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Transcript of Βοήθημα μαθηματικών Α' λυκείου | frontistiria.edu.gr
,
, -
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 .