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,

, -

yzM

x

(2)

(1)

(3)

2

,

x

2

( x + 3 )( x 1)( x 4)2

-

x2

x2 + 3

(4)

2

-2

0

3.

, < 0

, .

1 = = 1 1

+ ( ) = 0 = ( ) + 0

,-

( + ) = +

+0 = = 0+ 0

,

, , .

+(+) =( +) + =++

( ) = ( ) = . , :

++ ... + , * =

..., * =

= 1 : 1 = 1 = .

1 : () . 2 3 : ( ) () .

11

R

4.

i. = = + = + ii. = = =

i ii : .iii. + + iv. ,, , > 0 iii v : . ! . ! .5.

i. 0 = 0

ii. = 0 = 0 = 0

:

iii. 0 0 06.

7.

, , , 0 =

ii. =,

, , , , 0: = = , . :

=

i. > 0 > 0 + > 0ii. < 0 < 0 + < 0

8.

i. ,

ii. ,

9.

= , 0 .

i. = ,

> 0

10.

R : 2 0

11.

, > 0 * :

ii. , , : <

ii. < <

>0

< 0

i. , , : <

i. = =

, .

1 1<

1.

= =

2.

= = = =

3. = = 4. = = 1 + 2 + ... + 1 25. = = ... = = + + ... + 1212 R K

12

A - 1

1

, + ,

, 0 .

( )

( ) ( ) 0 ( )( 1) 0

0

+ , ,

< 1, < 1 < 1 1 < 0 .

: + = = , . ,

= , =

= , = , = . 0 < < 1 :+

11 +

, , :. x > 1 > x > x . 0 < x < 1 > x < x

. > , = + , .:x + > x x x x > 0 x ( x 1) > 0

x > 0 x > 0

+

3

.

2

( )( 1) 0 .

x > 1 x > 1 x 1 > 0

. ().

111 1 +

A 1.

) , :i) + , ii) ) ,, , 3.

A 2. x = 3 x 2y = 1 .y3x y 8A 3. A

A 4.

< 0 < < :

A 5.

11<

32) x > 2 x > 2x x + 2

) x < 1 < y xy + 1 < x + y

A 6.

,,

> > 0 > 0 .

+ + == = :3 2

3

:

+ 3, y > 2 : 2x 1 5y 1 = 20 .

A8.

() :

( 1)

A 9.

+ ( 1)

+1

+ ( 1)

+2

+ ( 1)

+3

=0

.

5ii) ( 1)

A 12.

:

2,5 ( 52 26 ) ( 0, 4 )1

=

( 8)

1

A 10. ,y ,

A 13.

x

x+y

y

y+

= 1

x

+ x

=1

y

A14. :

2x +10

1 = y

35y( 1) + .22

A 11.

( 1)

0, x, y x = x x = y :

= 2 ( 1)

= 1 iv)

A * x, y, :

x y

2

0,11

x +1

2 + + 1 = 0, 0 :

i) 3 = 1ii) 2007 + 2007 = 2

15

.3

. ( ).

.

( + )

2

= 2 + 2 + 2

( )

2

= 2 2 + 2

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

3

= 3 + 3 2 + 3 2 + 3

3

= 3 3 2 + 3 2 3

3 + 3 = ( + ) ( 2 + 2 ) 3 3 = ( ) ( 2 + + 2 )

: = ( ) ( 1 + 2 + ... + 2 + 1 )

: Euller

3 + 3 + 3 3 =

1222( + + ) ( ) + ( ) + ( ) 2

: + + = 0 = = 3 + 3 + 3 = 3 Lagrange

(

2

+ 2 )( 2 + 2 ) ( + ) = ( )2

2

.

16

A - 3

1

ii) O 0 + + =

+ + ++++ 3 = + +

:

( x 2 ) + ( 3x 4 ) + ( 6 4x )= 3 ( x 2 )( 3x 4 )( 6 4x )3

3

3

=

:

+ + ++++3 = =+++3== 1 + 1 + 1 + 3 = + +

2

( x 2 ) + ( 3x 4 ) + ( 6 4x ) = 0

3 ( x 2 )( 3x 4 )( 6 4x ) = 0

x 2 = 0 3x 4 = 0 6 4x = 0x=2

i) N :

( ) + ( ) + ( )= 3 ( )( )( )3

3

3

=

3

x=

43

( + + )

2

x=

32

= 2 + 2 + 2

1 1 1+ + =0

0

ii) :

( x 2)

3

+ ( 3x 4 ) + ( 6 4x ) = 03

3

i) E ( ) + ( ) + ( ) = 0

( + + )

2

= 2 + 2 + 2 + 2 ( + + ) ,

, :

+ + = 0

Euller :

( ) + ( ) + ( )= 3 ( )( )( )3

3

3

=

0 :

1 1 1+ + =0

A 15.

A 16.

N :i) ( 4x + 3 ) ( 3x 4 )2

ii) ( 3 + 2 ) ( 3 2 )3

2

i)

iv) ( x + 1)( x 1) x 2 + 5x

( x + y)

2

( x y ) = 4xy2

ii) 2 + ( 2 + 5 ) ( 2 + 3) = ( + 4 )2

3

iii) ( 2 + 4x )( 2 4x ) ( 4x 1)

N :

2

2

2

iii) ( )( + ) 4 + 4 = 2 ( + )( )3

iv) ( 2 + 2 ) + 4 ( 2 2 ) = ( 2 2 + 2 )2

2

17

.4

-

. , .

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

.. 2x 4 = 2 ( x 2 ) . 2.

1. 1 :( )

. xy 3 3xy + 2x 6xy 2

. 2x 2 y 6xy + 8xy 2

:. x. :xy3 3xy + 2x 6xy 2 = x ( y3 3y + 2 6y 2 )

. 2xy.:

2x 2 y 6xy + 8xy2 = 2xy ( x 3 + 4y )2 :

2. . xy 3x + 2y 6

( )

. 3x 3 x 2 y + 6x 2y

. .

xy 3x + 2y 6 = x ( y 3) + 2 ( y 3) = ( y 3)( x + 2 )3232. 3x x y + 6x 2y = ( 3x x y ) + ( 6x 2y ) =

= x 2 ( 3x y ) + 2 ( 3x y ) = ( x 2 + 2 ) ( 3x y )

3 :

3.

( + ) = 2 + 2 + 22( ) = 2 2 + 22

3. i) 25x 2 20xy + 4y 2

ii) 9x 2 24xy + 16y 2

iii) 16x 2 + 40xy + 25y 2

18

A - 4

25x 2 20xy + 4y 2 = ( 5x ) 2 5x 2y + ( 2y ) = ( 5x 2y )2

2

2

ii) 9x 2 24xy + 16y 2 = ( 3x ) 2 3x 4y + ( 4y ) = ( 3x 4y )2

2

2

iii) 16x 2 + 40xy + 25y 2 = ( 4x ) + 2 4x 5y + ( 5y ) = ( 4x + 5y )2

2

2

223. = ( + )( )

3. i) 25x 2 4y 2

ii) 16x 4 y 8

i) 25x 2 4y 2 = ( 5x ) ( 2y ) = ( 5x 2y )( 5x + 2y )2

2

ii) 16x 4 y8 = ( 4x 2 ) ( y 4 ) = ( 4x 2 + y 4 )( 4x 2 y 4 ) =2

2

22= ( 4x 2 + y 4 ) ( 2x ) ( y 2 ) = ( 4x 2 + y 4 )( 2x + y 2 )( 2x y 2 )

3. - 3 + 3 = ( + ) ( 2 + 2 ) 3 3 = ( ) ( 2 + + 2 )

3. i) x 3 64

ii) 8x 3 + 27

i)

x 3 64 = x 3 43 = ( x 4 ) ( x 2 + 4x + 42 ) = ( x 4 ) ( x 2 + 4x + 16 )

ii) 8x 3 + 27 = ( 2x ) + 33 = ( 2x + 3) ( 2x 2 ) 2x 3 + 32 = ( 2x + 3) ( 4x 2 6x + 9 )3

23. 2 x + ( + ) x + = ( x + )( x + )

, , 2 , x 2 + x + , 0 . : x 2 + x + , : x 2 + ( + ) x + = ( x + )( x + ) , , .

19

3. i) x 2 5x + 6

ii) x 2 + 4x + 3

iii) x 2 x 2

i) 2, 3 ( 2 ) + ( 3) = 5 ( 2 ) ( 3) = 6 . x 2 5x + 6 = ( x 2 )( x 3)ii) O 1, 3 1 + 3 = 4 1 3 = 3 .: x 2 + 4x + 3 = ( x + 1)( x + 3)iii) O 2,1 ( 2 ) + 1 = 1 ( 2 ) 1 = 2 .: x 2 x 2 = ( x 2 )( x + 1) 4. 4 :

2i) 2x 18

ii) x 2 + 2xy y 2 + 9

iii) x y + x 2 2xy + y 2

iv) x 9 x 7 x 5 + x 3

i) 2x 2 18 = 2 ( x 2 9 ) = 2 ( x 2 32 ) = 2 ( x 3)( x + 3)ii) x 2 + 2yx y 2 + 9 = 9 ( x 2 2xy + y 2 ) = 9 ( x y ) = 32 ( x y ) =2

2

= 3 ( x y ) 3 + ( x y ) = ( 3 + x y ) ( 3 x + y )

iii) x y + x 2 2xy + y2 = ( x y ) + ( x y ) = ( x y ) 1 + ( x y ) = ( x y )(1 + x y )2

iv) x 9 x 7 x 5 + x 3 = x 3 ( x 6 x 4 x 2 + 1) = x 3 x 4 ( x 2 1) ( x 2 1) =

= x 3 ( x 2 1)( x 4 1) = x 3 ( x 2 1)( x 2 + 1)( x 2 1) == x 3 ( x 1) ( x + 1) ( x 2 + 1)2

2

4 ( ) i) x 2 + 2xy 3y 2

ii) x4 + 5x2 + 9

i) 3y 2 y 2 4y 2

x 2 + 2xy 3y 2 = x 2 + 2xy + y 2 4y 2 = ( x + y ) ( 2y ) = ( x + y + 2y )( x + y 2y ) =2

2

= ( x + 3y )( x y )

ii) x 2

20

A - 4

x 4 + 5x 2 + 9 = x 4 + 5x 2 + 9 + x 2 x 2 = x 4 + 6x 2 + 9 x 2 = ( x 2 + 3) x 2 =2

= ( x 2 + 3 + x )( x 2 + 3 x ) = ( x 2 + x + 3)( x 2 x + 3)

ii) (13x 2 5y 2 ) (12x 2 + 4y 2 )2

A 17. :i) 4x ( x 2y ) x + 2yii) 5 x + y x 5 y2

2

iii) 36x 49y2

2

2

42 222 2iii) (1 + ) +

2

iv) x 3 10x 2 + 9x

2

v) x 7 + 8x 4 x 3 8

iv) 16x 4 y 4

vi)

v) 25x 2 20x + 4

( x + 5)

6

+ ( x 6 ) 2 ( x 6 ) ( x + 5)4

2

A20. N -

vi) 3 3 27

:

vii) 2 2viii) y 2 + 6y 40

i) ( 3y ) + ( x ) x ( 3y )

ix) x 4 + x 2 y 2 + y 4

ii) 2 2 2 + 2 3 4

x) x 2 + 6xy + 8y 2

iii) 375x 3 3iv)

A 18. N :2

3

ii) 5x + 10x 15

(

iii) ( x + 5 ) ( x 2 ) + 4 x 2

) ( x + 5)

iv) x 5 + x 22

) (

)

vi) 3 1 2 2 1 ( 1)

2

3

3

( x y ) ( z + ) ( x y ) ( z ) +22+z2 ( z ) z2 ( z + )2

2

ii) 2 + 2 + 2 + 2 + 2 + 2 + 2iii) 2 2 + 2 2 + 2 2

A19. O:i) 2 +

3

A 21. N :i)

v) ( 2x + 1)( x 1) 9 ( 2x + 1)

(

3

vi) ( x y ) + ( y z ) + ( z x )

2

2

( x + 2y ) ( 2x y )

v) x 3 x 2 + xy + x y 1

i) y + 2x x 12

14

3

(

iv) x 2 y 2 + x 2 + y 2

) ( x + y)

2

21

( )

( )

.5

, 0 . ( 0 )

=

..

2x 1 x + 1 0 . x 1 .x +1

() . . (). . ... ( ), .

..

345345++ 2=+=2x + 2 3x 3 4x 4 2 ( x + 1) 3 ( x 1) 4 ( x + 1)( x 1)

( ... 12 ( x + 1)( x 1) )

=

=

=

18 ( x 1)

+

16 ( x + 1)

12 ( x + 1)( x 1) 12 ( x + 1)( x 1)18 ( x 1) + 16 ( x + 1) 1512 ( x + 1)( x 1)

=

15=12 ( x + 1)( x 1)

18x 18 + 16x + 16 15=12 ( x + 1)( x 1)

17 ( 2x 1)34x 17=12 ( x + 1)( x 1) 12 ( x + 1)( x 1)

! , .

