNGG SSNN ((FFAACCEEBBOOOOKK));; XXYYZZ11443311998888@@GGMMAAIILL..CCOOMM ((GGMMAAIILL))

TTHH HH NNII MMAA TTHHUU 22001133

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L THUYT S DNG N PH CN THC (PHN 4) _______________________________________________________________________________________________________________________________________________________________

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CREATED BY GIANG SN; [email protected] 4 3 6D E F QUN ON B BINH

2

CCHHUUYYNN PPHHNNGG TTRRNNHH VV BBTT PPHHNNGG TTRRNNHH

LL TTHHUUYYTT SS DDNNGG NN PPHH CCNN TTHHCC ((PPHHNN 44))

------------------------------------------------------------------------------------------------------------------------------------------- Trong chng trnh Ton hc ph thng nc ta, c th l chng trnh i s, phng trnh v bt phng trnh l mt ni dung quan trng, ph bin trn nhiu dng ton xuyn sut cc cp hc, cng l b phn thng thy trong cc k thi kim tra cht lng hc k, thi tuyn sinh lp 10 THPT, thi hc sinh gii mn Ton cc cp v k thi tuyn sinh i hc Cao ng vi hnh thc ht sc phong ph, a dng. Mc d y l mt ti quen thuc, chnh thng nhng khng v th m gim i phn th v, nhiu bi ton c bn tng dn n mc kh thm ch rt kh, vi cc bin i p kt hp nhiu kin thc, k nng vn lm kh nhiu bn hc sinh THCS, THPT. Ngoi phng trnh i s bc cao, phng trnh phn thc hu t th phng trnh cha cn (cn gi l phng trnh v t) ang c ng o cc bn hc sinh, cc thy c gio v cc chuyn gia Ton ph thng quan tm su sc. Chng trnh Ton i s lp 9 THCS bc u gii thiu cc php ton vi cn thc, k t cn thc xut hin hu ht trong cc vn i s, hnh hc, lng gic v xuyn sut chng trnh Ton THPT. S a dng v hnh thc ca lp bi ton cn thc t ra yu cu cp thit l lm th no n gin ha, thc t cc phng php gii, k nng, mo mc hnh thnh, i vo h thng. V c bn lm vic vi lp phng trnh, bt phng trnh v t chng ta u tin kh hoc gim cc cn thc phc tp ca bi ton. Php s dng n ph l mt trong nhng phng php c bn nhm mc ch , ngoi ra bi ton cn tr nn gn gng, sng sa v gip chng ta nh hnh hng i mt cch n nh nht. i khi y cng l phng php ti u cho nhiu bi ton cng knh. Tip theo l thuyt s dng n ph cn thc (cc phn 1 n 3), kt thc tng s dng mt cn thc duy nht, tc gi xin trnh by ti qu c gi l thuyt s dng n ph cn thc (phn 4), ch yu xoay quanh mt lp cc bi ton cha cn thc c gii thng tng s dng hai n ph a v phng trnh ng bc ng cp bc hai c bn kt hp phn tch nhn t phng trnh tch. K nng ny ng hnh cng vic gii h phng trnh hu t ng bc ng cp, h phng trnh cha cn quy v ng cp, ngy mt nng cao k nng gii phng trnh h phng trnh cho cc bn hc sinh. Mc cc bi ton nng cao mt cht, do kh tng dn so vi cc phn 1 n 3, ng ngha i hi s t duy logic, nhy bn kt hp vi vn kin thc nht nh ca c gi. Ti liu nh ph hp vi cc bn hc sinh lp 9 THCS n thi vo lp 10 THPT i tr, lp 10 h THPT Chuyn, cc bn chun b bc vo cc k thi hc sinh gii Ton cc cp v d thi k thi tuyn sinh i hc Cao ng mn Ton trn ton quc, cao hn l ti liu tham kho dnh cho cc thy c gio v cc bn tr yu Ton khc. II.. KKIINN TTHHCC KK NNNNGG CCHHUUNN BB

1. Nm vng cc php bin i i s c bn (nhn, chia a thc, phn tch a thc thnh nhn t, bin i phn thc i s v cn thc).

2. K nng bin i tng ng, nng ly tha, phn tch hng ng thc, thm bt. 3. Nm vng l thuyt bt phng trnh, du nh thc bc nht, du tam thc bc hai. 4. Nm vng kin thc v a thc ng bc, cc thao tc c bn vi phng trnh mt n ph. 5. Bc u thc hnh gii v bin lun cc bi ton phng trnh bc hai, bc cao vi tham s. 6. S dng thnh tho cc k hiu logic trong phm vi ton ph thng.

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IIII.. MMTT SS BBII TTOONN IINN HHNNHH VV KKIINNHH NNGGHHIIMM TTHHAAOO TTCC BBii ttoonn 11.. GGiiii pphhnngg ttrrnnhh 2 6 3 4 2 1x x x x x . Li gii 1.

iu kin 12

x .

Nhn xt 2 16 3 0,2

x x x x . Phng trnh cho tng ng vi

4 2 3 2 4 3 2

2 2 22

22

30 12 36 9 16 2 1 20 46 36 9 0

1 18 1 9 1 0

18 9 1 0 9 6 2;1;9 6 2

x x x x x x x x x x

x x x x x

x x x x

i chiu iu kin thu c nghim 9 6 2;1;9 6 2S . Li gii 2.

iu kin 12

x . Phng trnh cho tng ng vi

222

2

4 14 2 1 3 6 3 3 1

2 11

1 3 2 1 03 2 1

x xx x x x x x

x xx

x x xx x

Ta c 20

9 6 2;9 6 218 9 0

xx

x x

. i chiu iu kin ta thu c ba nghim.

Li gii 3.

iu kin 12

x .

Phng trnh cho tng ng vi 2 4 2 1 3 2 1 0x x x x . t 2 1 0x y y thu c 2 24 3 0 3 0 3 0x xy y x x y y x y x y x y

22

000 2 1 1

2 1 0 1 0

xxx y x x x

x x x

.

20

3 0 3 2 1 9 6 2;9 6 218 9 0

xx y x x x

x x

i chiu vi iu kin 12

x , kt lun tp nghim 9 6 2;1;9 6 2S . Li gii 4.

iu kin 12


2 22 3 2 14 2 1 4 2 1 2 1 2 2 1 2 12 1

x xx x x x x x x x

x x

Vi 20

3 2 1 9 6 2;9 6 218 9 0

xx x x

x x

.

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Vi 22

002 1 1

2 1 0 1 0

xxx x x

x x x

.

i chiu vi iu kin 12

x , kt lun tp nghim 9 6 2;1;9 6 2S . Nhn xt.

Li gii 1 v 4 s dng php bin i tng ng thun ty, trong li gii 1 nng ly tha trc tip c km theo iu kin hai v khng m thng qua nhn xt da trn iu kin. Li gii 4 thm bt hng t a v hiu hai bnh phng cng cho kt qu nhanh chng.

Li gii 2 da trn php nhm nghim, s dng ng thc lin hp a phng trnh cho v dng tch, tc gi trnh by ti L thuyt s dng i lng lin hp trc cn thc h tm thi.

Li gii 3 l hng trng tm ca ti liu, mc d ch s dng mt n ph y nhng thc t a phng trnh cho v phng trnh hai n x v y. Cc bn c th thy a thc hai n 2 24 3x xy y d dng phn tch thnh hai nhn t, c th l 3x y x y .

S d nh vy v y l dng phng trnh hai n ng bc hai 2 24 3 0x xy y . Ngoi cch gii trn, cc bn c th tham kho thm cch trnh by cng bn cht sau Bin i v..... 2 24 3 0x xy y .

Xt 102

y x , khng nghim ng phng trnh ban u.

Xt trng hp 0y th ta c 2

2 24 3 0 4 3 0x xx xy yy y

t x ty ta c 2

1 2 14 3 0 1 3 0

3 3 2 1

t x xt t t t

t x x

BBii ttoonn 22.. GGiiii pphhnngg ttrrnnhh 23 1 4 4 4 3x x x x x . Li gii 1.

iu kin 34

x . Phng trnh cho tng ng vi 23 4 3 4 4 3x x x x .

t 4 3 0x y y thu c 2 23 4 0 3 03x y

x xy y x y x yx y

20

4 3 1;34 3 0

xx y x x x

x x

.

2

03 3 4 3

9 4 3 0x

x y x xx x

(H v nghim).

So snh iu kin 34

x ta thu c tp nghim 1;3S .

Li gii 2.

iu kin 34


22 2 2 2 4 33 4 3 4 4 3 4 4 4 3 4 3 2 4 33 4 3

x xx x x x x x x x x x x x

x x

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5

20

4 3 1;34 3 0

xx x x

x x

.

2

03 4 3

9 4 3 0x

x xx x

(H v nghim).

So snh iu kin ta thu c tp nghim 1;3S . Li gii 3.

iu kin 34

x . Nhn xt 23 3 4 3 04

x x x x . Phng trnh cho tng ng vi

4 3 2 2 4 3 2

2

9 24 2 24 9 16 4 3 9 40 46 24 9 0

11 3 9 4 3 0

3

x x x x x x x x x x

xx x x x

x

Kt hp iu kin thu c hai nghim, 1;3S . Li gii 4.

iu kin 34


2

2 2 24 4 3

4 4 3 4 3 4 3 4 3 4 3 3 04 3

x x xx x x x x x x x x x x

x x

.

21

4 3 03

xx x

x

2

03 4 3

9 4 3 0x

x xx x

(H v nghim).

i chiu iu kin ta thu c tp nghim 1;3S . BBii ttoonn 33.. GGiiii bbtt pphhnngg ttrrnnhh 22 3 2 3 2x x x x x . Li gii 1.

iu kin 23

x . t 3 2 0x t t , ta thu c

2 22 2 0 2 0x t xt x x t t x t x t x t (*).

