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T.s Nguyn Ph Khnh Lt http://www.toanthpt.net

PHNG TRNH V T

Dng cbn

Gii phng trnh :2

4 3 2 1

x 4 x 2

=

22

2 22

2 1 4 x0 0 x 40x 24 3 2 1 2x x 22

4 3 4 2 1x 4 x 2 4 3 2 1 1 0x

x 4 x x 4x 4 x 2

< = = = = + =

Gii phng trnh :x 6

x 6 x 9 x 6 x 923

++ =+

t 2t x 9, t 0 x t 9 9= = +

Phng trnh cho vit li :

2

2

2

t 4 0t 2

0 t 36 t 3 6 t 3 t 32 t 4

t 12t 32 0 t 8t 3

== nn ( ) ( ) ( )* t 2 x 1 3 x 2 x 1 3 x 0 x 1, x 3 = + + = + = = =

Ch : Cho hai s a 0,b 0 nu t a b= + th ( )a b t 2 a b+ + ( i s 9)

D thy

( ) ( )AM GM

2 2t a b t a b 2 ab a b t a b 2 ab 2 a b a b t 2 a b

= + = + + + = + + + + +

AM GM vit tt bt ng thc trung bnh cng trung bnh nhn.
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Gii phng trnh : ( ) ( )2 24x 1 x 1 2x 2x 1 1 + = + +

( ) ( ) ( )2 2 2 24x 1 x 1 2x 2x 1 4x 1 x 1 2 x 1 1 + = + + + = + +

t 2t x 1, t 1= + Phng trnh ( ) ( ) ( ) ( ) ( )2 21 4x 1 t 2t 2x 1 2t 4x 1 t 2x 1 0 2t 1 t 2x 1 0 = + + = + =

( )22

2

11 2x 1 0 xt 1 4x22

3x 1 2x 1t 2x 1 3x 4x 0

> >= < = + = = =

Gii phng trnh : ( ) ( )42 2 21 2x x 1 2x x 2 1 x 2x 4x 1+ + = +

iu kin phng trnh c ngha : 22 0 0 2 x x x .

( ) ( )42 2 21 2x x 1 2x x 2 1 x 2x 4x 1+ + = +

( ) ( ) ( ) ( )( )42 2 21 1 x 2x 1 1 1 x 2x 1 2 1 x 2 x 2x 1 1 + + + + = +

( ) ( ) ( ) ( ) ( )2 2 4 2

1 1 x 1 1 1 x 1 2 1 x 2 x 1 * + + =

t ( ) [ ] [ ] ( )2

t x 1 , x 0;2 t 0;1 a=

Phng trnh ( ) ( ) ( )2

* 1 1 t 1 1 t 2t 2t 1 ** + + =

iu kin phng trnh c ngha : ( )1

2 1 02

t t b .T ( ) ( )1

, ;12

a b t .

Vi1

;12

t , bnh phng 2 v phng trnh ( )** ta c

( ) ( )2 2

4 4 31 11 t 2t 2t 1 2 2t 1t t t+ = + =

( )

4 3

2

1 121

;1 22

2 2 1 2

= + = = =

VTt t tt VT VP

VP t

xy ra khi 1 2= =t x

Vy phng trnh c nghim 2=x .

Gii phng trnh : 2 4 23

x 3x 1 x x 13

+ = + +

( ) ( ) ( )( )

2 4 2 2 2 2 23 3x 3x 1 x x 1 2 x x 1 x x 1 x x 1 x x 13 3

+ = + + + + + = + + +

( )2 2

2 2

x x 1 3 x x 12 1 0 *

x x 1 3 x x 1

+ + + =

+ + + +

t2

2

x x 1t ,0 t 1

x x 1

+= <

+ +

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Phng trnh ( )2

2 22

3t 0

3 x x 1 33* 2t t 1 0 x 2x 1 0 x 13 x x 1 33

t3

= x .

t ( )1

, 1 0 1= > < < x x y ay

( ) ( )2 22 2

x 35 1 1 35 35x 1 y 1 y y 1 y 2

12 y 12 12x 1 1 y+ = + = + =

t ( )2

2 2 11 1 32

= + =

tt y y y y vi 0 1 1 2< < < y t

Phng trnh ( )2 vit li :