2x 2, R {1,1} x2 1 ..

2 ( x 1)2x 22, . R {1} .==2+x1x1xx 1 ()( ) + 1( )

22

A - 5

1

x 2xx2 + 1A=+x + 1 x ( x 2)

2i)

x 2 + 6x + 9x 2 + 3x

ii)

2 8 + 16 2 16

2i) x + 6x + 9 = ( x + 3) = x + 3x ( x + 3)xx 2 + 3x2

H

x + 1 0 x ( x 2 ) 0 x + 1 0 x 0 x 2 0 x 1 x 0 x 2 .

2( 4)4ii) 8 + 16 ==2 16( 4 )( + 4 ) + 42

x R {1, 0, 2} ( )x ( , 1) ( 1, 0 ) ( 0, 2 ) ( 2, + ) .

A 24. A 22. :

( 5 - 9 ) + 2 ( - 3 )

2 ( 4 - 5 ) - 3 ( 3 - )

2

i)

4xx2y++,24x 9y 3y 2x 3y + 2x3x y22

x y+ +2iv) y x, x, y 0 x y1 1+x y

A 25. :A=

A 26.=

A 23. N :i)

ii)

x2 + xx 2 + 5x + 6x 2 + 3x + 2 x 2 + 3xx 2 36 x 2 + 6x: 2 y 2 y 2

333++( - )( - ) ( - )( - ) ( - )( - )

2 - 2 - 2 2 - 2 - 2 2 - 2 - 2++=0 +++

A 27.=

+ + = 0 , :

+ + = 0 , :

444+ 3+ 3=033 + - 3 + - 3 + 3 - 33

23

.6

, x : x, x 0x = x, x < 0

1 1= , 0, 4 = 0, 4 ...2 2 :

.. 3 = 3,

x = 0 x = 0 , x 0 x > 0 x 0 .

, 0 .

xx x x .

.. 4 -1 4 ( 1) = 5 . d o , , - ( ) ( ). : , , 0=d= , < , < 0

24

A - 6

, , :

d = = ,

.1)

x = x , x R

4)

x y = x y , x, y R . : x1 x 2 ...x = x1 x 2 ... x

5)

xx= , x R, y R *yy

7)

x y x + y x + y ( )

2)

x x x

6)

3)

2

x = x2

x = x , *

: x1 + x 2 + ... + x x1 + x 2 + ... + x 8)

x = , > 0 x = x =

9)

x , > 0 x

10) x , > 0 x x

i) E 0 0 , : + = 0 , = 0 = 0 , : + = 0 = 0 = 0 . + .ii) : x + y = x + y x,y , x y = x + y .iii) O x = x < 0 , x

xR .

25

1

:

A = 1 + x + x 1 = 2x 2

+ = = 0

0 , x 1 A = 2x 2, x > 1

+ = ( )

- =

. + =

- , d ( , ) = 0 = 0 .

1 2x 1 = 2x 12 B = 2x 1 + 3x 1 = 5x 2

ii) A 2x 1 0 x

1 2x 1 = 2x + 12 B = 2x + 1 + 3x 1 = x 2x 1 < 0 x 1 1 x = 1 + x

+ 4 = 2 + 2

+ 4 = 2 2

= 2 = 2 = 2

A 28.

:

i) A = x +ii) B =

A 29.

1 x 4, 1 x 0x + 2 x 3

x2 + 2 x2 x+ 2x2 4x 4 x +4

x -

.i) A =

x 1x 1 1

ii) B =

1x 2 3 x

A30. d ( x,

2007 ) > 3

x.

26

A - 7

.7

0 .

, = .

-

,

2

= 2 = , , 0 , 0

x = . :2

0 ,

= , .

,

= = , , 0 .

, x = . :

= ,

1

2

, 0

= , ..

0 = 0,

4 = 2,

3

27 = 3 ...

x 2 2 , x 2y + 5 ... x 3, , ( ) .

x +1 ,

. .. :

1. 0 * 2.

( )

2 = , R . :

3. , 0 *

2

=

= .

2 = , R

=

:

2 +1 = , 0 .

.

=

= ( ) , k N* .

=.

4. 0 , > 0 * :

5. 0 , , * :

=

2 +1

= .

> 0, : =

= 0 , 0 = 0 .

27

1

4

:

x2x 2 + 4x + 4, 2 < x < 0i. =xx+224ii. B = 5 ( x 2 ) 3 4 ( x + 3 ) + 10 x 2 + 2x + 1 ,

3

3

3

13

x+2

x x+2= x x+2

x < 0 x = x

x > 2 x + 2 > 0 x + 2 = x + 2x x + 2= 1 1 = 2x x+2ii. B = 5 x 2 3 x + 3 + 10 x + 1 1 < x < 2 . A =

( 3)

2

=

4 33

22 :

xi. A = x

3 3

4 3

=

ii.

1 < x < 2

( x + 2 )2

4 3

=

2

=

3

3

22

2 3 22

=

3

222

3

3

=

222

iii. 2 + 1 . :

52 15(

=

5 ( 2 + 1)

(

2 1) ( 2 + 1)

2 + 1)

=

5 ( 2 + 1)

(

2 ) 122

=.

= 5 ( 2 + 1)

x < 2 x 2 < 0 x 2 = ( x 2 ) x > 1 :

=

x > 3 x + 3 > 0 x + 3 = x + 3

2 + 1 , , -

x > 1 x + 1 > 0 x + 1 = x + 1 B = 5 ( x 2 ) 3 ( x + 3) + 10 ( x + 1) == 5x + 10 3x 9 + 10x + 10 = 2x + 11

2

2 1

2 1 .

2 + 1 2 1 .

. -

iv.

:

3 2 .

i.

43

ii.

13

iii.

2

5

iv.

2 1

33+ 2

3 2 :

. :

43

+

52 1

3

3

3+ 2

3+ 2

.i. 3 :

=

=

(

3 ( 3 2 )

3 + 2)( 3 2)

3 ( 3 2 )

( 3) ( 2 )= 3 ( 3 2 )2

2

=

=

3 ( 3 2 )=3 2

28

A - 7

. :

43

+

5

3

2 1

3+ 2

4 3+ 5 ( 2 + 1) 3 ( 3 2 ) =35=8 23+53=

=

A 31.

:i.

i)

3

3, , > 02

4 2

ii)

4 3

3 3

16x 7 x + 25x

ii.

4 2 + 6 2 7

iii.

x 2 y x 4y + 16x 2 y

32 2 3 8 2

x 3 y 2 4 y3, x, y > 0y x2 x3

iv)

x 4 2x 2 + 1

iv.

5

iii)

A 37. N :

A 32. :i. A =

(2

ii. B =

(

2) +2

3 2)

2

(

+

2 2)

(

A = 18 + 27 3 + 3 + 3 3 3 + 3

2

3 + 2)

2

A 33. -

A 38. :i) A = 2

( x + 3)

2

4 ( x + 2 ) + 3 x 2 4x + 4 ,4

x 2

:

x 2 2x + 1 2 6 x 6 , x R {1}x 1

A = x 2 4xy + y 2 x = 3 + 2

ii) B =

y = 2 3

A 39. N :

A 34. i. :

(2 + 3 5 )

( 2 3 5 )ii. :2

2

i)

3 + 2 > 1+ 2

ii) A > 0 2 + 1 > + 2 + iii) A x, y > 0 :

49 12 5 + 49 + 12 5

iv)

A 35. :

3

112 7 3 + 16 8 3 = 3 12

A 40. N

i. 2 8 3 18 + 4 32 5 50 + 72ii. 5

3 48312 +27 104 3916

A 36. :

x 2 + y2 x + y22

:i)

iv)

4

ii)

3 2

3 34

13 + 2 1

27

iii)

22 2 33

v)

3

6

33 2

29

.8

1 360, 1 1.

1 ( )

1 rad ( ) . 1rad 1rad.

=, : , : 3,14 180( ).

:

: =.. 300 =

= 180180

30 = , 41806

= 4 180 = 45 . .:

:

=

=

:

=

=

:

=

=

:

=

=

30

A - 8

( ) :.

2 = 1 2 2 + 2 = 1 22 = 1

.

1 = = 1 = 1

.

=

=

. 1 + 2 =

11 1 + 2 = 22

() , .i. 2 =

21 +

iii. 2 =

2

11 + 2

11 + 2

ii. 2 =

iv. 2 =

21 + 2

300 (. 1) = =

30, 45 60

3, = 450 (. 2) 2

2. , 2 300, 450 600. :==

-

2

31

()

(1, 0) .: * -

* ,

, i .

A , ( ).

.A () A ( ) A rad , A A :2 + 360 0 + 0 , .

x rad ( ) A : A

x = 2 + , 2.

: i. (1, 0)

ii. ' ( 1, 0)

iii. ( 0, 1)

iv. ' ( 0, 1)

i. 0, 2, .ii. , 2 + , .iii. :

32 + , 2 + , iv.22

* ( ).

32

A - 8

( ) A, ( x, y ) . ( x, 0 )

x ' x , ( 0, y ) y ' y, E (1, ) (1, 0) ( , 1) ( 0, 1) , :

= y, = x, = = ( ). : 1, 1, R, R

.2 , . +

. (), : ()

0o

30

45

60o

90o

180o

270o

360o

(rad)

0

6

4

3

2

32

2

0

12

22

32

1

0

1

0

1

32

22

12

0

1

0

1

0

33

1

3

0

0

3

1

33

0

0

.

33

. :

33 , + , , + , ,+ , 2 , 2 + , 2 2222 :,

.

.

1.

0 , , 2 , 2

2.

3 ,22

1

1 . . 2 ( ), ( ). .(. ).

.. ( ) . 1 , , , . ( ). ( ) = . : 3 3 = , ( + ) = , + = , + = ... 2 22

, , :

...

34

A - 8

750 0 = ( 2 3600 + 300 ) = 30 0 =

12

172 17 16 = + = 4 + = = = 4 4 44442133 13 12 = + = 2 + = = ... = 66663 6 6

-

+ :

( + ) = -

( - ) = +

( + ) = +

( - ) = -

( + ) =

+ 1

( ) =

1 +

( + ) =

1 +

( ) =

+ 1

.45 0 + 30 0000=1. 75 = ( 45 + 30 ) =1 45 0 30 0

323 = 3 + 3 = (3 + 3) = 2 + 3933 3 313

1+

2. : ( + ) ( ) = 2 2 , () , : ( + ) ( ) = ( ) ( + )

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

-

, = , :

2 = 2 2 = 2 2 = 2 2 1 = 1 2 2

, =

2 =

2 2 12=21 2

4< < 2 .25

35

2 = 1 2 ,

= 1 2 = 1

163=255

4 324 2 = 2 = 2 = 5 525

2 = 2 2 1 = 1 2 2 , :

-

2 =

1 22

2 =

-

1 + 22

2 =

1 21 + 2

2 =

1 + 21 2

:

2 =

-

2

1 + 2

22

2 =2 1 + 21 2

=

=

2

1 2

2

1 2

2

2

2

.i. : 2ii. =

3+ 2= 1 88

1 : 2 + 22

3=i. + 2882

31 2 + 4+4 =44 =1222

1

21 2 2 + 1 2 ii. 2 + 2 =+==1 + 2 1 + 2 1 + 2

11 72 +124= 4=715 51+44

36

A - 8

1

. :

=

i. + = 1 2 4

4

2

2

ii. 6 + 6 = 1 3 2 2 . R

= 6 + 6 + ( 4 + 4 ) x.

. i. 4 + 4 = ( 2 ) + ( 2 ) =2

2

= ( 2 + 2 ) 2 2 2 =2

= 1 2 2 2 ii. 6 + 6 = ( 2 ) + ( 2 ) =

= ( + ) 3 2

3

2

( x + x )x x

2

2 =

2 x + 2x x + 2 x 2x x=x x1=x x=

2()2ii. 1 + x + x = 1 + x + x =x1 + xx (1 + x )

=

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

=

2 + 2x2 (1 + x )2==x (1 + x ) x (1 + x ) x

3

:

3

3

2

=

x + x x + x2 =xx

2 ( 2 + 2 ) = 1 3 2 2 . = 1 3 2 2 + (1 2 2 2 ) =

= + 1 ( 2 + 3) 2 2

32 + 3 = 0 = .2

2

:

x 1i. (1 + x ) 1 + 2 = xxxii.

x1 + x2+=x1 + x x

x i. (1 + x ) 1 +2 =x x x = 1 +2 = 1+x x

i. =

( 180 x ) ( 180 + x ) ( x ) ( 360 x )

( 270 x ) ( 90 + x ) ( 810 x ) ( 180 + x )

3 7 + x ( x ) x22ii. = 9 x ( x 4 ) ( x 3 ) 2

i. (180 x ) = x

( 270 x ) = x

(180 + x ) = x ( 90 + x ) = x (810 x ) = x ( x ) = x

(360 x ) = x =

(180 0 + x ) = x

x ( x ) ( x )( x )= 1 x ( x ) x x

37

4

3ii. + x = x 2() x = ( x ) = x

0 < 0 )

.. = =

x =

.