Ta c 2 ; 0 2 03

x t x t . Do 2

20 3 2 1 23

3 2 0

xx t x x x

x x

.

Vy bt phng trnh cho c tp nghim 1;2S . Li gii 2.

iu kin 23

x . Bt phng trnh cho tng ng vi

2 2 2

22

2

8 12 8 4 3 2 9 4 3 2 4 3 2

3 2 3 2 2 3 2 3 2 0

3 23 2 0 1 2

3 2 0

x x x x x x x x x

x x x x x x x

xx x x

x x

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iu kin 23

x .

Nhn xt 22 2 3 2 03

x x x x . Bt phng trnh cho tng ng vi

24 2 2

24 2

2 2

4 3 2 4 3 2 3 2

4 5 3 2 3 2 0

3 2 4 3 2 0 1

x x x x x x

x x x x

x x x x

Ta c 2

2 3 234 3 2 4 0,8 16

x x x x

nn 21 3 2 0 1 2x x x .


iu kin 23

x . Bt phng trnh cho tng ng vi

22 2

2

3 23 2 3 2 0 3 2 0

3 2

3 2 2 3 20 2

3 2

x x xx x x x x x x

x xx x x x

x x

Nhn xt 2 2 3 2 0; 3 2 03

x x x x x . Do 22 3 2 0 1 2x x x .

Vy bt phng trnh cho c tp nghim 1;2S . BBii ttoonn 44.. GGiiii bbtt pphhnngg ttrrnnhh 24 3 3 8 1x x x x x . Li gii 1. iu kin 1x . Bt phng trnh cho tng ng vi 24 8 1 3 1 0x x x x . t 1 0x y y thu c 2 24 8 3 0 2 2 3 2 3 0 2 2 3 0x xy y x x y y x y x y x y

22

02 0 2 1

4 1 02 3 0 2 3 1 4 9 9 0

xx y x x

x xx y x x x x

(H v nghim).

22

02 0 2 1 1 174 1 0 32 3 0 82 3 1 4 9 9 0

xx y x x

x x xx y x x x x

.

Kt lun tp nghim 1 17 ;38

S

.

Li gii 2. iu kin 1x . Bt phng trnh cho tng ng vi

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2 224 8 1 4 1 1 2 2 1 1 2 3 1 2 1 0x x x x x x x x x x x x Xt hai trng hp

22

02 1

4 1 02 3 1 4 9 9 0

xx x

x xx x x x

(H v nghim).

22

02 1 1 174 1 0 3

82 3 1 4 9 9 0

xx x

x x xx x x x

.

Kt lun tp nghim 1 17 ;38

S

.

Li gii 3. iu kin 1x . Nhn xt rng 24 3 3 0,x x x . Bt phng trnh cho tng ng vi

2 24 2 2 4 2

2 2

0 0

16 9 1 24 1 64 1 16 40 1 9 1 0

0

3 1 170 1 17 34 84 1 4 9 9 0 81 17 3

8

x x

x x x x x x x x x x

x

x x xx x x x

x

So snh iu kin, kt lun tp nghim cn tm 1 17 ;38

S

.

BBii ttoonn 55.. GGiiii bbtt pphhnngg ttrrnnhh 4 22 xx x xx

.

Li gii. iu kin 0 2x . Bt phng trnh cho tng ng vi

2 2 2 22 2 2

0 0x x x xx x x x

x x

Xt hai trng hp

0 2 2 2 0x x x . Khi 20 2

2 0 1 22 0

xx x x

x x

.

0 2 0x x x ; 20

2 2 0 2 2 2 2 3 04 8 0

xx x x x x

x x

.

Kt lun nghim 2 2 3;0 1;2S . BBii ttoonn 66.. GGiiii bbtt pphhnngg ttrrnnhh 2 23 2 7 3 1 3x x x x x . Li gii 1.

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iu kin x . Phng trnh cho tng ng vi 2 2 21 3 1 3 2 3 0x x x x . t 21 ; 3 0x a x b b . Phng trnh trn tr thnh

2 23 2 0 2 0 2 02

a ba ab b a a b b a b a b a b

a b

22 2

11 3 1

2 1 3x

a b x x xx x x

.

22 2 2

1 12 1 2 3

2 1 4 12 3 2 11 0x x

a b x xx x x x x

(H v nghim).

Vy phng trnh cho c nghim duy nht 1x . Li gii 2. iu kin x . Phng trnh cho tng ng vi

2 2 2

222

3 1 3 1 3 4 4 3 1 1 3 4 1

16 1 14 1 1 2 3 1 0

1 2 31 3

x x x x x x x x

xx xx x x x

x xx x

Vi 22 2 2

1 11 2 3

2 1 4 12 3 2 11 0x x

x xx x x x x

(H v nghim).

Vy phng trnh cho c nghim duy nht 1x . Li gii 3. iu kin x . Phng trnh cho tng ng vi

2 2 2 2 2 2

22 22 2

2

12 8 28 12 1 3 4 2 1 12 1 3 9 3 3

1 2 32 2 3 3 3

1 3

x x x x x x x x x x

x xx x x

x x

22 2 2

1 11 2 3

2 1 4 12 3 2 11 0x x

x xx x x x x

(H v nghim).

22 2

11 3 1

2 1 3x

x x xx x x

.

Vy phng trnh cho c nghim duy nht 1x . Li gii 4. iu kin x . Nhn xt 22 23 2 7 2 1 6 0,x x x x x . Phng trnh cho tng ng vi

24 3 2 2

23 2

1 0

9 12 46 28 49 9 1 3

111

1 3 2 11 03 5 13 11 0

x

x x x x x x

xxx

x x xx x x

.

Vy phng trnh cho c nghim duy nht 1x .

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BBii ttoonn 77.. GGiiii pphhnngg ttrrnnhh 227 1 7 2x x x xx

.

Li gii 1. iu kin 0x . Phng trnh cho tng ng vi

2 2 2 2 27 2 7 2 2 7 2 6 0x x x x x x x x x x x (1).

t 2 2 0x x t t , phng trnh (1) tr thnh 2 2

2 22

2

2 2

7 6 0 6 0 6 0

022 1 281

7002 62 36

t xt x t t x x t x t x t x

xx x xx x x

xxx x xx x x

Kt lun phng trnh cho c tp nghim 1 28170

S

.


2 2 2 2

2 22 2 2 2 2

7 2 7 2 28 4 8 28 2

4 2 28 2 49 25 2 2 7 5

x x x x x x x x x x

x x x x x x x x x x x

2 22

2

2 2

022 1 281

7002 62 36

xx x xx x x

xxx x xx x x

.

Vy phng trnh cho c nghim duy nht, hay 1 28170

S

.

Li gii 3. iu kin 0x . Phng trnh cho tng ng vi 2 27 2 7 2x x x x x (*). Nhn xt 27 2 0x x x nn

4 3 2 2 2

23 2

0

49 14 29 4 4 49 2

00 1 2812 35 2 0 7035 69 4 4 0

x

x x x x x x x

xxx

x x xx x x

Kt lun phng trnh cho c tp nghim 1 28170

S

.


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2 2 2

222

2 27 7 2 7 2

2 227 2 2 0 1 2812 62 35 2 0 70

x xx x x x x x xx x

x xxx x xx xx x xx x x x x

Th li nghim, kt lun 1 28170

S

.

BBii ttoonn 88.. GGiiii pphhnngg ttrrnnhh 2

26 4 8 5 2 31

x x x xx

.

Li gii. iu kin 1x . Phng trnh cho tng ng vi

2 2

2 2 2

2 2 2

6 4 8 5 1 2 3

2 2 1 5 1 2 3 2 2 3 0

2 1 5 1 2 3 2 2 3 0

x x x x

x x x x x

x x x x

t 21 ; 2 3 0x u x v v thu c

2 22

2 5 2 0 2 2 02

u vu uv v u v u v

v u

Xt cc trng hp

2 2 2

1 12

2 1 8 12 7 2 11 0x x

u vx x x x x

(H v nghim).

2 2 21 1 4 142

2 3 4 2 1 22 8 1 0

x xv u x

x x x x x

.

i chiu iu kin kt lun phng trnh bi c duy nht nghim 4 142

x .

BBii ttoonn 99.. GGiiii pphhnngg ttrrnnhh 2 5 2 3 4 2 3 0x x x x x . Li gii.

iu kin 32

x .

t 2 3 0x y y th phng trnh cho tr thnh

2 25 4 0 4 04

x yx xy y x y x y

x y

22

002 3

2 3 0 1 2

xxx y x x

x x x

(V nghim).

20

4 4 2 3 16 4 13;16 4 1332 48 0

xx y x x x

x x

.

i chiu iu kin ta thu c hai nghim k trn.

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BBii ttoonn 1100.. GGiiii bbtt pphhnngg ttrrnnhh 2 25 2 2 5 1x x x x x x . Li gii. iu kin x . Bt phng trnh cho tng ng vi 2 2 23 5 1 2 1 0x x x x x x . t 2 21 0

3x x y y y y . Thu c

2 2 2 23 5 2 0 3 3 2 2 023 2 0 3 2 03

x xy y x xy xy y

x x y y x y x y x y y x y

Nhn xt 20; 03

y y x y x . Xt hai trng hp

o 2 2 24 1 9 5 4 4 0 2 2 62 3

500

x x x x xy x x

xx

.

o 2 2

00

1x

x y xx x x

.

Kt hp hai trng hp ta c nghim 0x . Nhn xt.

