Shmeiwseis Algevra a Lykeiou Taexeiola.gr ΜΥ-ΝΕΤ-LAND.BLOGSPOT.GR

7/30/2019 Shmeiwseis Algevra a Lykeiou Taexeiola.gr --LAND.BLOGSPOT.GR

1/333

taexei
http://www.taexeiola.gr/


2/333

v 04

LATEX

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3/333

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4/333

2011-

2012

(MEd ).

.

1 - . -

.

4.2.25 . -

.

.

.

( -

)

.

pattern

. -

1 (2011-2012)

. -

,

.

2012

.. ,

(PhD, MSc)

i

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i

1 1

1.1 . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.2.1 . . . . . . . . . . . . . . . . . . . 4

1.2.2 . . . . . . . . . . . . . . . . . . . . . . . 4

1.2.3 . . . . . . . . . . . . . . . . . . . . . . 6

1.2.4 . . . . . . . . . . . . . . . . . . 6

1.2.5 . . . . . . . . . . . . . . . . . . . . . . 7

1.2.6 , . . . . . . . . . . . . 13

1.3 . . . . . . . . . . . . . . . . . . . . . . . . . . 161.3.1 . . . . . . . . . . . . . . . . . . . 16

1.3.2 . . . . . . . . . . . . . . . . . . 17

1.3.3 . . . . . . . . . . . . . . . . . . 18

1.3.4 . . . . . . . . . . . . . . . . . . . . . 19

1.4 . . . . . . . . . . . . . . . . . . . . . . 20

1.4.1 . . . . . . . . . 20

1.4.2 . . . . . . . . . . 21

1.4.3 . . . . . . . . . . . . . 22

1.4.4 . . . . . . . . . . . . . . . . 24

1.4.5 . . . . . . . . . . . . . . 27

1.4.6 . . . . . . . . . . . . . . . . . 271.4.7 . . . . . . . . . . 29

1.4.8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

1.4.9 . . . . . . . . . . . . . . . 31

1.4.10 . . . . . . . . . . . . 32

iii

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1.4.11 . . . . . . . . . . . . . . . . . . . . 33

1.4.12 . . . . . . . . . . . 37

1.4.13 . . . . . . . . . . . . . . . . . . . . . 43

1.5 . . . . . . . . . . . . . . . . . . . . . . . . 44

1.5.1 Pons Asinorum . . . . . . . . . . . . . . . . . . . . . . . . 44

1.5.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . 471.5.3 . . . . . . . . . . . . . . . . . . . . . . 48