: > 0 , =

, 0 = 0 .3

..

16 4 = 4 163 = 23 = 8

4

12

= 4 1 =

1 1=4 2

, , x , > 0 x R . ( )., , x, y R :

i) x y = x + y

ii) x : y = x y

iii) ( x ) = x y

iv) ( ) = x x

y

x

x

x v) = x

A B , :

, : .: 0 = 1 0 , 1 =

1

= > 0 .

39

1

:

3A = 2 2

(

12

) ( )

1

23

3

i) 27

32,5

1

1

1

1

, > 03

3

1 3 2

2

6

1

1

1

13 9 3 3 2 1 3 2iv) 8 + 12 2 3 2 2 24 4 16

6

126242= = 4 == 2 = 4 =1422 242 2 4

:

i) 27

56

5

3 2 = 6 27 5 35 = 6 315 6 315 =

= 6 30 = 6 1 = 12422ii) 3 + 12 = 3 + 2 3 =

1112 1 2i) x 3 + y 3 x 3 x 3 y 3 + y 3 = x + y1112 1 2ii) x 3 y 3 x 3 + x 3 y 3 + y 3 = x y

x, y > 0

= 3 2 3

iii) 3 8 6 + 4 3 3 4 3 125 6 4 == 2 2 3 + 4 2 3 5 2 3 = 2 3

3 13 3312 3 2 2+=4 2416

iv) 8

8 33 3 2 33 + 3=24493 933=4 3+ 3=424=

112112i) x 3 + y 3 x 3 x 3 y 3 + y 3 =

x+3y

3

3

)(

3

2

x 3 x3 y+3 y

2

)=

3

= 3 x + 3 y =x+yii) x y x + x y + y =13

(

3

13

x3 y3

23

)(

13

3

2

N

4i)

=

56

iii) ( 8 6 ) 3 + 4 ( 3 4 ) 3 ( 125 6 4 ) 3 ,

21 3 4 = 2 2 3 = 3 2 =

(

1

=

N :

ii) ( 3 2 ) 2 + ( 12 4 ) 2 , , > 0

21 = 322

=

3

13

23

x +3 x3 y+3 y

2 3 42 + 2 3 241

2

)=

, > 0

1

24ii) x 2x + 1 , x > 011

x 4 2x 8 + 1

3

= 3 x 3 y =xy

40

A - 9

1

i)

3

3

2 4 2 + 2 2 4 =4

2

2

ii)

2

2

13

2

3

=

x 28 x +1

2

8

2

2

2

8

2

8

2

8

2

=

2

8

2

2

4

8

8

53

= + 2

2

8

= + = 3 5 + 3 5 =3

4

2

8

5 1 1 5= 3 3 + 3 3 =13

x 24 x +1

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

10

2

=

x 24 x +1

4

= 3 3 + 3 3 =

53

18

14

x 2x + 1

4

= 2 3 3 + 2 3 3 =10

1

x 2 2x 4 + 1

A 47. :23

A 50.

35

12

i) x = 27 + 4 3 32 + ( 0, 25)

1ii) y = 16

A 48.

1

+ 2 4 24 4 54 4 5

( )

ii)

3

53

6

2 1 2i) = x 2 y y 3 x 3 ,

2

7

3

x =

8 3 + 25 2

1 y = 3 22

21ii) B = x 3 y 1 x 1 y 2

(

i) ( 0,5)

A 49.

( )

0,25

52

:

164

x =

7 4 7 6 7 2

:

A 51.

1

)

43

2 ,

11 y = 83

6

:3

i) =

xy34

12

x +x y3

ii) B =

14

1

12

14

14

x y +x y1

1

x2 + y21

1

3

2 2 2 2 2 4 + 2 41

1

2 2

12

i) ( 6 2 5 ) 2 = 8 ( 5 2 )

(

1

1

)(

ii) 2 2

12

+

12

)=

( )

41

.10

, 1 > 0 . x = . log : , > 0, 1 > 0 : log = x x = ., .. log 2 8 = 3 23 = 8, log 3 9 = 2 32 = 9 ... :i) = log

ii) log x = x

iii) log = 1

iv) log 1 = 0

= 10 log10

log .

. log = x 10 x = . .. log100 = 2, log 0,1 = 1 ...

= e , e 2, 71 , log e n ( ) .x. n = x e = . .. ne3 = 3, n

1= 1 ...e

, > 0, , 1, ,1 , 2 > 0 R ( ):

1) log ( 1 2 ) = log 1 + log 2

2) log

1= log 1 log 22

3) log = log 4) log =

log log

( )

42

A - 10

:

1log , , 2

i)

log =

ii)

log

iii)

log log = 1

iv)

log =

1= log

nlog , log =log n

1

i) log = log =ii) log

1log

1= log 1 log = 0 log = log

iii) log =

log 1 log = log log = 1log log

iv) = 10 = e .

43

1

2

:

log 3 2 =

log 8 12 .

24

. log 4 32

. log 0,1 100

. log 8

. log 1 3 3

. log 0,2 625

. log 10 10

9

log 3 2 = 3 = 2 . log8 12 = x .

8x = 12 23x = 22 3 ( 3 )

. log 4 32 = x . 4 x = 32 22x = 25 2x = 5 x =

52

. log 0,1 100 = x .

( 0,1)x = 100 (101 ) = 100 10 x = 102

3x

=

= ( 3 ) 3 33x = 32 +1 2

0

3x = 2 + 1 x =

2 + 13

x

x = 2 x = 2. log8

3

:

. log x 1000 = 6

2= x . 4

312 23x = 2 2 : 22 23x = 2 2 431 3x =x=22

. log x

8x =

. log 1 3 3 = x . 9

x

11112x3 = 3 3 = 33 2x = x = 369

. log x 16 =

23

16=481

x 6 = 1000. log x 1000 = 6 0 < x 11 1 61 6 3= 103 = 10x = x x100 < x 10 < x 10 < x 1

. log 0,2 625 = x .

( 0, 2 )2 = 625 = 625 5 x

x

x

21 = 625 = 10 54= 5 x = 4 x = 4

. log 10 10 = x .

10 = 10 10 10x

= 102 10 104x

=3= 103 4x = 3 x =42x

= 10 10 10

4x

x =

10.10

3 322 x = 16 2=x16. log x 16 = 30 < x 1 0 < x 1

( )

1 3 x = 16 2. x = 64 .0 < x 1

44

A - 10

4 1616x ==4. log x81 810 < x 1 x =

4

42x 4 = 30 < x 1

2.3

2 3log 3 2 + 2 log 3 6 log 3 32 == 3log 3 2 + 2 log 3 ( 2 3) log 3 25 == 3log 3 2 + 2 log 3 2 + 2 log 3 3 5log 3 2 == 2 log 3 3 = 2 1 = 2

R :. log x ( 3 x )

. log x

. 2 + 3log 5 2 2 log 5 10 =

= log 5 52 + log 5 23 log 5 102 == log 5 ( 52 23 ) log 5 102 =

1+ x5 x

= log 5

25 8= log 5 2100

2

3 x > 0x 0, 0 < a 1

= 2 + log 5 2 2 1 = log 5 2

y R

1 + x>0(1 + x ) ( 5 x ) > 0. 5 x0 < x 1 0 < x 1( x + 1) ( x 5 ) < 0 1 < x < 5 1, > 1 2 + 2 = 7 ,

log 2 0, 3

+ log = log ( ) 3 + 2 log= log + log 3. + 1 log= ( log + log )32:

:

(

)

11A = log 2 + log 2 + 2 +2211log 2 + 2 + 2 + log 2 2 + 222

(

)

(

)

1( log + log ) log log 2 log + log 2 log log

( log + log ) 4 log log 2 ( log log ) 02

.

46

A - 10

A 52.

x -

A 57. :

:. log 1 3 3 = x

. log 0,1 5 100 = x

1

. 103log 2 + log 2 = 4

9

. log x 27 =

32

1. log 8 x = 3

. log x 4 =

23

3. log 4 ( log x 25 ) =2

A 53. log 2x

11 log 254

. 100

x R

x +1.3 x

log 2 = log 3 = , :724, 5, 6, 12, 15, 30, 36,.50

A 55. :

1 : log = log .. 45log32 17 .

75532 2 log 2 + log 2=1169243

A 56. :

(

)

7.log 3 + 2 2 4 log1625log 2 1=8

(

. 3 = log 2 log 2

2

. = log 2 log 2... 2

> 1 > 1 , :

A 60.

5340105=0. 2 log + log log log2117732

(

)

2 +1 =

)

. log 3 2 + log 2 3 > 2.

*, R * , 1 R

A 59. :

A 54.

. log 2

A 58. .

= 20

log 125 + log 27 log 8 3=log15 log 22

22 = log ( 2 1) + log ( 2 1) log ( + 1) ( + )

A 61.

,, R * 1 1 :

log =

log 1 + log

47

1

1

.1

x + = 0

:

0 x + = 0 , x =

1

x + = 0 x = x =

,:

= 0 0 x + = 0 ,

x, :

x + = 0 0x = 0 = 0 = 0 x + = 0 , x, :

x + = 0 0x = 0 :

0

=0

0

x =

=0

A2A3

1

. : ( 2x 7 )( 6x + 5) = ( 4x 3)( 3x + 1)

:. ( 2x - 7 ) ( 6x + 5) = ( 4x - 3)( 3x + 1)

.

( x + 1)

2

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

2

12x 2 + 10x 42x 35 = 12x 2 + 4x 9x 3 2

.

32x 35 = 5x 3 27x + 32 = 0 27x = 32 x =

3227

. :

-

( x + 1)

=

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

2

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

2

1

48

B - 1

= 4x 2 + 4x + 1 ( 9 12x + 4x 2 )

4

:

x + 2x + 1 x 4x 4 =2

2

= 4x + 4x + 1 9 + 12x 4x2

2

2x 3 = 16x 8 2x 16x = 3 8

:x + 2 2x + 1 2x - 3=-x326

.. .

3

3

A3

+ + = 0, :

5 18x = 5 x =18

2

A2

( 1 x ) + ( 2x 3 ) + ( x + 2 ) = 03

3 + 3 + 3 = 3 = 0 = 0 = 0

(1 x ) + ( 2x 3) + ( x + 2 ) = 0333: (1 x ) + ( 2x 3) + ( x + 2 ) =3 (1 x ) ( 2x 3) ( x + 2)333 (1 x ) + ( 2x 3) + ( x + 2 ) = 0 3 (1 x ) ( 2x 3) ( x + 2 ) = 0

1 x = 0 2x 3 = 0 x + 2 = 03x =1 x = x = 22

. =

x + 2 2x + 1 2x 3=x 326x+22x + 12x 36=6 6x 6326

2 ( x + 2 ) 3 ( 2x + 1) = 2x 3 6x

5

:

x224=+ 22x2 x x 2x

2x + 4 6x 3 = 2x 3 6x 0x = 4

.

.

:

3

:

( 4x 5 ) ( x 2 ) ( 1 x ) = 0

= 0 =0 =0 =0

( 4x 5) ( x 2 )(1 x ) = 0 4x 5 = 05 x = 4 x 2 = 0 x = 2 x = 11 x = 0

... ( ) . .. . .. . .. .

A5

49

1

x 0... : 2x ( x 2 ) 0 x 2 0 x 2

(1) 2x ( x 2 ) x 2 =2x242x ( x 2 )+ 2x ( x 2 )x2x ( x 2)( x 2 )( x 2 ) = 4x + 8

x224=+ 22x2 x x 2xx224=+2x2 x x ( x 2)x224(1)=+2xx 2 x ( x 2)

( x 2 ) = 4x + 8 x 2 4x + 4 = 4x + 8 2

x 2 4x + 4 + 4x 8 = 0

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

2

1.

4.

:

(

)

. ( 3x + 5) - 9x 2 - 25 + 6x + 10 = 02

. x + 4 -

2.

3.

. 1-

x + 3 2x + 3=33

.

:.

( x -1) - ( x - 2 )

.

( x -1)

2

3

2

+

2

( x + 3)

= (1- 2x ) - ( 3 - 2x )2

2

=

6

:

2

5.

( x - 2 )( x -1)

x3=x 3 9 x2

35

x264+= 1+2x2x 4 x+2

:.

x 3 x 2+=2x 2 x 1

.

11=x+2 x+5

2

:. 4x .

(x

2

19 + 2x7x + 11= 15 54

- 4 ) - ( x + 2 ) ( 5x + 4 ) = 02

6.

:.

xx+4=x 3 x 5

.

x1x4=x 2 4 x 2 2x x 2 + 2x

2

1

50

B - 1

, 1

. ,

3x + 1 = 7,

4 x = 7-1,

( + )x = 1

. (, , , ...) . x + = 0 , , ., : 0 . . = 0 0 . . = 0 = 0 . .