Cc bi ton t 2 n 10 u c gii bng kh nhiu phng php, bao gm bin i tng ng (nng ly tha trc tip, thm bt a v hiu hai bnh phng), s dng ng thc lin hp v trng tm l t n ph khng hon ton.

im c bit trong cc bi ton trn, khi t n ph hon ton (hoc khng hon ton) u a v cc phng trnh (hoc bt phng trnh) bc hai c tnh cht ng bc bc hai 2 2 0ax bxy cy , thao tc phn tch nhn t tr nn n gin. Cc bn c th la chn mt trong cc phng n sau

Tnh nghim, a trc tip v nhn t 0mx ny

mx ny px qypx qy

Xt trng hp 0y (hoc 0x ) c l nghim ca phng trnh ban u hay khng. Xt trng hp 0y (tng ng 0x ), chia hai v cho 2 0y thu c

2

0x xa b cy y

(tng ng

2

0y yc b ax x

).

t x ty (tng ng y t

x ) quy v phng trnh c bn 2 0at bt c ( 2 0ct bt a ).

Quan st thy tnh cht ng bc, t trc tip x ky a v

2 2 2 2 2 2 20

0 00

yak y bky cy y ak bk c

ak bk c

Suy ra hai trng hp Gii phng trnh bc hai n k s thu c t l gia x v y. Lu do vai tr ca x v y bnh ng nn cc bn c th chia cho x hoc y m khng nh hng ti kt qu ca bi ton. Nu bi ton l bt phng trnh th trc khi chia cn xt du ca y (tng ng x). Ty theo tng trng hp c th chn php chia hp l v tit kim nht, s dng cc nh gi thng thng m bo cho li gii c gn gng (in hnh bi ton 10).

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BBii ttpp ttnngg tt.. GGiiii cccc pphhnngg ttrrnnhh vv bbtt pphhnngg ttrrnnhh ssaauu ttrrnn ttpp hhpp ss tthhcc

1. 24 12 9 7 4 3x x x x .

2. 22 4 2 5 1x x xx .

3. 2 24 10 5 4 4 2x x x x x . 4. 25 4 4 4 6 2 4x x x x x

5. 2 27 4 10 7 2 1x x x x . 6. 21 2 1x x x x

7. 2 26 6 5 5 1 2 2 1x x x x x . 8. 22008 4 3 2007 4 3x x x x .

9. 2

22 4 5 3 12

x x xx

.

10. 2 26 21 3 6x x x x x .

11. 42012 2011 5 4 5x xx

.

12. 2 11 42 2 11 42x x x x . 13. 24 12 1 27 1x x x x . 14. 24 1 5 1x x x x . 15. 3 1 3 1 8 1x x x x x . 16. 27 3 1 2 2x x x x .

17. 226 3 6 3 2x x xx

.

18. 43 4 7 1x xx

.

19. 2 25 5 4 2 5 0x x x x x .

20. 23 22 47 7 5

3x x x

x

.

21. 22 3 2 3 2x x x x . 22. 22 6 5 6 2x x x x . 23. 22 1 7 1 4 4x x x x .

24. 2 25 5 1 1x x x x x .

25. 33 5 1 8 3x xx

.

26. 2 29 8 9 9 1 2 1x x x x .

27. 2 212 5 2 3 5 20x x x x x .

28. 213 2 3 4 2x x xx

.

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BBii ttoonn 1111.. GGiiii pphhnngg ttrrnnhh 2 35 7 7 1x x x x . Li gii 1. iu kin 1x . Phng trnh cho tng ng vi 2 21 7 1. 1 6 6 0x x x x x x .

t 2 1 ; 1 0; 0x x u x v u v ta thu c 2 27 6 0 6 06

u vu uv u u v u v

u v

22

1 01 1

21 1x x

u v x x xxx x x

2 21 37 1509 37 15096 1 6 1 ;

2 237 35 0x

u v x x x xx x

.

Vy phng trnh cho c nghim 37 1509 37 15090;2; ;2 2

S

.

Li gii 2. iu kin 1x . Nhn xt 2 5 7 0x x x . Phng trnh cho tng ng vi

4 3 2 3 4 3 2

2

10 39 70 49 49 1 39 39 70 0

37 1509 37 15092 37 35 0 0;2; ;2 2

x x x x x x x x x

x x x x x

Vy phng trnh cho c nghim 37 1509 37 15090;2; ;2 2

S

.

Nhn xt. Li gii 1 t n ph a v phng trnh ng bc bc hai vi hai n u v v. i vi cc cn thc c th khai phng theo hng n thc, cc bn ch

3 3 2 2a b a b a ab b v 3 3 2 2a b a b a ab b . BBii ttoonn 1122.. GGiiii bbtt pphhnngg ttrrnnhh 2 31 3 2 1x x x . Li gii 1. iu kin 1x . Bt phng trnh cho tng ng vi 2 3 2 22 4 2 1 1 2 1. 1 3 1 0x x x x x x x x x .

t 2 1 ; 1 0; 0x x u x v u v thu c

2 2 2

2 2

2 3 0 3 0 3 1 3 1

1 14 6 4 6

1 9 9 8 10 0

u uv v u v u v u v x x x

x xx

x x x x x

Kt lun tp nghim 4 6;4 6S .

Li gii 2. iu kin 1x . Nhn xt 21 3 0x x . Bt phng trnh cho tng ng vi

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14

4 3 2 3 4 3 2

2 2

4 12 16 16 4 4 8 12 16 20 0

8 10 2 0 4 6 4 6

x x x x x x x x x

x x x x


Li gii 3. iu kin 1x .

Xt trng hp 1x khng tha mn bt phng trnh ban u. Xt trng hp 1x , bt phng trnh cho tng ng vi

3 22 3 2 3 2

3

2 2

3 3 2

2 2 22 2 2 2 1 2 2 1 1 2

1 11 2 2 2 12 2 2 1 1

1 1 1 1 1 1

x x xx x x x x x x

x xx x xx x

x x x x x x x

Nhn xt: 2

2

2

21 1 0 11 1

xx x x xx x x

. Do

2 2 21

2 1 1 1 3 1 1 4 6;4 68 10 0

xx x x x x x x x

x x

.


Nhn xt. Bi ton 12 thuc lp bt phng trnh gii c thng qua php t n ph, a v phng trnh ng bc

bc hai, kt qu phn tch nhn t rt p mt. Trong thao tc gii bt phng trnh, cc bn cn ch iu kin xc nh (hoc iu kin c nghim), iu kin ca n ph gim thiu cc trng hp xy ra, gim nh tnh ton v lm cho li gii tr nn sc tch.

Li gii 2 s dng php nng ly tha trc tip (sau khi nhn xt hai v khng m). Li gii 3 s dng ng thc lin hp, nhm hng t phn tch thnh tha s, gin c a v bt phng

trnh cha n mu thc. Tuy nhin, s dng linh hot ng thc lin hp "thm mt ln", h qu thu c tr nn n gin.

BBii ttoonn 1133.. GGiiii bbtt pphhnngg ttrrnnhh 2 313 3 2 3 9x x x x x . Li gii 1. iu kin 3 22 3 0 1 3 0 1x x x x x x . Bt phng trnh cho tng ng vi 2 23 3 1. 3 10 1 0x x x x x x

t 2 3 ; 1 0; 0x x a x b a b thu c 2 2

22

3 10 0 5 2 5 0 2 5 0 2 0

13 2 1 1

3 7 0

a ab b a a b b a b a b a b a b

xx x x x

x x

Vy bt phng trnh cho c nghim 1;S . Li gii 2. iu kin 3 22 3 0 1 3 0 1x x x x x x . Bt phng trnh cho tng ng vi 3 23 2 3 9 13x x x x (1).

Xt 2 9 13 0x x , bt phng trnh (1) nghim ng.

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15

Xt 2 9 13 0x x , ta c

2 2

3 4 3 2 4 3 2

2 2

2 2 2

9 13 0 9 13 01

9 2 3 18 107 234 169 27 107 252 196 0

9 13 0 9 13 03 7 24 28 0 24 28 0

x x x xx x x x x x x x x x

x x x xx x x x x x

Ta c 2 2 29 13 0; 1 9 13 15 1 0 24 28 0x x x x x x x x . Vy (*) nghim ng vi 2 9 13 0x x .

Kt hp hai trng hp, (1) nghim ng vi mi gi tr x thuc tp xc nh, hay 1x . Li gii 3. iu kin 3 22 3 0 1 3 0 1x x x x x x . Bt phng trnh cho tng ng vi

3 22 3 2

3

22 2

3 3

3 4 10 73 7 3 2 3 2 2 0 3 7 0

2 3 2 23 1 3 7 3 1

3 7 0 3 7 1 0 22 3 2 2 2 3 2 2

x x xx x x x x x x

x x xx x x x

x x x xx x x x x x

Nhn xt 23

3 13 7 0 ; 1 1 0

2 3 2 2

xx x x x

x x x

. Vy (2) nghim ng vi 1x .

Kt lun tp nghim 1;S . Nhn xt.

Li gii 1 s dng php t n ph a v bt phng trnh ng bc (ng cp) bc hai. Khi vi iu kin mi ca n, chng ta d dng lp lun loi b mt trng hp.

Li gii 2 nng ly tha trc tip, thu c bt phng trnh a thc bc 4, s dng h s bt nh a v nhn t. Cc bn ch kt hp iu kin xc nh trnh c cc php bin i cn thc phc tp.

BBii ttoonn 1144.. GGiiii bbtt pphhnngg ttrrnnhh 2 33 27 7 10x x x x . Li gii 1. iu kin 3 210 0 2 2 5 0 2x x x x x x . Bt phng trnh cho tng ng vi 2 23 2 5 6 2 7 2 5. 2x x x x x x t 2 2 5 ; 2 13; 0x x u x v u v , quy v

2 2

22

3 7 6 0 3 3 2 3 0 3 3 2 0 3

22 5 3 2 2

7 23 0

u uv v u u v v u v u v u v u v

xx x x x

x x

Kt lun tp hp nghim 2;S . Li gii 2. iu kin 3 210 0 2 2 5 0 2x x x x x x . Bt phng trnh cho tng ng vi

4 2 3 4 3 2

2 2

9 162 729 49 49 490 9 49 162 49 1219 0

7 23 9 14 53 0 1

x x x x x x x x

x x x x

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16

Ta c 2 27 23 0 ;9 14 53 0x x x x x x nn (1) nghim ng vi mi gi tr x thuc tp xc nh. Kt lun tp hp nghim 2;S . Li gii 3. iu kin 3 210 0 2 2 5 0 2x x x x x x .