(

22

7t35 t 1 5t . 35t 24t 35 0

512 2t 1; 2

7

== =

=

( ) ( )

22

2 2 2 4 2

2

16 44911 12 144 144 25 5251 1 0

9 32 2 25 625 625

25 5

= = = = = = + =

= =

y yt

y y y y y y b

y y

T ( ) ( )a v b suy ra ( )5 4 5 3

; ; , ;

4 5 3 5

=

x y

Vy phng trnh cho c nghim :5 5

,4 3

= =x x

Ch : Vi iu kin 1>x gi lin tng bi ton ny c cch gii lng gic , vi1

cos=x

thoc

1

sin=x

t

Gii phng trnh : 2x 4x 3 x 5 = + iu kin phng trnh c ngha : 5 0 5+ x x

( )22x 4x 3 x 5 x 2 7 x 5 = + = +

t ( )2

y 2 x 5, y 2 y 2 x 5 = + = +

Ta c h :

( )

( )

( )

( )( )

( )

( )

2

2 2

22

x 2 y 5x 2 y 5 x 2 y 5 x y 0 5 29

xy 2 x 5 x y x y 3 0 2x 2 y 5

x 1y 2 y 2 x y 3 0

y 2

= + = + = + = + = = + + + = = + = + + =

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Gii phng trnh : 22x 15 32x 32x 20+ = +

iu kin phng trnh c ngha :15

2 15 02

+ x x .

( )222x 15 32x 32x 20 2x 15 2 4x 2 28+ = + + = +

t ( )21

4y 2 2x 15, y 4y 2 2x 152+ = + + = + Ta c h :

( )

( )

( ) ( )

( )

( )

( )

2

2

22 2

4x 2 2y 15

x y 14y 2 2x 15 x y 8x 8y 9 0 x24x 2 2y 154x 2 2y 15 4x 2 2y 15

9 2218x 8y 9 0 x1 1y y 16

2 2 1y

2

+ = + = + = + + + = = + = + + = + + = + + + = =

Dng tng hiu bnh phng

Gii phng trnh : ( ) ( )4x 1 x 2 x 1 x 2 x 1 x 1+ + =


0 11 0

xx

x.

( ) ( ) ( )( ) ( )( )4 4x 1 x 2 x 1 x 2 x 1 x 1 x 2 x 1 x 1 x x 2 x 1 x 1 x 0+ + = + + =

( ) ( ) ( )( )2 2

4 4 4 4 4 4x 1 x x 1 x 0 x 1 x x 1 x x 1 x x 1 x 0 = + + =

( )

( )

4 4

4 4

x 1 x x 1 x 0 1

x 1 x x 1 x 0 2

+ = + =

Phng trnh

( )4 4 4 41 1

x 1 x x 1 x 0 1 1 x 1 x x x 04 4

+ = + + =

( )( )2 2

4 4 4 4 4 41 11 x x 0 1 x x 1 x x 1 02 2

= + =

( )( )

4 4

4 41 x x 0 a1 x x 1 0 b

= + =

( )4 4 4 41

1 x x 0 a 1 x x 1 x x x2

= = = =

( ) 4 42 34 4 4 4 41 x x 1 0 b 1 x 1 x 1 x 1 4 x 6 x 4 x x + = = = + +

( ) ( )( )4 4 43 2 24 4 4 4 4x x 2 x 3 x 2 0 x x 1 x x 2 0 + = + =

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4

44

4

4 2 4

x 0x 0 x 0

x 1 0x 1x 1

x x 2 0

= = =

= == + >

Phng trnh( )4 4 4 4

1 1x 1 x x 1 x 0 2 x x 1 x 1 x 0

4 4 + = + + + + =

( )( )2 2

4 4 4 4 4 41 1x 1 x 0 x 1 x x 1 x 1 02 2

+ + = + + =

4 44 4

4 4

x 1 x 0 1x 1 x x 1 x x

2x 1 x 1 0

= = = =

+ + >

Vy phng trnh cho c 3 nghim1

x 0, x , x 1.2

= = =

Dng dng bt ng thc

Gii phng trnh : 2 2 2x x 1 x x 1 x x 2+ + + + = +


2

x x 1 0

x x 1 0

+

+ + .