6

:. ( 1) x = -12

. ( 2 ) x = + 22

. 2 x 2 = 4x +

. x = , ( = 1, = 2 1) , :1 1 0 1 , :

x=

2 1(1)( + 1)x= x = +11 1

2 1 = 0 = 1 , ,:

(1 1) x = 12 -1 0x = 0 , .. x = , ( = 2, = 2 + 2) :

51

1

1

1

2 0 2 ,

( 2 ) ( + 2 ) 0 2 0

2 + 2 : x =2

+ 2 0 2 2 :

2 2 = 0 = 2 , :

0x = 2 2 + 2 0x = 6 .. x = , .

2 x 2 = 4x +

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

x=

+21=( 2)( + 2) 2

2 ( 2) ( + 2) = 0 = 2 = -2 , :i. = 2, (1), :

0x = 2 + 2 0x = 4 .ii. = 2, (1), :

0 x = 2 + 2 0 x = 0 , .

7.

.

11. :.

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

8.

. ( 4 ) x = 2

. ( x 2 ) + 6 = ( x + 1) :.

xxxx+=21 =+ 1 . + 1 +1

10.

13.

x + 3 + 2 ( x -1) = + 8x , .2

+ 3

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

.

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

235

2

12. :. ( 3x + ) + 7 - 2 = 2 + 3 (1 + x )

. 2 ( x 1) = ( x + )

9.

2

.

2

. 2 ( x ) = 3x 5 6

2

. 3 ( + 1) x + 4 = 2x + 5 ( + 1)

:3

( - 9) x =

2

.

(

.

x ( + 9)9xx+-1 =-20 2020

2

- 4 ) x = 2 - 2

x - x+ = 3x - 32i. , ii. 1

52

- 2

1

.2

x + > 0, x + < 0 ,

. . x + > 0, , , : > 0 x + > 0 x > x >

1

x + > 0 x > < 0 x + > 0 x > x <

.

x + > 0 x <

.

= 0 > 0 :

0 x + > 0 > 0 . x. = 0 0 :

0 x + > 0 > 0 . . .>0

x0

0

x x+0

( 2 - ) x .

x + > 0 , = 5 3, = 6 . : 5 3 > 0 , >

3, 5

( 5 3) x 6 > 0 (5 3) x > 6 x > 65 3

. > < .

:

x + > 0 x + < 0 . ( 1) x + 3 ( x 2) > ( 2 ) x

( 1) x + 3x 6 > ( 2 ) x [( 1) + 3 ( 2 )] x 6 > 0

(5 3) x 6 > 0

5 3 < 0 , 0 ( 5 3) x > 6 x < 65 33 5 3 = 0 , = , 5318 : 0 x 6 > 0 > 0 55

. .

:

14. :

18.

6 x 3x + 2.86

3x7x5 1.43

15. :. 3 ( x 1) ( x + 3) 4 2 ( x 3)

.

19.

16. :

.

x 1 x + 1 2x + 4 x + 6++455206 3x 2x + 84x 21x + 14+x+343312

17. x :

x+2 x 143

(1),

x x +1< 0 (2)5 15

.. x 2 x + . 3 ( x 1) + 2 ( 2 x ) < x

x 2 x +13x. 15210

.

: ( x + 1) 1 2x

20.

:

21.

x 1 3x + 2 5x+244

2x 5 ( x + 2 ) 3 2 ( x + 3)

, x.

1

56

.3

B - 3

E

:

1. x = , > 0 x = x = 2. x = x = x = 2

3. x = x 24. x = , < 0 .

6

1

:

x 4014 x = 0

x 0 . :5 + x 1 = 2x 5 + x 1 = 2x

. :

x 1 = 2x 5 x 1 = 2x 5 :

2x 5 0 x

x 4014 x 4014.

52

x x 0. x 4014 x = 0

x 4014 = x x 4014 0,

, ,

d(0, 4014) x = 2007 .

55, x .22 x 0, x

x 1 = 2x 5

2

:5 + x 1 = 2x

-

5 2 , + .:x 1 = 2x 5 x 1 = 2x 5 x 1 = 2x + 5 x = 4 3x = 6 x=4x=2

57

x = 2

5 2 , + . x 1 = 2x 5 x = 4 .

3

:

(1) x + 2 = 2x + 1x + 2 = 2x + 1 x + 2 = ( 2x + 1) (2) . :

:

= = =

2 x3 +1 2x = x 3 + 73

= , , .

-

x = 1 x = 1

.. . 3.:

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

2 x 3 +1 2x = x 3 + 7 3 2 x 3 + 1 6x = 3 x 3 + 21 x 3 = 6x 20

3x = 3 x = 1 x = 1 x = -1.

5

:

:

6x 20 0 6x 20 x

103

:

x 3 = 6x 20 x 3 = 6x 20 x 3 = 6x + 20 7x = 17 5x = 23

x 2 1 = x+ 3 + x1

x 2 1 = x + 3 + x 1 x 2 1 x + 3 x 1 = 0

x 2 1 x + 3 x 1 :

1723x=75 , x=

1017 x 37 . x=

x =

4

23.5

:

:

x + 2 = 2x + 1

x + 3, x < 3 x 3, 3 x < 1x 2 1 x + 3 x 1 = 3x 1, 1 x 2 x 5, x > 2

58

B - 3

,

3x 1 = 0 x =

x 2 1 x + 3 x 1 = 0 : , , -

13

1 x 2 , . x 5 = 0 x = 5 x > 2, .

. :

x + 3 = 0 x = 3 x < 3 . x 3 = 0 x = 3

x 2 1 = x + 3 + x 1 , x = 3 .

3 x < 1 .

22.

ii. 3 x + 1 = x + 6 :. 4x + 8 = 2

23.

:. x = x 3

24.

iii. x + 1 + 2x + 4 = 0

. x + 2 = 2x + 3

B27.

x 1

. x 1 = x 2

:

x 1.1 = 2 x 1 +12. 3 x + 1 + 2 x 1 = x + 2234

:

3

B28.

+

1 x2

= 2x + 2

:i. x 3 = 2x 4

B29.

B26.

ii. x 2 2 = x 2

:. x 1 + 2 = 3

B25.

(1)

. x x = 4 3 x

:i. 2x + 1 = 3

ii. 2x + 1 + 4 = 0

iii. 2x + 1 = 0

iv. 2x + 1 3 = 0

:i. x + 2 = 2x + 1

B30.

:

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

59

:1. x , > 0 x 2. x , > 0 x x

< 0 : x < ( x 0 ) x > , x R .

6

6

:. 3 2x < 8

. 7x + 1 12

: 5x 6 < 8 8 < 5x 6 < 8 2 < 5x < 14 214 x > 22. : 7x + 1 12 7x + 1 12 7x + 1 12 7x 13 7x 11 x

7

1311x77

x :2 5x - 6 < 8

5x 6 25x 6 < 8

: 5x 6 2 5x 6 2 5x 6 2 5x 4 5x 8 48x x55

2 5x 6 < 8 x :42148214x , 2 , 2 < x 5 . :

1 x < 0 0 x 2 2 < x 5,3, , , .

9

:

x+3 x2 x

1 x 5 .3

10

:5 (4 x) < x 2

( 1)

. x 2 0 x 2

(1) 5 ( 4 x ) < x + 2 20 5x < x + 2

: x + 3 x 2 x x + 3 x 2 x 0

5x + x < 20 + 2 4x < 18 x>

189x>42

x 2 .. x 2 > 0 x > 2

(1) 5 ( 4 x ) < x 2 20 5x < x 2

61

5x x < 20 2 6x < 22 x>

11

2211x>63

12

x 3 1x 3 1x 3 +5+ 12< 12 2 + 12646

2 ( x 3 1) + 3 ( x 3 1) < 24 + 2 ( x 3 + 5 )

:

2 x 3 2 + 3 x 3 3 < 24 + 2 x 3 + 10

x 3 1 x 3 1x3 +5+< 2+646

3 x 3 < 2 + 3 + 24 + 10 3 x 3 < 39 x 3 < 13 13 < x 3 < 13

x 3 1 x 3 1x 3 +5+< 2+646

13 + 3 < x 3 + 3 < 13 + 3 10 < x < 16

31. :i. x -1 < 3

ii. x + 2 < -2

iii. x + 7 0

i. x + 1 2

x 1x2

x + 1 x 2 > 2x (1)

36. i. : < x <

32. :

iii.

6

35. :

ii. x + 3 > -1

.ii. .

>1

33. :2 < x 1 4

34. :2 x 1 < 3

62

B - 4

B.4

x 2 + x + = 0, 0

x 2

x 2 + x + = 0 , 0, , R

.. 2x 2 3x + 4 = 0 2 = 2, = 3 = 4 .

x 2 + x + = 0 x 2 +

x+ =0

[ 0 ]

x2 +

x = x2 + 2 x = 2

x 2 + 2x

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

2

2

[ 0 ]2

2

2 4x + =2 4 2 = 2 4 :2

x + = 22 4

(1)

: > 0 , (1) :

x+

=2

x=

4 2

x+

=24 2

+ +x=x= x=2 22 222

( ) :

x1 =

+ , x2 =22

= 0 , (1) :2

=0x=x + = 0 x +2 22

63

x 0 =

.2

< 0 , (1) x 2 + x + = 0 , . :

x 2 + x + = 0,

0

= 2 4 . =

4

. < 0 , > 0 x 2 + x + = 0 . , . , , ..4x 2 16x = 0 4x ( x 4 ) = 0 x = 0 x = 4

3x 2 + 27 = 0 3x 2 = 27 , .4x 2 64 = 0 4x 2 = 64 x 2 = 16 x = 4 x = 4 x = 46x 2 96 = 0 6x 2 = 96 x 2 = 16 x = 16 x = 4 x = 4

:

2

x 2 + x + = 0 x + x + = 0

x = y : y 2 + y + = 0 ... x 2 8 x + 15 = 0 x 8 x + 15 = 0 y 2 8y + 15 = 0 x = y 02

y =

8 4 8 2 5== . x = 5 x = 5 x = 3 x = 3 .223

64

B - 4

1

:

, = 0.

x + ( 2 1) x 2 = 02

. = ( 1) 4 ( 2 2 ) = 72 + 6 + 1 .2

= ( 2 1) 4 1( 2 ) = 2 2 2 + 1 + 4 22

2

= 2 + 2 2 1 + 12 = ( 2 + 1) > 02

2

( 2 1) ( 2 + 1) 2 + 1 ( 2 + 1)==2 12 2 +1+ 2 +1=12= 2 +1 2 1 = 22

72 + 6 + 1 = 0 . , = 1 =

2

x1,2 =

1.7

. = ( 1) ( 2 2 + 1) = 0 . .2

. : = 9 ( 3) + 4 ( 2 + 10 ) = 172 14 + 812

2

;. x 2 ( 1) x + 2 2 = 0. x 2 2 ( 1) x + 2 2 + 1 = 0

172 14 + 81 = 0 . R. .

. x2 3 ( 3 ) x ( 2 + 10 ) = 0

B37. :

. 4x 2 2 ( 5 + 1) x + 5 = 0. 3x 2 ( 3 ) x = 0

B38.

B40.

5 .

B41.

:( - 1)x2 + (2 + 1)x + = 0. :. ; ;. ; ;

B42.

2 x + 2x + = 0, 0

x 2( + )x + ( ) = 0 .2

2

2

2

2 2

, ( ).

B39. x 2 2( + 2)x + 2 2 17 = 0 .

. .

x 2 3x + + 2 = 0 .

, :x 2 + 2( + + )x + 2( + ) + 3 = 0

65

1, 2

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

:

S = x1 + x 2 =

P = x1 x 2 =

.. x1 + x 2 =

[ Vieta]

x1 x 2 = x 2 + x + = 0

, x1 , x 2 ( x x1 )( x x 2 ) = 0 :

x 2 ( x1 + x 2 ) x + x1 x 2 = 0x2 +

x+ =0

x 2 + x + = 0

.. 3 -4. 3 + ( 4 ) = 1 3 ( 4 ) = 12 x 2 + x 12 = 0.. -3 4 :

x 2 ( 3) x 4 = 0

x 2 + 3x 4 = 0x 2 + 3x 4 = 0

x1 =

3 + 32 4 1 ( 4 ) 3 + 5==122

x2 =

3 32 4 1 ( 4 ) 3 25 3 5=== 4222

66

B - 4

R , x 2 + x + 8 = 0 .

3

x 2 + ( + 2)x + 2 = 0 . 3;

4

:

> 0 2 4 1 8 > 0 2 32 > 0 ( , 32 ) ( 32, + ) ( )

: = ( + 2 ) 4 2 = 2 + 4 + 4 8 =2

= 2 4 + 4 = ( 2 )

: x1 + x 2 = (1) x1 x 2 = 8 ( 2 ) x1 = x 22 ( 3)

2

-2, , .