Nhn xt 2x khng l nghim ca bt phng trnh ban u. Xt trng hp 2x , bt phng trnh cho tng ng vi

3 22 3 2

3

22 2

3 3

22

2 2

7 9 37 463 21 69 7 10 3 6 3 7 23

10 3 67 2 7 23 7 2

3 7 23 7 23 3 010 3 6 10 3 6

7 2 7 23 32 5 3 22 2 5 3 2

7 2 3 2 5 9 2 3 2 5

x x xx x x x x x x

x x xx x x x

x x x xx x x x x x

x xx x xx x x x

x x x x x x

2 2 0 2x

Bt phng trnh (2) nghim ng vi mi gi tr x thuc tp xc nh. Do ta c tp nghim 2;S . Nhn xt.

Li gii 2 s dng php bnh phng trc tip v h s bt nh, phn tch phng trnh bc bn h qu v hai phng trnh bc hai, ht sc may mn khi hai tam thc bc hai lun lun dng vi mi gi tr ca bin, suy ra tp nghim chnh l tp xc nh ca phng trnh ban u. Li gii 3 s dng ng thc lin hp kt hp iu kin xc nh, trnh c vic bin lun du mu thc ca phng trnh h qu, v cho kt qu hon ton tng t.

Li gii 1 ngn gn, sc tch da trn quan st 3 210 2. 2 5x x x x x . C th thy pha ngoi cn thc l 23 27x , d dng t n ph v phn tch nhn t. Trong mt s trng hp, iu ny khng n gin, mi cc bn theo di cc th d tip theo.

BBii ttoonn 1155.. GGiiii pphhnngg ttrrnnhh 2 4 25 5 5 1x x x x x . Li gii 1. iu kin x . Nhn xt 24 2 4 2 2 2 2 2 21 2 1 1 1 1x x x x x x x x x x x . Phng trnh cho tng ng vi 2 2 2 22 1 3 1 5 1. 1x x x x x x x x t 2 21 ; 1 0; 0x x u x x v u v thu c

2 22 3 5 2 3 2 3 02 3u v

u v uv u u v v u v u v u vu v

2 21 1 0u v x x x x x .

2 2 2 13 69 13 692 3 4 4 4 9 9 9 5 13 5 0 ;10 10

u v x x x x x x x

.

Kt lun tp nghim 13 69 13 690; ;10 10

S

.

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17

Li gii 2. iu kin x . Nhn xt 25 5 0x x x . Phng trnh cho tng ng vi

4 2 2 2 4 2

3 2 2

25 50 25 10 1 25 1

13 69 13 6910 26 10 0 5 13 5 0 0; ;10 10

x x x x x x x

x x x x x x x

Kt lun tp nghim 13 69 13 690; ;10 10

S

.

Nhn xt.

Li gii 1 t n a v phng trnh ng bc da trn quan st 4 2 2 21 1. 1x x x x x x . Tuy nhin c c biu th p mt 2 2 25 5 2 1 3 1x x x x x x l mt vn khng n gin, nguyn do c hai nhn t u c dng tam thc bc hai. Ngoi cp h s 2;3 , cc cp s khc cng kh kh thi, chng hn 4;1 , 1;4 , 3;2 , 6; 1 , 2;7 ,...

Cc bn c th s dng ng nht thc tm c cc h s 2 v 3. t n ph 2 21 ; 1 0; 0x x u x x v u v , gi nh

2 2 2 2 2 25 5 1 1x x mu nv m x x n x x m n x m n x m n .

ng nht

52

13

5

m nm

m nn

m n

Lu mt s php bin i ng nht quen thuc sau y

4 4 2 2 2 2

4 4 2 2 2 2

4 4 2 2 2 2

1 2 1 2 2 1 2 1

4 1 4 4 1 4 2 2 1 2 2 1

64 16 64 16 4 8 4 8 ...

x x x x x x x x

x x x x x x x x

x x x x x x x x

BBii ttoonn 1166.. GGiiii pphhnngg ttrrnnhh 2 42 1 4 1x x x x . Li gii 1. iu kin x . Phng trnh cho tng ng vi

4 3 2 2 4 3 2 2 20

4 4 2 2 1 4 1 4 5 2 0 4 5 2 04 5 2 0x

x x x x x x x x x x x xx x

Phng trnh 24 5 2 0x x v nghim do 0 . Kt lun tp nghim 0S . Li gii 2. iu kin x . Phng trnh cho tng ng vi

2 2 2 2 2 2 28 4 4 4 2 2 1 2 2 1 2 2 1 3 2 2 1 4 2 2 1. 2 2 1x x x x x x x x x x x x x x t 2 22 2 1 ; 2 2 1 0; 0x x u x x v u v ta thu c

2 23 4 3 03

u vu v uv u v u v

u v

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18

2 22 2 1 2 2 1 0u v x x x x x . 2 2 2 2 23 2 2 1 3 2 2 1 2 2 1 9 2 2 1 16 20 8 0u v x x x x x x x x x x (V nghim)

Vy phng trnh cho c tp nghim 0S . BBii ttoonn 1177.. GGiiii pphhnngg ttrrnnhh 2 48 20 1 64 1x x x x . Li gii 1. iu kin x . Nhn xt 24 2 2 2 264 1 8 1 16 8 4 1 8 4 1x x x x x x x . Phng trnh cho tng ng vi

2 2 2 23 8 4 1 2 8 4 1 8 4 1. 8 4 1x x x x x x x x t 2 28 4 1 ; 8 4 1 0; 0x x a x x b a b ta thu c

2 2

2 2 2 2

3 2 3 2 0 3 2 0

8 4 1 8 4 1 8 4 1 8 4 1 0

a b ab a a b b a b a b a b

a b x x x x x x x x x

Vy phng trnh cho c tp nghim 0S . Li gii 2. iu kin x . Phng trnh cho tng ng vi

2 2

4 2 2 2 4 3 2

22

22

8 20 1 0 8 20 1 064 16 1 40 8 1 400 64 1 320 416 40 0

08 20 1 0

08 20 1 040 52 5 0

40 52 5 0

x x x xx x x x x x x x x

xx x

xx xx x x

x x

Vy phng trnh cho c tp nghim 0S . BBii ttoonn 1188.. GGiiii bbtt pphhnngg ttrrnnhh 4 23 81 4 27 42 6x x x x . Li gii 1. iu kin x . Nhn xt 2 24 4 2 2 2 2 281 4 81 36 4 36 9 2 6 9 6 2 9 6 2x x x x x x x x x x . Bt phng trnh cho tng ng vi

2 2 2 23 9 6 2. 9 6 2 5 9 6 2 2 9 6 2x x x x x x x x . t 2 29 6 2 ; 9 6 2 0; 0x x u x x v u v quy v

2 2

2 2 2 2

3 5 2 5 2 5 2 0 5 2 0

9 6 2 9 6 2 9 6 2 9 6 2 0

uv u v u u v v u v u v u v u v

x x x x x x x x x

Kt lun nghim ;0S . Li gii 2. iu kin x . Xt hai trng hp

227 42 6 0x x , bt phng trnh cho nghim ng.

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19

227 42 6 0x x , bt phng trnh cho tr thnh

2 2

4 4 2 2 3 2

2 2

2

27 42 6 0 27 42 6 09 81 4 729 324 36 84 27 6 2268 324 504 0

27 42 6 0 27 42 6 063 9 14 0 0

x x x xx x x x x x x x

x x x xx x x x

Kt hp hai trng hp thu c nghim ;0S . BBii ttoonn 1199.. GGiiii bbtt pphhnngg ttrrnnhh 2 44 2 4x x x x . Li gii 1. iu kin x . Nhn xt 2 24 4 2 2 2 2 24 4 4 4 2 2 2 2 2 2x x x x x x x x x x . Bt phng trnh cho tng ng vi

2 4 2 2 2 22 8 4 2 4 3 2 2 2 2 2 2 2. 2 2x x x x x x x x x x x . t 2 22 2 ; 2 2 0; 0x x a x x b a b ta thu c

2 2

2 2

2 2

3 2 3 3 0 3 0

2 2 2 22 2 2 2 0

a b ab a a b b a b a b a b

a b x x x xx x x x x

Kt lun: Bt phng trnh ban u c tp nghim ;0S . Li gii 2. iu kin x . Bt phng trnh cho tng ng vi

2 2 2

4 2 2 2 4 2

4 2 0 4 2 0 4 2 00

4 4 8 2 16 4 2 5 4 0 0

x x x x x xx

x x x x x x x x x x

.

Kt lun: Bt phng trnh ban u c tp nghim ;0S . BBii ttoonn 2200.. GGiiii pphhnngg ttrrnnhh 2 43 4 23 3 8 63x x x x x . Li gii 1. iu kin 4 8 63 0x x . Nhn xt 2 24 4 2 2 2 2 28 63 16 64 16 8 1 8 4 1 4 9 4 7x x x x x x x x x x x x . Phng trnh cho tng ng vi

2 2 2 22 4 7 4 9 3 4 7. 4 9x x x x x x x x t 2 24 7 ; 4 9 0; 0x x u x x v u v ta thu c

2 22 2

2 2

4 7 4 9 12 3 2 0

2 2 4 7 4 9 2

x x x xu vu v uv u v u v

u v x x x x

2 2 11 4 7 4 94

x x x x x .

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20

2 2 2 10 43 10 432 4 4 7 4 9 3 20 19 0 ;3 3

x x x x x x x

.