p dng bt ng thc trung bnh cng trung bnh nhn ,

( )

( )

2 22 2

2 2

2 22 2

1 x x 1 x xx x 1 1. x x 1

2 2 x x 1 x x 1 x 11 x x 1 x x 2

x x 1 1. x x 1 2 2

+ + ++ = + = + + + + +

+ + + + + + + = + + =

Phng trnh : ( )22 2 2 2x x 2 x x 1 x x 1 x x 2 x 1 x 1 0 x 1 + = + + + + + + =

Vp phng trnh cho c nghim x 1=

Gii phng trnh : 2 2 22x x 3x 3x 1 x 2x 3 + + + = +


2

2x x 0

3x 3x 1 0

+ + .

p dng bt ng thc trung bnh cng trung bnh nhn ,

( )

( )

( )

2

2 2

2 22 2

222 2

1 2x x2x x 1. 2x x 2

1 3x 3x 1 3x 3x 23x 3x 1 1. 3x 3x 1

2 2

x 1x 3x 2VT 2x x 3x 3x 1 2 2

2 2

+ =

+ + + + + + + = + + =

+ + = + + + =

( )22VP x 2x 3 x 1 2 2= + = +

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2

2

x 1 0

VT VP 2 khi 1 2x x x 1

1 1 3x 3x

== = = = = +

Vy phng trnh c nghim x 1= .

Dng khcGii phng trnh :

2 2a) x 4 x 2 3x 4 x+ = + ( ) ( )b) x 1 x 4 x 1 x 4 5+ + + + = 2c) 4x 1 4x 1 1 + = Hng dn :

2 2a) x 4 x 2 3x 4 x+ = +

t 2;4 2 += xxxt c 20';4

1'2

==

= xtx

xt t 2;2 2 Phng trnh :

2 2x 4 x 2 3x 4 x+ = + 3

142,2,00823 2

==== xxxtt

( ) ( )b) x 1 x 4 x 1 x 4 5+ + + + =

t [ ]t x 1 4 x; x 1;4 t ' 0 t 5; 10 = + + =

( ) ( )x 1 x 4 x 1 x 4 5+ + + + = 3052

52===

+ xx

tt

2c) 4x 1 4x 1 1 + =

>+=

0)(';1414)(

2

1

2xfxxxf

x

2

1)

2

1(1)( === xfxf

Nhn lng lin hp

Gii cc phng trnh :

a) ( )( )x 1 1 x 1 2x 5 x+ + + + = b) 2 22x 3x 5 2x 3x 5 3x+ + + + =

a) ( )( )x 1 1 x 1 2x 5 x+ + + + =

Nhn c hai v phng trnh vi x 1 1+ ta c phng trnh h qu

( ) ( ) ( ) ( )x x 1 2x 5 x x 1 1 x x 1 2x 5 x 1 1 0 + + = + + + + =

( ) ( )x 0 x 0

x 1 2x 5 x 1 1 0 x 2

= = + + + = =

Th li ta thy x 2= tha mn .

b) ( )2 22x 3x 5 2x 3x 5 3x 1+ + + + =

Nhn c hai v phng trnh vi 2 22x 3x 5 2x 3x 5+ + + ta c phng trnh h qu :

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( )( )

2 2

2 2

x 06x 3x 2x 3x 5 2x 3x 5

2x 3x 5 2x 3x 5 2 2

== + + +

+ + + =

Ly ( ) ( )1 2+ ta c ( ) ( )22 22 2x 3x 5 2 3x 4 2x 3x 5 2 3x+ + = + + + = + phng trnh h qu

2 2 2 x 4

8x 12x 20 4 12x 9x x 16 x 4

=

+ + = + + = =

Kim tra li cc nghim x 4; x 4; x 0= = = ta thy x 4= tha mnGii cc phng trnh :

a)2x

x 1 1 x 24

+ + = b)2 24x 1 2x 1 1 x 2x + = +

a)2x

x 1 1 x 24

+ + =

c gi thy qu quen thuc bi ton trn gii bng phng php s dng bt ng thc , nh gi ,lng gic nay ti gii thiu cch gii nhn lng lin hp .

iu kin phng trnh c ngha :1 0

1 11 0

+

xx

x.