( 3)

( 2 ) x 22 x 2 = 8 x 32 = 8 x 2 = 3 8 x 2 = 2 ( 4)( 4)

( 3 ) x1 = 2 2 x1 = 4 ( 5 )( 4 )( 5)

(1) 4 + 2 = 6 = = 6 ()

3

+21=3= .2

B43.

x 2 - ( + 1) x = 0 .

x1 = 2, :. x2 x1 .

x1, x2 :3x12 -7x12 x 2 -7x1 x 2 2 + 3x 2 2 = 2

x 2 3x 2 + 1 = 0 . , .

B44.

B45.

.

B47.

. 2 2 = 2 + 2. 2

x 2 + x + 3 = 0 2 2 3 2 + 2 2 3 24 3 5 + 4 3

B46. 2x 2 + ( 3 19 ) x 8 = 0 .

x 2 + x + = 0 .

,

A=

. .

+ 2 = 0

B48.

x 1 , x 2 x 2 + 2x + = 0 x12 , x 22

x 2 + x + 3 = 0 =2 - 4.

67

f ( x ) = x 2 + x + , 0 .

1, 2 x 2 + x + = 0 -

= 2 4 -

(1)

. , 0 .: > 0 1 , 2 , x 2 + x + = x 2 + x + == x 2 ( 1 + 2 ) x + 1 2 = ( x 1 )( x 2 )

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

= 0 , 1 = 2 = =

(2)

, :2

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

2

(3)

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

.

3

5

:. f ( x ) = x 2 ( k + + 2 ) x + 2 ( k + )

. g ( x ) = ( x + 3 ) + ( x + 4 ) + ( x + 5 ) ( x + 6 )2

2

2

2

. f ( x ) = 0 .

=

f ( x ) = ( x 1 )( x 2 ) == ( x 2 )( x k )

: 1,2 =

k + + 2 (2 k )=2k++2+2k=21 =2= = k + + 2 2 + k + = k + 12

=

2k + + 2 ( k + + 2) 8 ( k + )=2

2k + + 2 (2 k )=2

. g ( x ) = x 2 + 6x + 9 + x 2 + 8x + 16 + x 2 ++10x + 25 x 2 12x 36 = 2x 2 + 12x + 14

g ( x ) = 0 , :

68

1,2 =

B - 4

6 36 2 14 6 8 6 2 2===222

( + 1) x 2 + x 1 =

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

= 3 + 2= 1 2 = 3 2

g ( x ) = 2 ( x + 3 2 )( x + 3 + 2 ) .

6

7

:x 2 x + x x 2 3x + 2 2

( + 1) x 2 + x 1 .

:

( + 1) = 0 = 0 = 1 -1 . ( + 1) 0 0 1 2 = 1 4 ( + 1) ( 1) = 1 + 4 2 + 4 == 4 2 + 4 + 1 = ( 2 ) + 2 2 1 + 1 =2

= ( 2 + 1) 0

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

= ( 3 ) 4 1 2 2 = 9 2 8 2 = 2 02

x1,2 =

2

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

x1,2 =

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

1 +12 ( + 1)1=2 ( + 1)

3 2 3 2==2 12

x 2 3x + 2 2 = ( x 2 )( x ) .

=

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

( x ) ( x + )x += x x 2 .( x 2 )( x ) x 2

B49. :i. x 2 6x + 5

ii. 3x 2 4x + 1

iii. 5p 2 25

iv. x 2 ( + ) x +

A=

B52.

i.

x2 x 6x 2 + 3x + 2

ii.

x 2 + ( 2 + ) x + 2x 2 2

B51. :

:

=

B50. :B53.

6x 3 x 2 x4x 2 4x + 1

( x 2 + 3x 4 )2 ( x 2 x )2( x 3 1) ( x 2 + x 2 )

:

K=

4x 4 13 2 x 2 + 9 42x 2 + x-3 2

3A4

69

f ( x ) = x 2 + x + = 0, 0 . :1. > 0, f ( x ) = ( x x1 )( x x 2 )

(1) .

A x < x1 < x 2 x x1 < 0 x x 2 < 0 :

( x x1 )( x x 2 ) > 0

(1), f(x)

. A x1 < x < x 2 x x1 > 0 x x 2 < 0 , :

( x x1 )( x x 2 ) < 0

(1), f(x)

. A x1 < x 2 < x , x x1 > 0 x x 2 > 0 , :

( x x1 )( x x 2 ) > 0

(1) f(x)

.

f ( x ) = x 2 + x +

2

2. = 0 f ( x ) = x + f(x) 2 . x 2

2 3. < 0 f ( x ) = x + + 2 2 4

x R , f(x) R.

70

B - 4

x R x 2 + x + > 0 x 2 + x + < 0 ( 0 )

, x, x 2 + x + . .

.. x 2 + 6x 8 > 0 . x, x 2 + 6x 8 , ( = 1 ). = 36 4 ( 1) ( 8) = 4 > 0 , x 2 + 6x 8 = 1 2 4, .

x 2 + 6x 8

2

0

+

4+

0

(2,4).2 x + 6x 8 0 , , , . [2,4].

2.. x 6x + 12 > 0 .

x, , = 1 . = 36 4 12 = 12 < 0 , x R ... 4x 2 12x + 9 < 0 .12 3= = 36 4 9 = 0 = . 8 24

= 4 , , x

3. .2

71

8

10

N :ii. x + 2x 5 > 0

i. x 3x + 5 0

2

2

i. x 2 + 4x + 3 0

iii. 2x + x 2 < 02

i. = ( 3) 4 1 5 = 11 < 0 = 1 > 02

x 3x + 5 > 0 , x R .2

ii. E = 2 2 4( 1)( 5) = 16 < 0

N :

2iii. x 16 0

i. = 16 12 = 4 > 0 . = 1 > 0 x 2 + 4x + 3 0 . O

= 1 < 0 x 2 + 2x 5 < 0 x R . .iii. E = 12 4( 2)( 2) = 15 < 0 .

= 2 < 0 2x 2 + x 2 < 0 x R . x R .

9

2ii. 3x + 7x 2 < 0

x1,2 =

4 2 1=2 3

x [-3, -1]. :

N :i. 4x 2 8x + 4 > 0 ii. x 2 + 4x 4 0iii. x 2 + 2x 1 0

3x 2 + 7x 2 < 0 , -

i. E = ( 8) 2 4 4 4 = 0 . = 4 > 0

2

1. :3

4x 2 8x + 4 0 x R . 4x 2 8x + 4 > 0 x =

8== 1 28

. x R {1} .ii. E = 16 4 1 4 = 0 . = 1 < 0 x + 4x 4 0 x R . x R .iii. E = 4 4(1)(1) = 0 . = 1 < 02

1x , ( 2, + ) .3 x 2 16 0 4 4. :

x 2 + 2x 1 0 x R . x 2 + 2x 1 0 2x===122

x ( , 4] [ 4, + ) . 2

72

11

B - 4

R ( + 2 ) x 2 2x + 3 < 0

12

x R .

R x 2 + ( 3 ) x + = 0 .

+ 2 < 0 < 2 < 0 (2 2 ) 4 ( + 2 ) 3 < 0

= 0 : 3x = 0 x = 0 = 0 x = 0 . 0 2 :

< 2 < 2 2 224 12 24 < 08 24 < 0

= ( 3 ) 42 = 2 6 + 9 42 = 32 6 + 9

< 2 < 28 ( + 3) < 0 ( + 3) > 0

2 + 2 3 = 0 = 3 = 1

< 2 < 3 < 3 > 0

2

= 0 32 6 + 9 = 0 3 ( 2 + 2 3) = 0

. < 3 > 1 . 3 < < 1 0 .. = 3 = 1 .

B54.

B55.

-x 2 + (2-1)x + 3 + 10 = 0 R .

56.

( 2 ) x 2 + 2 ( 2 3) x + 5 6 > 0

.. f(x) = ( 2)x 2 + 2(2-3)x + 5-6

x R. f(x) = ( 2 ) x + 2 ( 2 3) x + 5 6 > 02

x R.. f(x) = ( 2 ) x + 2 ( 2 3) x + 5 6

x R .

B57.

2

. R x1, x2.. ) :x12 + x 2 2 3x1 x 2

x R.

( 2 ) x 2 + 2 ( 2 3) x + 5 6 < 0 x R .. f(x) = ( 2 ) x 2 + 2 ( 2 3) x + 5 6

x R

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

58.

, :

x 2 3x + 2 > 0x 2 5x 50 < 0x 2 2x 15 > 0

73

B.5

22K

x + y = . (x,y) .. 2x2 .

1x + 1 y = 1 2x2 2 x + 2 y = 2

. (x,y) . .

:

1. 2. 3. 4. Cramer

:

1

2x 3y = 7 4x + 6y = 14

( 1)( 2)

(1) 2x = 7 + 3y x =( 3)

(2) 4

7 + 3y2

7 + 3y+ 6y = 14 2

(3)

2(7 + 3y) + 6y = 14 14 6y + 6y = 146y + 6y = 14 + 14 0y = 0 , . . (3) : x =

7 + 3y.2

74

B - 5

: y = k :

() :(3)

(x, y)=(

7 + 3y, y) , y R .2

7 + 3k , k R .,k 2

- x + y = x + y = x + y = . (): x + y =

. D=

=-

(1)

Dx D x .Dx =

=

(2)

Dy D y .Dy =

=

(3)

. :1. D 0 : D Dy (x, y) = x , D D

2. D = 0 (Dx 0 Dy 0 ) .3. D = Dx = Dy =0 = = = = 0 0 0 , .. x + y = 0. .x + y = 0

75

. (0,0)., . , ., (x, y) = (0,0).. D 0 (x, y) = (0,0).. D = 0 ( ) , ( (x, y) = (0, 0) )

2

:

yx -1- 2x = - 12 3-2(1 + y) + 3 = -3(x - 2) + 2y ( Grammer): 22 :x + y = x + y =

(3) (4) : 10x 3y = 4 3x 4y = 5

D =

10 33 4

4 Dx = 5

:(1)

: D = 0 Dx 0 , Dy . . Dx = 0 Dy .

yx 1x 1 2x = 1 6 6 2x =323

Dy=

34

10 43

= 10( 4) 3( 3) = 40 + 9 = 49 0

5

= 4( 4) 5( 4) = 16 + 20 = 36

= (10) 5 3(4) = 50 + 12 = 38

y= 6 6 1 2(x 1) 12x = 3y 6 2x 2 12x =2

D = 49 0

= 3y 6 2x 12x 3y = 6 + 2

Dx 36x = D = 49 y = Dy = 38D49

-10x - 3y = -4

(3)

(2) 2(1 + y) + 3 = 3(x 2) + 2 y

2 2y + 3 = 3x + 6 + 2y 2 y + 3 x 2 y = 6 + 2 3

4y + 3x = 5 3x - 4y = 5

(4)

36 38 : (x,y) = , - 49 49

76

3

B - 5

x,y :( 1) 2D x + 3D y = D 4D x + 7D y = 11D ( 2 ) , .

DxDy y =.D 0 . x =DD (1) (2) D 0 : 2Dx 3Dy D D + D = D 2x + 3y = 1 4D7D11Dxy 4x + 7y = 11+= DDD

2 3= 2 7 ( 4) 3 = 14 + 12 = 26 04 7

1

4 11

Dy =

Dx 26 x = D = 26 = 1 y = Dy = 26 = 1D26

(x,y) = (1,-1)

4

( 2)x + y = 2() : 3x + ( + 2)y = 12

R ().

2 = 2 ( + 2) 12 =12 + 2

( 2)

- 2 2= 12( 2) 3 2 = 12 24 6 =312= 6 24 = 6( 4)

( 3)

:1 :(1)

: D 0 ( 4)( + 1) 0

4 0 4 + 1 0 1 () (x , y ) Dx2( 4)2x = D = ( 4)( + 1) = + 1 y = Dy = 6( 4) = 6D ( 4)( + 1) + 1

= 2(11) (4)(1) == 22 4 = 26

= ( 2)( + 2) 3 =

= 2 2 8 = 2 ( 4)

= 7 + 33 = 262

+2

3

Dx =

1 3 Dx == (1) 7 (11) 3 =11 7

Dy =

2 4 3 = 2 3 4 = ( + 1)( 4 ) (1)

: D=

2

D=

2 : D = 0 = 4 = -1. = 4 :( 2)

Dx = 2 4(4 4) = 8 0 = 0( 3)

Dy = 6(4 4) = 6 0 = 0 .

= 4 D = Dx = Dy = 0 x 2 0 ( 3 0 ) () .

. = 4 ()

77

.

. = 4 () ::2

(4 2)x + 4y = 2 4 2x + 4y = 8

1. 4 1 ()

x + 2y = 4 x = 4 2y

6 2, (x,y) = +1 +12. = 4 () (x,y) = (4 2 y,y), y R

(x,y) = (4 2 y,y), y R .. = -1 :( 2)

Dx = 2( 1)( 1 4) = 2( 5) = 10 0

3. = 1 () .

, = -1 D = 0 Dx 0

B59. ()

B62.

x y = 24 x y =

2D x + D y = 4DD x 3D y = 5D .

R

B60. .

x-y = 3 x + y = -2

22 x,y :

, .