So snh iu kin, kt lun nghim 1 10 43 10 43; ;4 3 3

S

.

Li gii 2. iu kin 4 8 63 0x x . Nhn xt 23 4 23 0x x x . Phng trnh cho tng ng vi

4 3 2 2 2 4 3 2

3 2 2

9 24 16 46 3 4 23 9 8 63 24 154 112 38 0

10 43 10 4312 77 56 19 0 4 1 3 20 19 0 0; ;3 3

x x x x x x x x x x

x x x x x x x

So snh iu kin, kt lun nghim 1 10 43 10 43; ;4 3 3

S

.

BBii ttoonn 2211.. GGiiii pphhnngg ttrrnnhh 2 35 2 4 8x x x x . Li gii. iu kin 2x .

o Xt 2x khng tha mn phng trnh ban u. o Xt 2x , phng trnh cho tng ng vi

2 2

2 2 2 4 2 42 4 3 2 4 2 4. 2 3 42 2

x x x xx x x x x xx x

t 2 2 4 0

2x x t t

x

thu c 21

4 3 0 1 3 03

tt t t t

t

Vi 2 21 2 4 2 6 0t x x x x x (V nghim). Vi 2 23 2 4 9 18 7 22 0t x x x x x (V nghim).

Vy phng trnh cho v nghim. BBii ttoonn 2222.. GGiiii bbtt pphhnngg ttrrnnhh 2 32 5 1 7 1x x x x Li gii. iu kin 1x . Khi bt phng trnh cho tng ng vi

2 23 1 2 1 7 1. 1x x x x x x . t 21 ; 1 0; 0x a x x b a b thu c

2 2 2 22 2

2 2 2 2 2 2

22

9 43 7 2 0 3 2 0 . 03 2

9 4 0 8 19 8 3 6 3 0

19 105 19 1058 19 8 1 016 16

a b a ba ab b a b a ba b a b

a b a b x x x x

x x x x

Kt hp iu kin ta c nghim 19 105116

x .

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21


1. 2 35 6 4 6 1x x x . 2. 2 34 7 1 7 1x x x . 3. 2 33 5 5 1x x x .

4. 2

3

5 2 8 81

x xx

.

5. 3

2

10 8 111 14x

x x

.

6. 2 43 2 3 3 1x x x .

7. 2 411 6 22 11 4x x x .

8. 4

2

5 4 1 110 6 5

xx x

.

9. 2 472 4 9 9 64 1x x x . 10. 2 35 6 28 9 8x x x .

11. 2 34 15 45 7 27x x x . 12. 2 321 5 27x x x .

13. 2

3

36 664

x xx

.

14. 3 264 3 10 56x x x .

15. 2 4 24 1 1x x x x . 16. 2 4 27 5 7 7 1x x x x .

17. 4 2

2

1 13 5 3 3

x xx x

.

18. 2 4 24 4 1 2 16 4 1x x x x . 19. 2 36 1 5 3 14x x x x .

20. 2

3

6 14 35 52 5 26x x

x x

.

21. 2 3 24 17 99 4 24x x x x . 22. 2 42 2 1 1x x x . 23. 2 44 4 31 4 8 63x x x x .

24. 4 2

2

413 7 5 6

x xx x

.

25. 2 4 24 7 1 2 4 3 1x x x x . 26. 2 4 220 3 5 5 16 1x x x x .

27. 2

2

7 1 21

5 12 8x xx x

.

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22

BBii ttoonn 2233.. GGiiii pphhnngg ttrrnnhh 42 1 2 2 3 2 1 2x x x x x . Li gii.

iu kin 1 22

x .

Phng trnh cho tng ng vi 4 42 1 2 2 3 2 1. 2x x x x .

t 4 42 1 ; 2 0; 0x u x v u v ta c 2 22 3 2 02

u vu v uv u v u v

u v

2 1 2 1u v x x x .

4 4 112 2 1 2 2 2 1 16 26

u v x x x x .

So snh iu kin 1 22

x ta thu c tp nghim 11;16

S

.

BBii ttoonn 2244.. GGiiii pphhnngg ttrrnnhh 243 3 2 4 4 3 7 12 17 6x x x x x . Li gii.

iu kin 34

x .

Phng trnh cho tng ng vi 4 43 3 2 4 4 3 7 3 2. 4 3x x x x (*) t 4 43 2 ; 4 3 0; 0x u x v u v th (*) tr thnh

2 23 4 7 3 4 03 4u v

u v uv u v u vu v

4 43 2 4 3 3 2 4 3 1u v x x x x x .

4 4 7143 4 3 3 2 4 4 3 27 3 2 256 4 3943

u v x x x x x .

i chiu iu kin 34

x ta c tp hp nghim 714 ;1943

S

.

BBii ttoonn 2255.. GGiiii bbtt pphhnngg ttrrnnhh 242 5 2 3 30 17 2 6 1x x x x x . Li gii.

iu kin 25

x .

Bt phng trnh cho tng ng vi 4 42 5 2 3 5 2. 6 1 6 1x x x x . t 4 45 2 ; 6 1 0; 0x a x b a b thu c

2 22 3 2 2 0 2 0a ab b a a b b a b a b a b (1)

Ta c 4 42 0 5 2 6 1 5 2 6 15

x x x x x a b .

Do 4 4 311 2 2 5 2 6 1 16 5 2 6 174

a b x x x x x

i chiu vi iu kin 25

x , kt lun tp nghim 2 31;5 74

S .

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23

BBii ttoonn 2266.. GGiiii pphhnngg ttrrnnhh 3

38 3 42

x x x xx

.

Li gii 1. iu kin 0 2x . Phng trnh cho tng ng vi 2 3 2 22 4 3 4 4 2 3 . 4x x x x x x x x .

t 2 4 ; 0; 0x u x v u v ta thu c

2 22 3 2 0 2 02

u vu v uv u u v v u v u v u v

u v

2 2 24 4 4 0u v x x x x x x (V nghim).

22 22 4 2 4 4 2 0 2u v x x x x x x . So snh iu kin, kt lun phng trnh cho v nghim. Li gii 2. iu kin 0 2x . Phng trnh cho tng ng vi 2 32 4 3 4x x x x (1). Nhn xt 22 2 4 1 3 0x x x x nn

4 3 2 2 3 4 3 2

22 2 2 2 2

1 4 4 8 16 16 9 36 5 12 20 16 0

4 4 4 4 4 4 4 0 4 2 0 2


x x x x x x x x x x x x

So snh iu kin, kt lun phng trnh cho v nghim.

BBii ttoonn 2277.. GGiiii bbtt pphhnngg ttrrnnhh 22 42 1 2 1 5 1 1x x x x x . Li gii. iu kin 1 1x . Bt phng trnh cho tng ng vi 2 2442 1 2 1 5 1 . 1x x x x .

t 244 1 ; 1 0; 0x a x b a b ta c 2 22 2 5 2 2 2 2 2 0a b ab a a b b a b a b a b . Xt hai trng hp

2

2 244

244

1 151 1 16 15 0 1 12 0

2 1 1 16 15 0 15 152 0 11 1 16 161 2 1

1 1

xx x x x xa bx x x x

a b x xxx x

x

2

2 244

244

15 11 1 16 15 0 162 0 152 1 1 16 15 0 1 1 12 0 16

1 1 151 2 11

xx x xa b

x x x x x xa b

x xx xx

.

Kt lun tp nghim ca bt phng trnh l 1;1S . Nhn xt. Ngoi cch x l "th cng" phn cui bi ton 24, cc bn c th th sc vi cch s dng ng thc lin hp.

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24

BBii ttoonn 2288.. GGiiii pphhnngg ttrrnnhh 2 2 3 23 32 3 2 4 4x x x x . Li gii. iu kin x . t 3 32 ; 2x a x b , phng trnh cho tr thnh

2 23 4 3 0 3 03

a ba b ab a a b b a b a b a b

a b

2 2 0 4a b x x x (V nghim).

3 3 283 2 3 2 2 27 213

a b x x x x x .

Kt lun phng trnh c tp nghim 2813

S

.

BBii ttoonn 2299.. GGiiii pphhnngg ttrrnnhh 2 2 3 23 34 2 1 3 1 2 8 4 1x x x x . Li gii. iu kin x . t 3 32 1 ; 2 1x u x v , phng trnh cho tr thnh

2 22

4 3 8 2 2 3 2 3 0 2 2 3 02 3u v

u v uv u u v v u v u v u vu v

3 3 92 2 2 1 2 1 8 2 1 2 114

u v x x x x x .

3 3 352 3 2 2 1 3 2 1 8 2 1 27 2 138

u v x x x x x .

Phng trnh cho c hai nghim 914

x hoc 3538

x .

BBii ttoonn 3300.. GGiiii bbtt pphhnngg ttrrnnhh 3 3 32 2 26 9 4 6 9 5 9 0x x x x x x . Li gii. iu kin x .

Bt phng trnh cho tng ng vi 2 2 3 23 33 4 3 5 9x x x . t 3 33 ; 3x u x v ta thu c

2 24 5 4 0 4 0u v uv u u v v u v u v u v (1). Nhn xt 3 33 3 3 3x x x x u v .

Do 3 3 651 4 0 3 4 3 3 64 321

u v x x x x x .

Vy bt phng trnh cho c tp nghim 65;21

S .

Nhn xt. Cc bi ton t 23 30 v hnh thc gi chng ta t n ph a v phng trnh ng bc (bc hai), ngoi ra c th nng ly tha trc tip cng cho kt qu tng t. i vi lp bt phng trnh, cc bn ch chia cc trng hp chnh xc hoc linh hot s dng tp xc nh (iu kin c nghim) lp lun, nh gi nhn t, gim thiu cc nghim ngoi lai v mt s tnh ton cng knh, khng cn thit.