1.5.4 . . . . . . . . . . . . . . . . . . . . . . . . 48

1.5.5 . . . . . . . . . . . . . . . . . . . . . . . . . 50

1.5.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

1.5.7 . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

1.5.8 . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

1.5.9 . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

2 55

2.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

2.1.1 . . . . . . . . . . . . . . . . . . . . 55

2.1.2 . . . . . . . . . . . . . . . . . . . . . 55

2.1.3 , . . . 56

2.1.4 . . . . . . . . . . . . . . . . . . . . 57

2.1.5 . . . . . . . . . . . . . . 58

2.1.6 . . . . . . . . . . . . . . 58

2.1.7 Euler-Venn . . . . . . . . . . . . . . . . 58

2.1.8 R . . . . . . . . . . . . . . . . . . . . . . . 59

2.1.9 . . . . . . . . . . . . . . . . . . . 63

2.1.10 . . . . . . . . . . . . . . . . . . . . 64

2.1.11 . . . . . . . . . . . . . . . . . . . 65

2.1.12 . . . . . . . . . . . . . . . . . . 66

2.1.13 . . . . . . . . . . . . . . . . . . . 67

2.1.14 . . . . . . . . . . . . . . 67

2.1.15 . . . . . . . . . . . . . . . 69

2.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

2.2.1 . . . . . . . . . . . . . . . . . . 71

2.2.2 . . . . . . . . . . . 73

2.2.3 . . . . . . . . . . . . . . . . . . . . . 75

2.2.4 . . . . . . . . . . 79

2.2.5 : A B . . . . . . . . . . 79

2.2.6 : A . . . . . . . . . . . . . . . . . . . 792.2.7 : A B . . . . . . . . . . . . . . . . . . . . . . 80

2.2.8 : A B . . . . . . . . . . . . . . . . . . . . . . . 80

2.2.9 : A B . . . . . . . . . . . . . . . . . . 81

2.2.10 . . . . . . . 82

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3 , , , 85

3.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

3.1.1 2 = 2 2+ 2 . . . . . . . . . . 853.1.2

3

=3

32

+32

3 . . . . . 87

3.1.3 2

2

=

+

. . . . . . . . . . . . . 88

3.1.4 . . . . . . . . . . . . . . . . 903.1.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

3.1.6 . . . . . . . . . . . . . 99

3.1.7 . 100

3.1.8 - . . 100

3.1.9 . . . . . . . . . . . . . 101

3.1.10 102

3.1.11 - . 103

3.1.12 . . . . . . . . . . . . 103

3.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

3.2.1 . . . . . . . . . . . . . . . . . . . . 105

3.2.2 . . . . . . . . . . . . 1073.2.3 . . . . . . . . . . . . . . . 109

3.2.4 . . . . . . . . . . . . . . . . . . 112

3.2.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

3.2.6 . . . . . . . . 115

3.2.7 . . . . . . . . . . . . . . 116

3.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

3.3.1 x + = 0 . . . . . . . . . . . . . . . . . . . . . 1213.3.2 x + = 0 . . . . 1223.3.3 . . . . . . . . . . . . . . . . . . . . . . . 124

3.3.4 x2

+x

+

=0 . . . . . . . . . . . . . . . . . 126

3.3.5 x2 + x + = 0 . 1293.3.6 Vieta . . . . . . . . . . . . . . . . . . . . . 1313.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

3.4.1 x + > 0 . . . . . . . . . . . . . . . . . . . . 1333.4.2 () . . . . . . . . . 136

3.4.3 x + , 0. . . . . . . . . . . . . . . . . . 1373.4.4 -. . . . . . . . . . . . . . . . 139

3.4.5 x2 + x + > 0, 0 . . . . . . . . . . . . 1423.4.6 x2 + x + > 0, 0 . . . . . 1423.4.7 x2 + x + , 0 . . . . . . 1443.4.8 x2

+x

+

>0,

0 . . . . . 147

4 1514.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151

4.1.1 . . . . . . . . . . . . 151

4.1.2 154

4.1.3 . . . . . . . . . 156

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4.1.4 . . . . . . . . . 159

4.1.5 . . . . . . . . . . . . . . . . . . 162

4.1.6 . . . . . . . . . . . 168

4.1.7 . . . . . . . . . . . . . . . . . . . 169

4.1.8 . . . . . . . . . . . 170

4.1.9 . . . . . . . . . . . 1734.1.10 . . . . . . . . . . . . . . . 175

4.1.11 . . . . . . . . . . . . . . . . . . . . . . . . . . . 176

4.1.12 . . . . . . . . . . . . . . . . . . . . . . . . . . . 177

4.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181

4.2.1 . . . . . . . . . . . . . . . . . . 181

4.2.2 . . . . 184

4.2.3 . . . . . . . . . . . . . . . . . . . . . 190

4.2.4 x . . . . . . . . . . . . . . . . . . . . . . . 191

4.2.5 - . . . . . . . . . . . . . . . . . 193

4.2.6 - . . . . . . . . . . . . . . . . . . 198

4.2.7 - . . . . . . . . . . . . . . . . . . . 2024.2.8 x = . . . . . . . . . . . . . . . . . . . . . . . . 2034.2.9 . . . . . . . . . . . . . . . . . . 204

4.2.10 fx = x + . . . . . . . . . . . . . . . . . 2084.2.11 , . . . . . . . . . . . . . . 210