B63.

. (x0,y0)

22 x,y ,

2x 0 + 3y0 9

D x D y 3D + 2D x + 3D y + 4D = 0 .

B61. 2x2 ( -

) :

D 2x = 2DxDy D 2y 2x-y = 3 .

B 64.

22 x,y :

2x + y = 18 D 2x + D 2y = 2D x D y

78

B - 5

33) 3 :

x + y + =

) ; 3 3

1 x + 1y + 1 = 1 2 x + 2 y + 2 = 2 x + y + = 333 3

.

) (x,y,)

5

2x + 3y 5 = 10

:

:

4x y + 7 = 4

2x + 3y 5 = 103x + 2y = 0 4x y + 7 = 4

4x + 6 y 10 = 20 4 x + y 7 = 4

3x3 . : ( ) .. x .:

2x + 3y 5 = 10 3x + 2y = 0

(2) (1)

(3) (2)

6x + 9 y 15 = 30 6x + 4 y 2 = 0 : 13y-17 = 24 (5) : .

: 7y -17 = 30 (4) : (4) (5) .

13y 17 = 307 y 17 = 24

(4) (5)

M : 6y = 6 y = 1, = -1. (y,) = (1,-1) : y = 1 = -1 x.:(1) 2x + 3y 5 = 10 2x + 3 1 5 ( 1) = 10 2x + 3 + 5 = 10 2x = 2 x = 1

(x,y,) = (1,1,-1)

79

B65.

x + 2y = 03x + 5y 2 = 02x y = 0

B66. x + y + z = 2y + z + = 5 y + z + = 3 + x + y = 8

-

B67. :1 1x + y = 21 1) (1 ): + = 3 ,y 1 1 + = 15 x

1 xyx + y = 21 y) ( 2 ) =y+ 3 x1= x + 15

2 .

.

6

2y 2 + 2y 24 = 0 y 2 + y 12 = 0

:

= 1 4 ( 12 ) = 49 > 0 :

y + xy + y 2 = 24y = x 1

y1,2 =

y=x-1 x=y+1 . :y + y ( y + 1) + y = 24 y + y + y + y = 24 .2

2

2

1 49 1 7 3===222 4

: y = 4y = 3 x = y + 1xy1=+

(x, y) = (4, 3) (x, y) = (-3, -4).

B68.

2xy + 1 = 11x + y = 1

B69.

x 2 + y 2 = 73 xy = 24

80

B.6

B - 6

- P(x) = 0 P(x) . :1. 0 0 .2. P(x) = 0 , :

P() = 0 P ( x ) = (x ) ( x ) x - P(x).3. Horner () P ( x ) = 2x 3 3x 2 x + 2 x 2 , Horner.2

-3

-1

-2

4

2

2

2

2110 P(x). 2 x2. P(x). / 2. ( ) . , . , Horner 0 P(x). :P ( x ) = 2x 3 3x 2 x + 2 = ( x 2 ) ( x 2 + x + 1)

, ( , , , , ...) , , . , P(x) :

P(x) = P1 (x) P2 (x) ... P (x)

81

-

T:

P(x) = 0 P1 (x) P2 (x) ... P (x) = 0 P1 (x) = 0 P2 (x) = 0 . . . P (x) = 0

P(x) = 0 ,

Pi (x) = 0 , i= 1 , 2 , . . . , , . : 1: P(x) = 0 , , . 2: i, . 3: P(x) . 4: Horner, 1 : P(x) = (x 1 ) 1 (x) , Horner. 5: 2 4 1 (x) P(x) . 6: i (x) , , . P(x) , .

1

:

x x 4x 4 = 04

Horner 1 = 1

2

1

0 = 4 .

1

0

-1

-4

-4

-1

1

0

4

0

-4

0

-1

-1

: 1, 2, 4

P(x) = (x + 1)(x 3 x 2 4)

: P(1) = 14 12 4 1 4 = 8 0

1 (x) = x 3 x 2 4 , :

( 1 P(x) )

: 1, 2, 4

P( 1) = ( 1) ( 1) 4 ( 1) 4 = 0 ( -

(: 1 P(x) ,

4

2

- 1 P(x) )

-

82

B - 6

1 (x) , 1.)1 (1) = 6 0 ,( -1 1 (x) )1 (2) = 23 2 2 4 = 0 ( 2

1 (x) ) Horner 2 = 21

1

-1

0

-4

2

2

4

1

2

0

2

2 (x) = x 2 + x + 2 ,

= 12 4 1 2 = 7 < 0 2 (x) R. :1 = 1 2 = 2 . . :

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

: P(x) = (x + 1)(x 2)(x 2 + x + 2)

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

70. o :i.

1 3 1 2 11x x + x = 03236

vi. x 4 4x 3 x 2 + 16x 12 = 0vii. 6x 4 5x 3 + 19x 2 15x + 3 = 0

ii. x 3 x 2 x + = 0

2517viii. x 3 x 2 + x = 0326

iii. 3x 4 8x 3 12x 2 + 32x = 0

ix. 2x 3 7x 2 3x + 18 = 0

iv. 3x

2

(x

2

3x + 2 ) = 27 ( x 1)( x 2 )

x. 2x 2 x + ( x 2 )( 2x + 1) = x 2 5x + 6

v. 2x 4 7x 3 3x 2 + 18x = 0

:

1: P(x) . 2: . 3: , , x, .

83

-

*: x , ' . (+) . (-). . . x .

2

:

P(x) = x (x 3) ( x 2 + x 1)

x + 4x 4x + 3x 04

3

2

2 (x) = x 2 + x 1

P(x) = x + 4x 4x + 3x .4

3

2

= 12 4( 1) ( 1) = 3 < 0 . :

P(x) = x (x 3) ( x 2 + x 1)

P(x) x. (1= 0)2 P(x) = x (x3 + 4x 4x +3)

1 (x )

1 ( x ) = x 3 + 4x 2 4x + 3 : 1, 3 .

1 (1) = 2 0 , 1 (1) = 12 0 , 1 (3) = 0 2 = 3 .

-1

4

-4

3

-3

3

-3

1

-1

0

3

0

x

-

0

x 3,

Horner 1 (x) 2=3.-1

x

-

x 2 + x 1 12345678P(x)

123456781234567812345678- 012345678

+

3+

+

-

0

-

123456789012312345678901231234567890123012345678901231234567890123

+

+

x ( , 0] [3, + ) .

B.71 :

iv. x 3 6x 2 + 11x 6 0

i. (1 x)(x + 2) (x 3)x

vi. x 3 2x 2 + x 2 > 0

ii. x 3 + 4x 2 3x 0

vii. x 3 ( x + 1) 2 > x ( x 1)

-

84

B - 7

B.7

:1.

2.

3.

4.

-

5.

x 4 + x 2 + = 0 ,, R 0

:1: x 2 = y , y 0 (1) y

y 2 + y + = 0 , .

2: ( ).3: (1) ( ).

1

(1) :

x 5x + 4 = 04

2

x 2 = 1 x = 1 x 2 = 4 x = 2

:

x 2 = y, y 0 (1) , -

x = 1 x = 2 .

: y 5y + 4 = 0 y = 1 y = 42

72.

75.

x + x + = 0, 0 R;

( + )x 4 + (2 - -10)x 3 + 2x 2 - ( - - 7)x + 6 - = 0

4

2

73. :

76. :

. 4x + 3x 1 = 0 . 4x 4x + 1 = 04

2

4

2

. 5x 4 2x 2 + 3 = 0

74. > 0, > 0.

.

64x 4 52x 2 + 9 = 0 (1)

77. :2

x 2 2 =

1 x2

x 4 10x 2 + 169 = 0 (1)

78. :x 4 + 3x 2 4 = 0 (1)

85

. :

... ( ).T .. . .. . .. .

5

2

-

:

x=

x+76+ 2x+1 x + x

:

1, 2, 3, 6 .

P( 1) = ( 1)3 7 ( 1) 6 = 0

1 = 1 .

... x ( x + 1) .

Horner -1

:

x + 1 0 x 1 1

x 02x + x 0 x (x + 1) 0 x 1 ...x+76x (x + 1) x = x (x + 1) + x (x + 1) 2x +1x +x

1

0

-7

-6

-1

1

6

-1

-6

0

-1

P(x) = (x + 1) (x 2 x 6) .

(x) = x 2 x 6 : 2 = 2

3 = 3 .

1 = 1 , -

x 3 + x 2 = x (x + 7) + 6 x 3 7x 6 = 0 (1) (1):

P(x) = x 3 7x 6 = 0

: x = 2 x = 3 .

79. .

.

:

23 + x2+=1x 2 x x 2 2x + 12xx +13=+4x 1 2 x x 2 3x + 2

.

.

2x 1 x 23x 6+= 2x 3 x 2 x 5x + 62 ( x 1)x

2

+

x 2 2x + 125=2x4x 4x

86

B - 7

, . : . ( ).

, ( , ) . , , .

1 : :

f ( x) = g ( x)

7

3

:

f ( x ) 0 , g ( x ) 0f (x) = g(x) 2 f ( x ) = g ( x )

5 x x = 1

:

5 x x = 1 5 x = x +1 :5 x 0 x 5 :

(1)

:

(

5x

)

x1 = 1 x 2 = 3 (1) x = 3 x1 = 1, x 2 = 3 .

x + 1 0 x 1

x [ 1,5]

(1) 2

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

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

2

x 2 + 3x 4 = 0 x = 1 x = 4 . x = 4 , , x = 1 , .: , . ( ) ( )... x 2 = 1 (1) ( x 2 ) = 122

.

4

: x 2 x + 1 = 1 2x

x R < 0 x 2 x + 1 01x121 2x 0 x 2

:1x 2x x + 1 = 1 2x x 2 x + 1 = (1 2x )22

1x 2 x 2 x + 1 = 1 4x + 4x 2

87

3x ( x 1)3x 2 3x = 0x = 0 x = 1x=0111x x x 222 :

F(x) 0

G (x) 0

2 :

F ( x) + G ( x) = H ( x)

H(x) 0

.

(

F(x) + G (x)

)

2

= H(x)

F ( x ) + G ( x ) + 2 F ( x ) G ( x ) = H2 ( x ) 2 F ( x ) G ( x ) = H 2 ( x ) F ( x ) G ( x ) 2H ( x ) F ( x ) G ( x ) 04F ( x ) G ( x ) = ( H 2 ( x ) F ( x ) G ( x ) ) ...2

5

:

x + 32 + x = 16

:

x + 32 0 x 32 x0x0x 0

M x 0 :

x + 32 + x = 16 x + 32 + x + 2 x ( x + 32 ) = 256 112 x 0 2 x ( x + 32 ) = 224 2x x ( x + 32 ) = 112 x 2 x ( x + 32 ) = (112 x ) x 112 x 112 222 x ( x + 32 ) = 12.544 224x + x x 32x = x 224x + 12544 x 112 x 112 x = 49256x = 12544 x = 49

88

B - 7

F(x) 0

:

G (x) 0

3 :

H(x) 0

F ( x) + G ( x) = H ( x)

2

F(x) + G (x) =

(

H(x)

)

2

F2 ( x ) + G 2 ( x ) + 2 F ( x ) G ( x ) = H ( x )

6

:

x1 + x 4 = x+ 4

(I)

7

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

:

x 1 + x 4 + 2

( x 1)( x 4 ) = x + 4 2 ( x 1)( x 4 ) = 9 x

9 x 0 x 92 2 4 ( x 1)( x 4 ) = ( 9 x )4 ( x 1)( x 4 ) = ( 9 x )x 9x 9 2 2224x 20x + 16 = 81 18x + x4x 20x + 16 = 81 18x + xx 9x 913 13 x = 5 x = 233x 2x 65 = 0 x = 5 x = 3

80. : x 281.

:

x 3 2 = 0

2 + x 5 = 13 x

82. :83. :

x + 3 x = 10x + 32 = 16 x

84. :.

x + 2 x 1 = 1

.

2x + 2 + 3 = x

.

2x + 8 x + 5 = 1

.

x 1 + x + 2 = 2x + 5

89

: x = , 0, R, 2

() C

C.( )() R = 2 + 1, : x 2 +1 = x = 2 +1 = 2 +1 , < 0 2 + 1 : 2 +1 , > 0x =( 2 ).

.. R x 5 = 32 .xR

x 5 = 32 x = 5 32 = 5 ( 32) = 5 32 x = 2 = 2 () : > 0 : x 2 = > 0 2 : x = ..

R x

16

2

= 2007xR

16

E x16 = 2007 x = 2007 < 0 : x 2 = < 0 2 () ... x 6 = 5 , R.