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25


1. 2 35 7 6 7x x x x . 2. 2 36 7 8 9 3 4x x x x . 3. 2 316 7 4 11 4x x x x .

4. 2 31 3 2 4x x x .

5. 2 47 10 14 5 4x x x .

6. 2 2 3 23 31 5 1 6 1x x x .

7. 2 3 22 2 3 5 5 3 2x x x x x . 8. 3 27 1 6 2x x . 9. 6 23 32 1 1 1x x x .

10. 2 4 23 9 3 3 1 0x x x x . 11. 2 35 4 3 5 5 3x x x x .

12. 22 41 2 1 3 1 1x x x x .

13. 243 2 8 3 11 5 6x x x x .

14. 245 5 6 11 1x x x x .

15. 2 2 3 23 31 3 1 4 1x x x .

16. 2 2 3 23 32 5 2 5 5 25x x x .

17. 44 11 3 7 3x x x x .

18. 2 2 3 23 310 3 2 7 3 2 3 9 4x x x .

19. 2 2 3 23 31 2 1 2 3 1x x x x .

20. 2 2 3 23 32 3 1 3 4 1 5 12 7 1x x x x .

21. 2 343 2 2 2 2 5 2 4x x x x x .

22. 22 32 33 32 2 2 4 8x x x x .

23. 22 32 33 35 1 8 1 3 1x x x x .

24. 22 32 33 317 3 4 3 9 13 27x x x x .

25. 2 2 32 2 4 23 36 1 5 1 1x x x x x x . 26. 42 3 2 2 3 3 2 2x x x x .

27. 4 2 25 3 3 8 1 1x x x x x .

28. 2 3 241 4 1 5 1x x x x x .

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26

BBii ttoonn 3311.. GGiiii pphhnngg ttrrnnhh 2 3 27 3 3 4 5x x x x x . Li gii. iu kin 3 2 24 5 0 1 5 5 0 1x x x x x x . Phng trnh cho tng ng vi 2 25 5 2 1 3 1. 5 5x x x x x x (1).

t 21 ; 5 5 0; 0x u x x v u v ta c

2 21 2 3 2 02

u vu v uv u v u v

v u

22

11 5 5

4 5 0x

u v x x xx x

(H v nghim).

22 2

1 12 2 1 5 5

5 5 4 4 9 0x x

u v x x xx x x x x

(H v nghim).

Vy phng trnh cho v nghim. BBii ttoonn 3322.. GGiiii pphhnngg ttrrnnhh 2 3 23 4 3 6 11 6x x x x x x . Li gii 1.

iu kin 3

1 2 3 01 2x

x x xx

Phng trnh cho tng ng vi 2 25 6 2 1 3 1. 5 6x x x x x x (1).

t 21 ; 5 6 0; 0x a x x b a b ta c

2 21 2 3 2 02

a ba b ab a b a b

a b

2 23 2

1 5 6 6 7 03 2

xa b x x x x x

x

2 2 21 41 21 412 1 4 5 6 4 21 25 0 ;8 8

a b x x x x x x

.

i chiu iu kin ta thu c bn nghim 21 41 21 413 2;3 2; ;

8 8S

.

Li gii 2.

iu kin 3

1 2 3 01 2x

x x xx

Nhn xt 1x khng l nghim ca phng trnh cho. Do phng trnh cho tng ng vi

2 2

2 2 5 6 5 65 6 2 1 3 1 5 6 2 31 1


t 2 5 6 0

1x x t t

x

ta c 21

2 3 1 2 02

tt t t t

t

2 23 2

1 1 5 6 6 7 03 2

xt x x x x x

x

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27

2 2 21 41 21 412 1 4 5 6 4 21 25 0 ;8 8

t x x x x x x

.

i chiu iu kin ta thu c bn nghim 21 41 21 413 2;3 2; ;8 8

S

.

BBii ttoonn 3333.. GGiiii pphhnngg ttrrnnhh 2 3 22 5 1 3 6 11 6x x x x x x . Li gii 1.

iu kin 3

1 2 3 01 2x

x x xx

Phng trnh cho tng ng vi 2 22 3 2 3 3 3 2 3x x x x x x (*) t 2 3 2 ; 3x x a x b th

2 2

2 01

2 02 02 3

4 04 4 9 2 02

4

a ba b a ba b

a b aba b a ba ab b ab a b

a b

2 2

2 2

2 5 1 0 2 5 1 01

3 2 3 4 5 0x x x x

x x x x x

(H v nghim).

2 2

2 2

2 5 1 0 2 5 1 02 1

4 3 2 3 4 13 11 0

x x x xx x x x x

(H v nghim).

Kt lun phng trnh cho v nghim. Li gii 2.

iu kin 3

1 2 3 01 2x

x x xx

Nhn xt 3x khng tha mn phng trnh ban u. Do phng trnh cho tng ng vi

2 2

2 2 3 2 3 22 3 2 3 3 3 2 3 2. 1 33 3


(*)

t 2 3 2 0

3x x t t

x

th (*) tr thnh: 21

2 3 1 0 1 2 1 0 12

tt t t t

t

Xt hai trng hp 2 21 3 2 3 4 5 0; 1 0t x x x x x , phng trnh v nghim.

2 21 4 3 2 3 4 13 11 0; 7 02

t x x x x x , phng trnh v nghim.

Kt lun phng trnh cho v nghim.

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28

Nhn xt. Cc bi ton 31 33 u c gii c bng php s dng n ph a v phng trnh ng bc v tm

t l gia hai n thng qua phng trnh bc hai. Ngoi ra cc bn c th s dng nng ly tha trc tip kt hp h s bt nh cng cho li gii "khe mnh, chp nhong, bt ng".

Quan st v thc hnh cc th d pha trc mt cch c h thng, dng ton ny c th tr nn quen thuc vi mt s bn hc sinh, hai bi ton 32 v 33 v hnh thc vn cha c iu g mi l, tuy nhin c mt s khc bit nho nh trong lp lun, im nhn trng tm ca hai th d ny l: a thc trong cn thc kh c th phn tch thnh cc nhn t c lp (nu khng chia trng hp theo iu kin xc nh).

C th trong hai bi ton 32 v 33 ta u c phn tch

2

3 2 2

2

1 2 3 1 5 6

6 11 6 1 2 3 2 1 3 2 4 3

1 2 3 3 2 3

x x x x x x

x x x x x x x x x x x x

x x x x x x

.

Xin nhc li kin thc c bn: 0

... . . ... ...0

AABC A B C

B

i vi bi ton 32 ta c quan st 2 23 4 5 6 2 1x x x x x nn hng xy dng n ph ng bc

s l 3 2 26 11 6 1 2 3 1. 5 6x x x x x x x x x (*).

Lu iu kin 3

1 2 3 01 2x

x x xx

D thy (*) xy ra hin nhin. Kt qu thu c li gii 1 bi ton 32. i vi bi ton 33 ta c quan st 2 22 5 1 2 3 2 3x x x x x nn hng xy dng n ph ng bc

s l 3 2 26 11 6 1 2 3 3 2. 3x x x x x x x x x (**).

Lu iu kin 3

1 2 3 01 2x

x x xx

Trong trng hp ny (**) khng ng, ni khc n ch xy ra khi 3x . iu ny t ra mt nghi vn: Phi chng chng ta ang gp mt chng ngi vt ?

Bi v hai nhn t ny lun dnh vi nhau nh "hnh vi bng", nu tch ra s rt phc tp. Vy c nn dng li hay khng ? Khng, trng k khng chin nht nh thng li ! on kt, on kt, i on kt ! Thnh cng, thnh cng, i thnh cng ! C mt s phng n la chn nh sau

Dng v lc, tch bit hon ton hai nhn t bng cch chia trng hp Trng hp 1: 3x , (**) xy ra, chng ta "mn nguyn" vi hai n ph

2 2 23 2 ; 3 0; 0 2 3x x u x v u v u v uv Trng hp 2: 1 2x , (**) khng xy ra nhng li c mi vu hi bt ng

3 2 26 11 6 1 2 3 3 2. 3x x x x x x x x x

Chng ta khng c "hi lng" lm vi hai n ph 2 3 2 ; 3 0; 0x x a x b a b . Phng trnh khi c dng 2 22 3 2 0 0a b ab a b a b a b S d nh vy v 0; 0a b . V tt nhin trng hp ny s v nghim. Phng n 1 rt kh thi, song cha phi ti u v phi chia hai trng hp, hai ln t n ph, mt khc mt trng hp mang tnh cht "th tc" v l do v nghim, song nu khng c n th bi ton coi nh khng trn vn v "t tng".

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29

Tha hip, tm l chin, gin tip: Khng tch bit ring bit hai nhn t, vn chng "dnh kp" vo nhau. S dng php t n ph pha trong cn, sau php bnh phng trc tip (ko theo iu kin) chng ta c quyn sinh quyn st, thc hin a v nhn t, lc ny c "dnh kp" cng khng quan trng na. Kt qu chng ta thu c li gii 1 bi ton 32. Tuy nhin vic gii cc h hn tp cng gy khng t tr ngi.

Chin tranh du kch, mm do, linh hot: Khng tch ring hai nhn t, nhng mc tiu trung gian l tm t l gia hai nhn t, vy ti sao khng chng "dnh kp" vi nhau theo "t l" y, khng nh hng nhiu n vic c lp hay ly khai phc tp. Chng ta hy tc thnh cho h !

Cc bn lu : AB xc nh th 0A BB

cng xc nh.