V 1 1 x nn2x

2 04

>

Phng trnh cho

( )4 2

2 2 2 2x x2 2 1 x 4 x 2 1 1 x x 116 16

+ = + =

( )( ) ( )

22 2 2 2x

2 1 1 x 1 1 x x 1 1 1 x16

+ = +

( ) ( )

2

22 2 2 2

2

x 0x

2x x 1 1 1 x x 0x16 2 1 1 1 x

16

= = + = = +

V x 0 nn ( )2

22

2

x1 1 x

16 1 1 1 x 216

1 1 x 2

< +

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Nu1

2x th phng trnh cho

( ) ( ) ( ) ( ) ( )2 1 2 1 2 1 1 2 1 2 1 2 1 1 1 + + + = + + + = x x x x x x x x

( ) ( )( ) ( )( )2 1 1 2 1 1 2 1 1 2 1 1 2 1 1 2 1 1 = + + + = + + + x x x x x x x x

( ) ( )( ) ( )12 1 2 1 1 2 1 1 1

2 2 1 2 1 1 0= = + + + =+ + + =

x x x x x x

x x

Vy phng trnh cho c nghim1

, 12

= =x x

Dng o hm

Gii phng trnh : 6 23 x 7 x 2x 1 2+ + + = 33

6 23 3 3

33

x 7 x 1 2

x 1

x 7 x 2x 1 2 x 7 x 1 2 x 7 x 1 2

x 1

+ + =

+ + + = + + = + =

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( ) ( ) ( ) ( ) ( )1 1

' 0 0; 4 :2 5 2 4

= + > =

h x x f x g x h x

x xng bin mi x D phng

trnh c nghim khi v ch khi ( ) ( ) ( ) ( ) 12453240 mfxff .

Bi tp :

Bi tp 1: Xc nh m phng trnh : ( )( ) 01562 =++ xxmxx c nghim.

Hng dn :( ) ( )2

t x 5 1 x ; 0 t 4 19m 17

4m t t 5

= = +

Bi tp 2: Tm m phng trnh : mxxxx =++ 22 sin2sinsin2sin c nghim.

Hng dn : [ ]2;02

2'

1||;sin

sin2sin2

22

=

=+=

tz

zzt

zxz

xxt

[ ]31

2;0)(222

22sin2sin

222

=+== m

t

tfttmtxx

Bi tp 3 : Cho phng trnh : mxxxx =++++ 22 cossin1sinsin21. Gii phng trnh khi 22=m 2. nh m phng trnh cho c nghim

2;

2

x

Hng dn :

=

=+=

4

9;021'1||;sin

sinsin2 2

tztzxz

xxt

22214)(

4

9;0

=+=

m

mttf

t

Bi tp 4: Xc nh theo m s nghim phng trnh : 4 444 4 6x x m x m m+ + + + + =

Hng dn : 4 44 4 ; ( ) 4 16t x x m f x x x m= + + = + = 19m > : v nghim ; 19m = : 1 nghim ; 19m < : 2 nghim

Tm m bt phng trnh : ( ) ( ) ( )21 2 3 2 5 3 x x m x x+ > + + tha mn1

;32

x

.

t ( )( ) ( ) ( )

1 5 4 1

1 2 3 ; ;3 c ' , ;32 22 1 2 3

= + = +

x

t x x x t xx x

5' 0

4t x= =

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x 12

54

3

t + 0

t7

2

1 7: ;3 0;

2 2x t

0 0

bt phng trnh cho ng 21

;3 th : 62

x t t m + > +

ng7

0;2

t

.

t 21

( ) '( ) 2 1 '( ) 02

f t t t f t t f t t = + = + = =

t 12

0 72

f(t) +f(t)

0

70;

26 min ( ) (0) 0 6 + < = = <

tm f t f m

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