: Ax + x = 0, , R, ,

: > , 0 . :

x = 0 x = 0 Ax + x = 0 x x + = 0 x + = 0 x =

90

B - 7

BBB , x= > 0 x =AAAi. () B < 0 RA

ii. () BB > 0 x =AA B < 0 x = B A A

7

: x 5 5x 2 = 0

: x 9 x 5 + x 4 1 = 0

x = 0x 5x = 0 x ( x 5) = 0 3x 5 = 05

8

2

2

3

x = 0x = 0 33x = 5x = 5

x 9 x 5 + x 4 1 = 0 x 5 ( x 4 1) + x 4 1 = 0 x 4 1 = 0 ( x 4 1)( x 5 1) = 0 5x 1 = 0x 4 1 = 0x 4 = 1 x = 1 5 5x = 1x 1 = 0x = 1

85.

87. , :

i. x 1 = 0

ii. x + 16 = 0

iii. x + 1 = 0 i

v. 8x + 125 = 0

6

3

4

3

x 3 24x 72 = 0 3

86.

i. 8x 3 125 = 0

ii. 3x 7 2x = 0

x x = .

91

:

x + x + x = 0

, , R, , , . :

= > >

= = = + = + = + + = + 2

x + x + x = 0 x + 2 + x + + x = 0 x=0 x ( x 2 + x + ) = 0 Ax 2 + x + = 0 ( )

A () x = y y.

9

: x10 26x 7 27x 4 = 0

x = 0x10 26x 7 27x 4 = 0 x 4 ( x 6 26x 3 27 ) = 0 63x 26x 27 = 0

y = 27 x 3 = y : x 6 26x 3 27 = 0 y 2 26y 27 = 0 y = 1x 3 = 27 x = 3 27 = 3x 3 = 1 x = 1

88. :89. :

x2x3 + 2=3x ( x 3 2)

90. :

x13 5x 7 + 6x > 0

x 4 4x 7 + 3x 3 = 0

92

B - 7

.

10

11

:

:

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

( x 2 6x + 4) 2 + ( x 2 6x + 3) + 9 = 0

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

y = x 2 6x + 4 :

( x 2 + 5x + 4 )( x 2 + 5x + 6 ) = 24

( x 2 6x + 4) 2 + 5 ( x 2 6x + 3) + 9 = 0 ( x 2 6x + 4) 2 + 5 ( x 2 6x + 4 1) + 9 = 0 y 2 + 5 ( y 1) + 9 = 0 y 2 + 5y + 4 = 0

y = 1 y = 4

x 2 + 5x + 4 = y

(1)

( 2 ) ,

x 2 + 5x + 6 = y + 2 (1) :

y ( y + 2 ) = 24 y 2 + 2y 24 = 0 ,

y = 6 y = 4 . (2) :

i. y = 1 x 2 6x + 4 = 1

x 2 6x + 5 = 0 x = 1 x = 5ii. y = 4 x 2 6x + 4 = 4

x 6x + 8 = 0 x = 2 x = 42

x 2 + 5x + 4 = 6 x 2 + 5x + 10 = 0 ()x 2 + 5x + 4 = 4 x 2 + 5x = 0 x ( x + 5) = 0 x = 0 x = 5 . : x = 0 x = 5 .

91. :

. 2 3 x + 5 2 x 4x 3 = 0

2 3 x 5 2 x + 4x 1 = 0

92. :

. x x 4x 19 x 14 = 0.

( x + 1)

.

(x

( x 2 5x + 8) 5 ( x 2 5x + 9 ) + 11 = 02

93. :6

3

11. x + 28 x + + 27 = 0xx

.

2

(x

3

2

+ 5x 2 x + 1 19 = 0

+ x + 1) 4 ( x 2 + x + 2 ) + 7 = 02

2

+ x + 1)( x 2 + x 9 ) = 21

93

x

1. x = , 1, 0 x

x =

1.

x 3 + x 2 + x + = 0

( 3 )

x 3 + x 2 x = 0

( )

x 4 + x 3 x = 0

( 4 ( x 2 ).(4 )

.. 2x 3 5x 2 = 5x 2 2x 3 5x 2 5x + 2 = 0.. 3x 4 + 2x 2 = 3 3x 4 + 0x 3 + 2x 2 + 0x + 3 = 0x 3 + x 2 + x + = 0

( )

( x

3

+ ) + ( x 2 + x ) = 0 ( x 3 + 1) + x ( x + 1) = 0

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

(1)

x +1 = 0

x 2 ( ) x + = 0

( 2)

(1) x = 1 (2) x =

( )2

2

4 2

, ( ) 4 2 0 .2

12

2x 3 + 7x 2 + 7x + 2 = 0

94

B - 7

-

:

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

( 2x

3

+ 2 ) + ( 7x 2 + 7x ) = 0

(1) x = 1 .

( 2x + 1) + 7x ( x + 1) = 02 ( x + 1) ( x x + 1) + 7x ( x + 1) = 0( x + 1) 2 ( x x + 1) + 7x = 0( x + 1) ( 2x + 5x + 2 ) = 03

(2) 2

2

1.2

:

2

x1 = 1, x 2 = 2, x 3 =

2

12

x 3 x 2 + x = 0

: ( x 3 ) ( x 2 x ) = 0 ( x 3 1) ( x 1) = 0

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

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

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

( )

(1)

x 1 = 0

x 2 + ( ) x + = 0

( 2)

(1) x = 1 . (2) x =

( )

( )

2

4 2

2

, ( ) 4 2 02

13

:3x 3 13x 2 + 13x 3 = 0

( x 1) ( 3x 2 10x + 3) = 0

x 1 = 0

:

( 3x 3) (13x 13x ) = 0 3 ( x 1) 13x ( x 1) = 0 3 ( x 1) ( x + x + 1) 13x ( x 1) = 0 3

2

(1)

3x 2 10x + 3 = 0

(1) x = 1 . (2) x = 3 x =

3

:

2

( x 1) 3 ( x 2 + x + 1) 13x = 0

1.3

x1 = 1, x 2 = 3, x 3 =

1.3

95

x 4 + x 3 + x 2 + x + = 0 , 0

x 2 , x 2 + x + + + 2 = 0 x

+ x + + = 0x

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

x +

111= y (2) x + = y 2 x 2 + 2 = y 2 2xxx

(1) :( )

2 x + 2x

( y 2 2 ) + y + = 0 y 2 + y + 2 = 0 ()

(2) y, y y ()

11= y ( 3) x + = y ( 4 )xx2 ( 3) x yx + 1 = 0( 3) x +

( 4 ) x 2 yx + 1 = 0

( 4)

( 3 ) ( 4 ) .

14

:6x4 + 5x3 38x2 + 5x + 6 = 0

x 2

5 6+=0x x25 2 6 6x + 2 + 5x + 38 = 0xx 1 1 6 x 2 + 2 + 5 x + 38 = 0 (1)xx 1 x + = y (2)x (1) :6 ( y 2 2 ) + 5y 38 = 0 6y 2 + 5y 50 = 06x 2 + 5x 38 +

52

: y = , y =

103

(2) y 1101 5x + = ( 3) x + = ( 4)x3x 2 (3) 2x 2 5x + 2 = 0

12 (4) 3x 3 + 10x + 3 = 0 x = 2, x =

13 :x = 3, x =

x1 = 2, x 2 =

11, x 3 = 3, x 4 = 23

96

B - 7

x 4 + x 3 x = 0

:

( x

4

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

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

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

( )

x2 1 = 0

(1) ,

x 2 + x + = 0

( 2)

(1) x = 1 . (2) x =

2 4 2, 2 4 2 .2

15

:

x2 1 = 0

4x4 17x 3 + 17x 4 = 0

4

4 ) (17x 3 17x ) = 0

(2) x = 4, x =

4 ( x 4 1) 17x ( x 2 1) = 0

( 2)

1.4

:

4 ( x 2 1)( x 2 + 1) 17x ( x 2 1) = 0 ( x 2 1) 4 ( x 2 + 1) 17x = 0

4x 2 17x + 4 = 0

(1) x = 1 .

:

( 4x

(1) ,

x1 = 1, x 2 = 1, x 3 = 4, x 4 =

1.4

( x 2 1)( 4x 2 17x + 4 ) = 0

94. :

96.

:

i. 2x + 7x + 7x + 3 = 0

i. 6x 4 + 5x 3 38x 2 + 5x + 6 = 0

ii. 3x 4 10x 3 10x 3 = 0

ii. 4x 4 17x 3 + 17x 4 = 0

3

2

iii. x 3 2x 2 + 2x 1 = 0

97. N :

95. :i. 2x 3 7x 2 + 7x 2 = 0ii. 3x 3 13x 2 + 13x 3 = 0

5x 4 16x 3 + 2x 2 + 16x + 5 = 0

98.

N :

6x 4 + 5x 3 38x 2 + 5x + 6 = 0

97

B.8

1. A ( x ) B ( x ) ... > 0 A ( x ) B ( x ) ... < 0 0 > 0 0

( ) .

1

:

( x 2 7x + 10) ( x 2 4x 4) ( x 2 + 3x 7 ) (1 x ) 0

. , , x, . x , '. (+) -

. (-). . . x .

(1) : x {2} [1, 2] [5, + )

AN

98

B - 8

B99. :

102. :(1 x ) ( x 2 10x + 21)( x 2 + x 5 ) < 0

(1 3x)(2x + 4)(4 x)(x 6)(7 x) 0

B100. :

103. :

(2x -1)(1- 5x)(x 2 - 5x + 6) 0

x ( x 2 7x + 10 )( x 2 5x + 4 ) ( x 2 9 ) 06

7

101. :( 3x 7 ) ( x 2 + 3x + 4 )( x 2 6x + 9 ) < 0

A ( x)>0B ( x)

A ( x)0B ( x)

A ( x) 4 x +1

3x 1 > 4 ( x + 1)

3x 1> 4 x +1

, x + 1 , x R . :i. 0 ( ) .ii. () . -

99

:

A(x)B(x)

> ( , , ( , , < ) 0 ( x ) B ( x ) > ( , , < ) 0, B ( x ) 0

.

2

:

(x

2

+ 3 ) ( x 1)

( x2 4 )

2

0 . x R =1>0 02 4 3 < 0 . ( x 1) > 0 . x 1 .

x +1 x + 3x +1 x + 30x 1 2x + 1x 1 2x + 1( 2x + 1)( x + 1) ( x + 3)( x 1)0( x 1)( 2x + 1)

2

x 4 > 0 . x ( , 2 ) ( 2, + ) .2

x . , .

2x 2 + 2x + x + 1 x 2 3x + x + 30( x 1)( 2x + 1)

x2 + x + 40( x 1)( 2x + 1)

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

:

x2 + x + 4 0 , .x 1 > 0 x > 12x + 1 > 0 x >

12

AN

100

B - 8

.(. .97)

1 x ,1 . 2

4

:

( x 3 ) ( x2 7x + 10 )0( x + 1)( x 4 )

.(. .97)

x ( , 1) [ 2,3] ( 4,5] .

104. :i.

x 5

x + 2 x 3 x2 x 6

108. :

2

B109. :

2x 3x 1+ 0x 7x 2 + 10x > 03

22. ( x 1) ( x x + 4 )( x + x 7 ) 0

(1) . .

(1) x [1, + ) .. : 2x + 1 > 0

( 2)

2x + 1 > 0 2x > 1 x >

12

1 (2) x , + . 232. : x 7x + 10x > 0 ( 3)

x 3 7x 2 + 10x > 0 x ( x 2 7x + 10 ) > 0 . .

(3) x ( 0, 2 ) ( 5, + ) AN

104

B - 8

K .

x [1, 2 ) ( 5, + ) .

111. x

112. :

: x 0. 6x 2 + 5x + 1 > 0 2 x + 5x 6 < 0

3 x>0ii. x1 3x 0 4

x 1>0. 2x + 1( x 2 4 )( x 2 + 2x + 4 ) < 0

(. ),

: 0 . ( ), : , > 0 :

> >

> , : > , < 0 : < , : .

8

N : x 2 > x 4

E,

x 2 0. x 2

( 1 ) : x 4 < 0 x < 4 2 x < 4 1

105

, 2 . x 4 0 x 4 :

x2 > x4

(

x2

)

2

> ( x 4)

3 x < 2 x 2 < 0 5 ,

2

5x 3 > 0

x 2 > x 2 8x + 16 x 2 9x + 18 < 0 3 < x < 6E x 4 4 x < 6 . x R 2 x < 6 .

x 2 , , :5x 3 > ( x 2 ) x 2 9x + 7 < 02

9 539 + 53 x 1

9 539 + 53 0 .x>

7. x 1 .3

2x x 1

(

3x + 7

) >(2

x 1

)

2

3x + 7 > x 1 2x > 8 . x > 4 x 1. x R x 1 .

10

5x 3 > x 2

9 + 53 3 , 2 2, , 2 5

5x 3 > x 2

, , 5x 3 0 x

,

35

3 9 + 53 ,.2 5

113. :

x 1 > x 3

114. :

2x x 2 < 3 x

115. :

3x + 5 > 1 x

116. :5 2 x > 3 x

117. :

x + 1 + 2x + 3 > 5 AN

106

B.9

B - 9

-

.1. 2 x + = 1 , 3 x 7x + 6 = 0 , .32. x + = 2 x 2 . ( ) . .

x = . , : x = 2 + x = = , x = 2 +

x = . : x = 2 + x = = , x = 2

107

( ) + (. ).

k + k . x = , R :

x = = x = + , x = .