Kt qu tng thu c li gii 2 bi ton 32. Nhn xt 3x khng l nghim, y s l "ci c" chia hai v cho biu thc 3x , h qu dn n n ph rt gn gng. C th ni vi dng ton ny, phng n 3 l ti u. Cc bn hon ton c th chia hai v cho 2 3 2 0x x , nhng khng nn, v nh vy s phi xt hai trng hp 1; 2x x c l nghim ca phng trnh hay khng. Gi nh c mt ch kin "phi" b t nh ny sang nh kia (hai nh ca hai gc nhn) ca mt tam gic vung thc, kin s chn b theo cnh huyn hay theo hai cnh gc vung, hay l theo ng gp khc hoc mt ng cong no ? Vi iu kin trn cc con ng ca ta khng c mt git mt no. Hy vng cc bn thng minh hn kin nh ! S linh hot ny cc bn c th p dng trong vic gii bt phng trnh cha cn, loi b i kh nhiu tiu tit khng cn thit. Mi qu c gi theo di cc th d tip theo.

BBii ttoonn 3344.. GGiiii bbtt pphhnngg ttrrnnhh 2 33 4 5 5 7 6x x x x x . Li gii.

iu kin 33

7 6 0 1 3 2 02 1

xx x x x x

x

Bt phng trnh cho tng ng vi 2 23 2 3 2 2 5 2 3. 2x x x x x x t 2 2 3 ; 2 0; 0x x u x v u v ta c 2 23 2 5 3 2 0u v uv u v u v Xt hai trng hp

2 2

2 2

3 292 3 20 3 5 0 2

3 2 0 9 2 3 4 2 9 22 35 0 3 292

xx x xu v x xu v x x x x x

x

2 2

2 2

2 3 20 3 5 0 11 2 109 11 2 1093 2 0 9 99 2 3 4 2 9 22 35 0

x x xu v x xx

u v x x x x x

.

Kt hp iu kin ta c nghim 3 29 11 2 109 3 292; ; 1 ;2 9 2

S

.

BBii ttoonn 3355.. GGiiii pphhnngg ttrrnnhh 2 32 13 36 7 24 32x x x x x . Li gii. iu kin 3 24 32 0x x . Nhn xt 4x khng l nghim ca phng trnh ban u.

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Phng trnh cho tng ng vi

2 2

2 2 4 8 4 82 4 8 5 4 7 4 8 4 2. 5 74 4


(1)

t 2 4 8 0

4x x t t

x

th 21

1 2 7 5 0 1 2 5 0 52

tt t t t

t

2 21 4 8 4 3 4 0 1 4 0 4;1t x x x x x x x x .

2 25 4 4 8 25 4 4 9 68 0; 1007 02

t x x x x x . Phng trnh v nghim.

So snh iu kin, ta thu c tp nghim 4;1S . BBii ttoonn 3366.. GGiiii bbtt pphhnngg ttrrnnhh 2 34 25 14 3 31 30x x x x x . Li gii.

iu kin 35

31 30 0 1 5 6 06 1

xx x x x x

x

Bt phng trnh cho tng ng vi

2 2

2 2

4 6 5 6 3 6 5 6

4 6 5 6 3 6 5. 6

x x x x x x

x x x x x x

t 2 6 5 ; 6 0; 0x x a x b a b ta c

2 24 0

1 4 3 4 0a b

a b ab a b a ba b

2

16 5 0

4 0 0 56 0

6

xx x

a b a b xx

x

(H v nghim).

2 2 7 53 7 536 5 6 7 1 02 2

a b x x x x x x .

Kt hp iu kin ta c nghim 7 53 7 53;1 5;2 2

S

.

BBii ttoonn 3377.. GGiiii bbtt pphhnngg ttrrnnhh 2 3 23 14 14 2 26 24x x x x x x . Li gii.

iu kin 3 24

26 24 0 1 6 4 06 1

xx x x x x x

x

Bt phng trnh cho tng ng vi 2 23 5 6 4 2 5 6 4x x x x x x t 2 5 6 ; 4x x u x v ta thu c

2 2 2 23 03 0 3 0

3 29 09 6 4 9 10 0

u vu v u vu v uv

u v u vu uv v uv u uv v

Xt hai trng hp

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31

2

2

2

22 9343 14 14 03 090 4 2 0

22 9349 0 9 44 50 09

x xu v xu v x xu v xx x

2

2

2

3 14 14 03 00 4 2 0

9 0 9 44 50 0

x xu vu v x xu v x x

(H v nghim).

Vy ta c nghim 22 9346; 4;9

S

.

BBii ttoonn 3388.. GGiiii pphhnngg ttrrnnhh 2 3 23 5 5 2 2 11 12x x x x x x . Li gii.

iu kin 4

1 3 4 03 1

xx x x

x


2 2

2 2 2 3 2 33 2 3 4 2 2 3 4 3. 1 24 4


(1)

t 2 2 3 0

4x x t t

x

ta c

2 2 21 3 1 2 1 3 1 0 1 2 3 4 1 0t t t t t x x x x x (V nghim). Vy phng trnh cho v nghim. BBii ttoonn 3399.. GGiiii pphhnngg ttrrnnhh 2 35 11 2 2 4x x x x x . Li gii.

iu kin 2 24 0 2 0x

x xx


2 2

2 2 2 25 2 2 6 4 5. 1 62 2

x x x xx x x x xx x

t 2 2 0

2x x t tx

ta thu c 21

5 1 6 1 5 1 05 1t

t t t tt

Xt hai kh nng xy ra 2 21 2 2 2 0t x x x x x . Phng trnh v nghim.

2 2 49 2201 49 22015 1 25 2 2 25 49 2 0 ;50 50

t x x x x x x

Kt hp vi iu kin ta thu c nghim 49 2201 49 2201;50 50

S

.

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32


1. 2 3 24 15 10 5 6 11 6x x x x x .

2. 2 24 19 5 4 5 1 5x x x x x . 3. 2 3 26 7 4 7 20x x x x x .

4. 2 25 22 12 3 4 2 1x x x x x . 5. 2 3 27 24 19 4 7 13 4x x x x x .

6. 2 22 3 4 1 3 5x x x x x . 7. 2 25 6 1 6 1x x x x . 8. 2 33 13 7 5 12x x x x .

9. 2 23 17 22 2 6 5 7x x x x x . 10. 2 3 23 10 5 5 6 11 6x x x x x . 11. 2 33 8 13 5 7 6x x x x . 12. 2 3 23 10 2 8 9 26 24x x x x x .

13. 2 22 4 2 2 1 3x x x x x . 14. 2 22 3 5 2 2 4 5x x x x x . 15. 2 25 12 4 3 2 3x x x x x . 16. 2 3 22 7 3 5 5 22 23 6x x x x x .

17. 2 26 7 2 7 3 2x x x x x . 18. 2 3 212 17 10 5 3 22 20 24x x x x x .

19. 2 3 22 9 22 9 3 12x x x x x .

20. 2 3 22 1 2 1 2 2x x x x x x . 21. 2 310 10 6 2 1x x x x .

22. 2 3 218 13 46 11 6 29 33 10x x x x x . 23. 2 3 24 19 4 7 7 15x x x x x . 24. 2 3 29 2 4 2 3 11 6x x x x x .

25. 2 23 4 2 4 1 2x x x x . 26. 2 3 24 7 17 7 4 5 6x x x x x . 27. 2 3 24 12 5 6 7 1x x x x .

28. 2 3 25 5 16 11 5 31 50 24x x x x x . 29. 2 3 24 22 6 4 5 47 12 11x x x x x . 30. 3 2 5 4 3 22 3 2 2 2 3 2 3 1x x x x x x x x .

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BBii ttoonn 4400.. GGiiii pphhnngg ttrrnnhh 21 2 1 3 8 4x x x x x . Li gii 1.

iu kin 2

2 1 03 8 4 0

xx x

Phng trnh cho tng ng vi 21 2 1 3 2 1 2 1x x x x x . t 1 ; 2 1 0; 0x a x b a b ta thu c

2 2 2 2 2 23 2 3 0a b a b a ab b a b a a b a b (Do 0a ).

Do 2

11 2 1 0

2 1 2 1x

x x xx x x

Kt hp iu kin ta thu c nghim 0x . Li gii 2. iu kin 23 8 4 0;2 1 0x x x . Phng trnh cho tng ng vi

2 2 2

2

2

2 1 2 1 2 1 2 1 3 8 4 2 1 2 1 2 4 2

11 0 1

11 2 1 102 1 1

2 1 2 1


xx x

xx x xxx x

x x x

So snh iu kin ta thu c nghim 0x . BBii ttoonn 4411.. GGiiii pphhnngg ttrrnnhh 21 2 3 5 12 8x x x x x . Li gii 1.

iu kin 2

32

5 12 8 0

x

x x

Phng trnh cho tng ng vi 21 2 3 5 2 1 2 3x x x x x (1) t 1 ; 2 3 0; 0x u x v u v th

2 2 2 2 2 2 2 2

2 2

1 5 2 5 2 0 2 0

1 11 2 3 2

2 1 2 3 4 4 0

u v u v u uv v u v u uv v u v u v

x xu v x x x

x x x x x

Kt hp iu kin, phng trnh c nghim duy nht 2x . Li gii 2.

iu kin 235 12 8 0;2

x x x .


2 2 2

2 2

2

2 1 2 3 2 1 2 3 5 12 8 2 1 2 3 4 12 10

1 2 3 2 6 5 1 2 3 2 2 1 2 3

2 3 2 31 2 3 2 1 2 3 2 21 1


x x x x x x x x x

x xx x x xx x

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34

t 2 3 01

x t tx

th 22 2 1 2 0 1 2 3 1 2t t t t t x x x .

Kt hp iu kin, suy ra phng trnh c nghim duy nht 2x . BBii ttoonn 4422.. GGiiii pphhnngg ttrrnnhh 22 2 1 1 4 2 4x x x x x . Li gii 1.

iu kin 12


222 1 2 1 4 4 1 3 2 1 2 1 2 1 2 1 3 2 1x x x x x x x x x (*).

t 2 1 ; 2 1 0x u x v v ta c 2 2

2 2 2 2

003

02 3

u vu vu v u v

v v uu uv v u v

Xt hai trng hp xy ra

0 2 1 2 1 0

0 2 1 0u v x xv x

(H v nghim).