( ) + (. ).

k + k . x = , R :

x = = x = + , x = .

. , 2 x 1 , 3 x 8x + 7 > 0 , .6 ( ) . .

108

B - 9

:

x < x > , 1 < < 1 [0, 2] . [0, 2] = . (. ) 1 2 = 2 1 1 < 2. 0 < 1 1 < 0

x > 2 < x 2 0 x < 1

( )

x < 1 < x < 2

( )

O .

O ..

x

15 x x 2333

x

1242 x x 2 0 x 2333

:

452xx3333

2 45 2k + 3 , 2k + 3 2k + 3 , 2k + 3 , k

:

x < x > , 1 1 [ 0, 2 ] =

x = . 1 , 2 [0, 2] , = . 0 < 1 1 , 2 [0, ] , = . 0 1 = 1 < x < 2

( )

109

x < = 2 < x 2 0 x < 1 ( )

1 < 0 , 1 , 2 [,2] , = . 1 = 2 < x 2 0 x < 1 ( ) = 0 = - 1 = 1 = 2 , = 3/2 , = /2 ..

x

:

32

(1)

32 2= = x = x = =.23333 3( 1, 2 /3 2/3). x [0, 2] , : x =

x [0, 2] (1)

2x33

2 2k + 3 , 2k + 3 , k Z .

1

: x =

32

3= 32

. 3 : x = 2 + ,3x = 2 + , Z .3

110

B - 9

: 3x + x = 1

2

x + x = , 0 : :(1)x + x = , = 2 2 (1) :

x + x = x + x =

x + x =

( x ) = , ,

1 2 + 2 2 , . = 3 :33 x + x = 1 x + x = 1 33

x + x = 333

x = 2 +33 x = , 33 x = 2 33

x = 2 +

2 x = 2 , .3

118. :

.122

. 2x + = x6

3 x =.42

. 2 x 2 x = 0

119. : 3x = 1 .4

120. :2 2 x + 3x 2 = 0 .

121..

:

123.

.

12x = , x = x = 322. i. 24x > 3

ii. x > 1

iii. x 0

iv. x > 1

( 2x + 1)

v. 2x 3

. 2x =

2

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

2 x [ 0, 2 ] .2

111

-

.10

-

. x . , . :

.

:1 : .

x = ,

, > 0 1.

x = x = x =

3x = 81 3x = 34 x = 3

..

2 : , . x = x = ...

.

f

(x)

=

f ( x )

. .1 :..

1.

III.

x ,>0 1.

= 32 2 x

2

5x +11

= 25 x 2 5x + 11 = 5

.. 63 2x = 1 63 2x = 60 3 2x = 0 2 x = 3 x =

33x=22

, : f ( x ) = g ( x ) f ( x ) = kg ( x ) f ( x ) = kg ( x ) ...x +1

5x +11

.

f(x) = g(x) f, g

2

x 2 5x + 6 = 0 x = 2 x = 3

, > 0

2x

..

3

27 x 2 = 32x 4 3

x +1x 2

= 32x 4 3

x +12= 2x 4 3x + 3 = 2 ( x 2 )x2

2x 2 8x + 8 = 3x + 3 2x 2 11x + 5 = 0 111 2 x ( x 5) = 0 x = 0 x 5 = 0 x = x = 5222

, . -

112

B - 10

xV. f(x) = g(x), > 0 1.

x = > 0 ... 3x + 2 + 5 3x + 3x 1 3x 2 = 128 ( 2 )1 2 x 23x + 2 + 5 3x + 3x 1 3x 2 = 128 3 3 + 5 + 3 = 128 3128 3x = 3x = 9 x = 21289

.. 9 x 4 3x + 3 = 0 (1) . 3x = > 0 g ( x ) = ( 32 ) = ( 3x ) = 2x

V. f(x) = g(x), f, g

:

(1) 2 4 + 3 = 0 = 1 = 3

3x = 1 3x = 30 x = 0 ,

x ... 3 2x 4 5x 2 = 2x 1 6 5x 3 2 5. 5x

2x 4 5x 2 2x 15x 312 1 1 2 1 x = x 6 3 4 2 = 6 3 x555555 2 55 2x

3

,>0

x

x

( f ( x ))

(f(x)) g(x) = 1,.. f(x)>0

x

4

3222= x=4 = 5 1250 5

.

f, g

x

1610 2 1 3 1 2 = =25 125 16 2 5 32 5 125

1

VI.

3x = 3 x = 1

g( x )

f ( x ) = 1=1 g ( x ) = 0 f ( x ) > 0

( x 2 + 2x 2 )x

2

3x

=1

(1)

x 2 + 2x 2 = 1 ( 2 )(1) g ( x ) = 0 f ( x ) > 0 ( 3) x 2 + 2x 2 = 1 x 2 + 2x 3 = 0 x = 1 x = 3 ( 3) x ( x 3) = 0 x 2 + 2x 2 > 0

x = 0 x = 3

x < 1 3 x > 1 + 3 . T x = 3 .

2

113

-

B. .i. 0 < < 1 :

f

(x)

( )

> g x f ( x ) < g ( x ) f

(x)

( )

< g x f ( x ) > g ( x )

f ( x ) , g ( x ) x.ii. > 1 :

f

(x)

( )

> g x f ( x ) > g ( x )

f

(x)

( )

< g x f ( x ) < g ( x )

, , ... .

1

:

3

x+ 2

+ 53 + 3x

x 1

3

x 2

= 128

3 9x 5 6x + 22x + 1 = 0

(1)

:

2 22x 5 2x 3x + 3 32x = 0

3x 3x = 1283 9 3x = y :3x 32 + 5 3x +

y y9y + 5y + = 128 128y = 1.152 y = 93 9

3x = 9 = 32 x = 2

2

32x :2x

x4

2

x 1

=5

x2

65

x3

:

x

3

x

x

2255= 6 32 4 2 525

6 3 1 12 x = 5x 16 2 25 125 x

x

4

1622 2 = x=4 =62555 5 :

(3)

3 : y1 = , y 2 = 1 .2 (2), (1) :x

x

(2)

2: = y , (2) :32y 2 5y + 3 = 0

x

x

222 5 + 3 = 033

:

32

3

(1)

1

3 22 = = ,2 33

4

x

22 =1= 33 = -1 , = 0.

0

:. 3 x + 3 x 1 >

457+ x ,x+ 233

. 32x 22x + 2 < 11 4x 1 9x -

114

B - 10

5

. :

4573x5 7+ x 3x + > x + x x +23 3 333

3x + 3x 1 >

3>14 3x 123x3> x > x 3x 1 > 31 x 33 33

x 1 > 1 x 2x > 2 x > 1. 32x 22x + 2 < 11 4 x 1 9 x 11 9x + 9x < 4x + 4 4x 427 x 2 9x < 4 8 9 x < 27 4 x 4x

2793 < 842 2x < 3 x 1

2x

3

3 4

3. e x e x < 0

x+

12

128. :x2 2

1ii. e

32x 4

1

x x + y = 26. x3 ( x + y ) = 1296

1i. e

10. 2 4x + 3 9x = 5 6x

27 x 2

x

3

3x 2

3x + 2 5y = 77 1. x +1 y + 23 + 5 = 544

7. 3 2x 4 5x 2 = 2x 1 6 5x 3

(

)

127. :

4. 32x +1 5 6 x + 2 4 x = 0 ,

6. x 2 7x + 10

(

4. 188 4x < 54 2

ex > 0x2 2

ex < 0

12

129. :32x 28 3x 1 + 3 0

115

- -

.11

- -

. :log f ( x ) = log g ( x ) f ( x ) = g ( x )

0 < 1 f(x) , g(x) x. :f (x) > 0

g(x) > 0

. :. 0 < < 1 : log f ( x ) < log g ( x ) f ( x ) > g ( x )

(1)

log f ( x ) < log g ( x ) f ( x ) < g ( x )

(2)

. > 1 :

> 0 1 f(x), g(x) x. :f (x) > 0

g(x) > 0

(1) (2). , , , , , . .. x ln x 1 = e 2 , x ln x < e .2 ,. ln x = 2ln x ln x = 2 ln x .

2

-

116

B - 11

. x =

, x = x = log

(

. 2 x = 7 x = log 2 7

).

f (x)

=

,

f ( x ) = f ( x ) = log x = 1 + log 5 3. 5 x 1 = 3 x 1 = log 5 3 x = 1 + log 5 3 x = 1 log 5 3.

f

(x)

( )

= g x

f , g x , > 0 1.

, :

f

(x)

= g x f ( x ) log = g ( x ) log( )

..

3x

2

+ 3x

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

= ( 3log 3) + 16 log 2 log 3 > 0 (1) :2

x=

3log 3

( 3log 3)

2

+ 16 log 2 log 3

2 log 3

. , . , .

117

- -

1

:

. log ( 4x 1) = 2 log 2 + log ( x 2 1)

(1)

. n ( x 1) + n ( 2x + 4 ) = 2n ( x + 2 ) .

1log ( x + 2 ) = log x + 2 1 = log102 (1) :

x + 2 > 0 , x 3 > 0 ,(2) log ( x + 2 x 3 ) = log (10 3 )

. (1) :

{4x 1 > 0, x

2

1 > 0, log ( 4x 1) = log 4 ( x 2 1)

:

4x 1 = 4 ( x 2 1) 31 4x 4x 3 = 0 x1 = , x 2 = 22 . (1) , :2

x=

32

. n ( x 1) + n ( 2x + 4 ) = 2n ( x + 2 ) (1).

x 1 > 0x > 1 2x + 4 > 0 x > 2 . x > 1 .x + 2 > 0 x > 2

(2) :x > 3 (3) : ( x + 2 ) ( x 3) = 10 3 ( x + 2 ) ( x 3) = 300 x 2 x 306 = 0 :

x1 = 18 , x 2 = 17 (3), (1) x = 18 .

3

:

x log

n [( x 1)( 2x + 4 )] = n ( x + 2 )

= 10

(1)

(1) :

{x > 0

(1) :

x

x log

x

= 100}

(2)

2

x log x = 100 log x log x = log100 12( log x ) = 2 log x = 2 log x = 22

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

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

log x = 2 :

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

2

:

1log ( x + 2 ) + log x 3 = 1 + log 32

log x = 2 = log100 , x = 100 log x = 2 :

log x = 2 = log 0, 01 x = 0, 01 (1) : {102 , 102 }

(1) -

118

4

B - 11

2 x > 00 < x < 2log 2 x < log x x > 022 x < x 2x < x

1log 3 x (1) 2 :

0 < x < 20 < x < 2 21< x < 2x < 2 x > 1x + x 2 > 0

:

log 3 x log 9 x = 2

(1)

log 9 x = log 32 x =

{

x>0 ,

}

12( log3 x ) = 22

(2)

6

{log x + log y = log14, 3x y = 1}

( log 3 x ) = 4 (2)2

, log3 x = 2}

{x > 0

, log3 x = 2}

, x = 32 x = 32

5

(1)

:

{x > 0

:

x y , :log ( x y ) = log14 x y = 14 (1) :

{3x y = 1 ,

: log 2 x < log x

xy = 14}

(2)

(2) x > 0, y > 0 -

:

( x, y ) =

7 , 63

:

B130. :

B134. :

. log log ( 2x + x 11) = 0

3

2

i. ( og2x 1) x + 4 R ;

. 2log x + 25log x = 12

x

nx 1 ii. R ; nx + 1

. log ( 2 + 2 3 ) + log81 = x log 3 + log178x

. log

B131.

2

x

x log 2 x log 2

2

x log 4 x = 54

. 2

nx

12

+2

nx +

12

1ogx 3 og x2

+2

nx +

B132. B133.

i. og1 3 ( x 2 + 5x 1) > 0

. 2og ( x 2 ) =

52

= 11

og

B135. :

x 1>1.x+2

i.

log5x ( 2x 2 + x + 1) R ;1 xR ;ii. n2+ x

ii. og ( 2 x + 2 3x ) + og81 > xog3 + og178B136. n [ n ( x 2 x + e ) ] = 0

B137.

:

og ( 3x 2x 2 ) R ;

B138. :nx ny = 2i. x + 2y = 10

ogx + ogy = og2iv. 223x + 2y = 10

119

y x : y x

H y x x . x

R ( A R ) y

R ( B R ) . ( y R ) ( x R ). : A R x A y B R f, g, h, . .. ... f : A R x, f, B R y () f f(A). :

f : A B = f (A) f

f (A) R . D(f) R(f) , , .

H

120

- 1

. x [ 0,1] , A = [ 0,1] . , :

y = 2x y = f ( x ) = 2x f ( x ) = 2x . .

( 1)

x [ 0,1] .

x = 0 f ( 0 ) = 2 0 = 0 x =

111 f = 2 = 1 , ( 1)222

f : A R x A ... f ( x ) = 2 2 , x R .

( x, f ( x ) ) x .