Do 0v nn 2

110 322 1 2 1 222 3 04 4 1 2 1

xxu vu v x x x

u v x xx x x

.

i chiu iu kin ta thu c nghim 32

S

.

Li gii 2.

iu kin 12


2

2 2

2

2 1 2 1 4 2 4

2 1 1 2 1 2 02 1 2 1 2 0

4 4 1 2 1 2 2 1 2 1 4 2 4 2 2 1 2 1 4 2

102 1 1 32

22 1 2 12 1 2 1 2 1 4 4 1 2 1

x x x x

x xx x

x x x x x x x x x x

xx xx

x xx x x x x x

i chiu iu kin ta thu c tp nghim 32

S

.

BBii ttoonn 4433.. GGiiii pphhnngg ttrrnnhh 21 2 1 2. 1 2 1x x x x x . Li gii 1.

iu kin 12

x .

t 1 ; 2 1 0x a x b b phng trnh cho tr thnh

2 2

22 2 2 2

00 02.

2 2 2 0

a ba b a ba b a b a b

a ba b ab a b a b

.

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35

211

1 2 1 44 02 1 2 1

xxx x x

x xx x x

.

Kt hp iu kin ta c nghim 4S . Li gii 2.

iu kin 12

x . rng ;a b ta c

2 22 2 2 2 2 2 2 20 2 2 2 2a b ab a b a ab b a b a b a b (*). p dng bt ng thc (*) ta c

2 2

2

2

1 2 1 2 1 2 1

1 2 1 2 1 2 1

1 2 1 2. 1 2 1 1

x x x x

x x x x

x x x x

Do phng trnh cho c nghim khi ti (1) xy ra du ng thc

2

1 2 1 011 2 1 0

1 44 01 2 1 2 1 2 1

x x xx xx x

x xx x x x x

.

Vy phng trnh cho c nghim duy nht 4S .

BBii ttoonn 4444.. GGiiii bbtt pphhnngg ttrrnnhh 21 2 1 3x x x x x . Li gii 1. iu kin 0x . Bt phng trnh cho tng ng vi

221 2 1 2 2 1 2 1 2x x x x x x x x x (1). t 1 ; 1; 0x u x v u v th (1) tr thnh

22 2 2 2 2 2

2 2 2

2 2 2 0

0 1 2 1 1 0

u v u v u uv v u v u v

u v u v x x x x x x x

Kt lun bt phng trnh cho v nghim. Li gii 2. iu kin 0x . p dng bt ng thc Bunyakovsky cho hai cp s dng 1;1 , 1 ;x x ta c

2 22 2 2 21 1 1 1 2 1 3 1 2 1 3x x x x x x x x x x (2) Bt phng trnh cho c nghim khi v ch khi (2) xy ra ng thc

2

001 312 41 1

xx

x x x

(H v nghim).

Vy bt phng trnh cho v nghim. Li gii 3. iu kin 0x .

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36

Nhn xt 0x khng l nghim ca bt phng trnh cho.

Xt 0x , bt phng trnh ban u tr thnh 1 11 2 3x xxx

t 21 10 2x t t t xxx

; ta c

22 2 21 2 1 2 1 2 2 1 0 1t t t t t t t 11 1t xx

(3)

Mt khc, p dng bt ng thc AM GM ta c 1 12 . 2x xx x . Dn n (3) v nghim.

Vy bt phng trnh cho v nghim. Nhn xt.

Cc bi ton t 40 n 44 u gii c bng cch s dng n ph a v phng trnh ng bc (bc hai), im nhn trng tm l khng th a v dng ng bc trc tip m trc tin phi thc hin php bin i tng ng v nng ly tha hp l. Mc d cng mt phng hng nhng mi li gii u c nhng nt ring ng lu (hai li gii ca bi ton 43). Ngoi ra cc bn c th s dng bin i tng ng, nng ly tha trc tip hay a v phng trnh tch, vn rt kh thi i vi lp bi ton dng ny.

Hai bi ton 43 v 44 tc gi trnh by phng php s dng nh gi hm s bt ng thc cc bn c cch nhn ton din, y v bao qut hn trong qu trnh la chn li gii. Hai li gii 2 tng ng ca bi 43 v 44 c cng bn cht s dng hnh thc bt ng thc Bunyakovsky, tuy nhin li gii 2 bi 40 c lp lun theo hng ng thc mang tnh cht "c bn", v l phn no c a chung hn.

BBii ttoonn 4455.. GGiiii pphhnngg ttrrnnhh

2

3 11 2 3 4

x x xx x

.


2 2 23 1 2 3 4 2 2 3 4 2 2 4 4x x x x x x x x x x x x x t 2 ; 0x u x v u ta thu c

2 222 2 2 2

0 02

2 2 0

2 1 2 0 1 1

u v u vu v u v

u v uv u v u v

u v x x x x x x

Kt hp iu kin ta c nghim duy nht 1x . Li gii 2. iu kin 0x . Bin i v 2 23 1 2 3 4 2 2 3 4x x x x x x x x .

t 0x t t ta c

2 4 2 2

4 2 2 4 2

02 2 3 4 2 0

4 4 2 2 2 3 4

tt t t t t t

t t t t t t t

4 3 24 3 2

0 0 21 2 0

4 2 2 3 02 3 4 4 0

t tt t

t t t tt t t t

.

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37

Nhn xt 0t khng l nghim ca h. Do vy ta c

2 22

2

2

0 2 0 20 24 2 2 2 22 3 0 2 1 0 1 0

0 2 0 20 21 12 1 2 01 0 2 0

t tt

t t t t tt t t t t

t ttt x

t tt t tt

So snh iu kin, kt lun nghim 1x . Li gii 3. iu kin 0x . Phng trnh cho quy v 2 23 1 2 3 4 2 2 3 4x x x x x x x x . Nhn xt 0x khng tha mn phng trnh trn.

Xt trng hp 0x thu c 2 41 2 3x xxx

. t 22 4 4x t x txx

. Suy ra

222 2

11 21 2 1 1 12 1 2 2 1 0

22 1 2 0 1

1

ttt t t x

t t t t x

xx x x x x

x

i chiu iu kin, kt lun nghim 1x . Li gii 4. iu kin 0x . a phng trnh v dng 2 23 1 2 3 4 2 2 3 4x x x x x x x x . p dng bt ng thc Bunyakovsky ta c

2 22 2 2

2 2

2 1 1 2 2 3 4

2 2 3 4 2 2 3 4

x x x x x x

x x x x x x x x

Phng trnh bi c nghim khi v ch khi du ng thc xy ra, ngha l

2 1 2 0 1x x x x x . Vy phng trnh cho c nghim duy nht. Nhn xt.

Bi ton 45 sau khi bin i c dng tng t bi ton 44. Li gii 1 s dng n ph a v phng trnh ng bc quen thuc, li gii 2 v bn cht nng ly tha trc tip c ko theo iu kin, vic t t n ph 0x t t ch lm cho bi ton gn gng v hnh thc. H qu a v hng trnh a thc bc bn, tuy nhin c mt s may mn xut hin, bi y l phng trnh i xng hi quy, phng php gii c l rt quen thuc vi mt s bn. Kt qu thu c hon ton trng hp vi li gii 1.

Ngoi ra hnh thc bi ton 45 cng c s c bit do y l trng hp xy ra ng thc ca bi ton p dng bt ng thc Bunyakovsky, cc bn c th gii tng t theo li gii 2 bi ton s 43. Phng php t n ph trong li gii 3 cc bi ton 44 v 45 thng xut hin trong cc k thi tuyn sinh i hc mn Ton nhng nm gn y, c th l thi tuyn sinh i hc khi A nm 2010 mn Ton chnh thc v thi tuyn sinh i hc khi B nm 2012 mn Ton chnh thc. V dng ton ny, tc gi ch xin nhc li, hin ti c trnh by cn k ti L thuyt t n ph cc phn 2 v 3.

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38

BBii ttoonn 4466.. GGiiii bbtt pphhnngg ttrrnnhh

2

2 12 5 9 1

x x xx x

.

Li gii 1.

iu kin

2

2

05 9 0 0

2 5 9 1

xx x x

x x

.

Nhn xt rng:

22

2

2 10 172 5 9 1 0,2 5 9 1

x xx x xx x

.


2 2 22 2 5 9 1 3 2 5 9 3 2 6 9x x x x x x x x x x x x x . t 3 ; 0x a x b b ta thu c 2 22a b a b (1) Ch rng ;a b ta c

2 2 2 2 2 2 2

2 2 2 2 2

0 2 2 2

2 2

a b ab a b a ab b a b

a b a b a b a b a b

Do (1) nghim ng vi ; 0a b . Kt lun nghim ca bt phng trnh l 0x . Li gii 2. iu kin 2 20; 5 9 0;2 5 9 1 0x x x x x x .

Nhn xt 2

2 2 25 92 5 9 1 2 10 17 2 0 2 5 9 1 02 2

x x x x x x x x

.


2 22 2 5 9 1 3 2 5 9x x x x x x x x (1)

D thy (1) tha mn vi 0x . Xt trng hp 0x ta c 3 91 1 2 5x xxx

t 23 9 6x t x t

xx ta thu c

2

22 2

111111 2 11

2 1 2 2 1 0

tttttt t tt

t t t t

.

Vy bt phng trnh cho c nghim 0x . Nhn xt. Xin nhc li hai dng tng qut ng bc hai v dng ng bc ba nh cc bi ton trc

2 2 0af x bf x g x cg x ; 2 2 2 3 0af x bf x g x cf x g x dg x .

p dng cho dng tng qut cha cn: 2 2af x bg x c df x eg x . Cc bn thc hin bnh phng hai v km theo iu kin quy v dng phng trnh ng bc bc hai c hnh thc r rng hn.

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