4.2.12 y = 1x + 1 , y = 2x + 2. 2124.2.13 x + = 0 x + > 0 . . . . . 2164.2.14 fx = x . . . . . . . . . . . . . . . . . . 2194.2.15 fx = x2 + x + . . . . . . . . . . . . . 2224.2.16 fx = x2 + x + > 0. . . . . . . 2234.2.17 f

x

=x2

+x

+

>0 ; . . . . . 226

4.2.18 f

x

=x2

+x

+

>0. . . . . . . . . 227

4.2.19 fx = x2 + x + . . . . . . . 2294.2.20 fx = x2 + x + > 0. . . . . 2304.2.21 fx = x2 + x + < 0 . . . . . . . . . . . . 2334.2.22 fx = x2 + x + . . . . . . . . . . . . . 235

5 237

5.1 1 . . . . . . . . . . . . . . . . . . . . . . 237

5.2 2 . . . . . . . . . . . . . . . . . . . . . . 248

5.3 3 . . . . . . . . . . . . . . . . . . . . . . 256

5.4 4 . . . . . . . . . . . . . . . . . . . . . . 279

6 289

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1

. .

1.1

0, 1, 2, 3, 4, ...

0, 1 0, 2

1 .. .

. 0 .

1. 2011; 20112011;

a;

:

2 + 3 = 5 5 + 11 = 16 32 + 39 = 71 32 39

61. ,

+ . .

.

2 3 = 6 5 11 = 55 32 39 = 1248 ,

. .

1

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11/333

2 1.1.

2. , 6.

;

:

7 2 = 5 = 2 7 = 5 ,

.

. : ,

.

3. ,

. , ;

:

8 4 = 84= 2 =

4 8 = 48= 1

2= 0, 5

8

5

=8

5

=1, 6

5 8 = 58= 0, 625

, .

, .

: 1

1 .

;

. = = .

= 0 1 (1)

0 (1) . -

: .

http://lyk-evsch-n-smyrn.att.sch.gr

..

www.nsmavrogiannis.gr

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12/333

1. 3

-

0. 0 :

0

=0

0

1.2

2 7 = 5 5 . - 5.

:

0, , 1, ,

1, , 2, ,

2, , 3, ,

3, , 4, ,

4, ,...

0, ,1, ,2, ,3, ,4, ,... (...) .

.

:

... 4, 3, 2, 1, 0, 1, 2, 3, 4,... -

. a a

+1 a

1.

4. .. 2.

5.

.

5

.

2.

2. ;

;

= 0 (1) . 0 = 0

. . !

.

. .

.

.

..



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13/333

4 1.2.

,

.

2

+7

=5

8

+7

= 15

2

7

= 14

8

7

=56

6 12 = 6 12 6 = 18 , , -

.

.

.

1.2.1

-

.

:

0

1 .

.

1.2.2 .

2

0 0, 2, 4, 6, ... 0 2, 4 6 .... :

... 6, 4, 2, 0, 2, 4, 6 ... .


..


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1. 5

:

2

2. 2

.

= 2 = 2

2.

( -

) 2

.

( : 2= )

2=

=2

= 2 . ( ) =2 .

. :

= 2 .

= 2 .

= 2. 5.

- : = 2 (2)

- = 2

..



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15/333

6 1.2.

- = 2.- .

- .

;

1.2.3

1

0 - 2

... 5, 3, 1, 1, 3, 5, 7, ... .

. :

.

.

:

+ 1. = 2

+ 1 = 2+ 1. :

= 2+ 1 .1.2.4

4

(, , )

. -

:


..


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16/333

1. 7

- : -

.

:

.

. .

180 -

180 .

2

.

. -

. :

.

-

.

. 3 6 .

.

.

1.2.5

.

.

- .

.

.

. :

2 1.5 44.

..



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17/333

8 1.2.

1. .

.

2 2 .

3 :

3 4 = 2 2 . 4 :

4 8

=2

2

2 .

5 ; ; 16

. :

5 16 = 2 2 2 2 .


..


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18/333

1. 9

2 2 2 ... 21

;

. 6 32 .

:

:

6 2 2 2 2 2 = 32 31 .3

Leo Moser

1921-1970 . .

, .

2. .

2 + 8 = 10 , 12 + 24 = 36, 18 + 24 = 42, 100 + 98 = 198 .

.

. -

. .

3

Moser.

463+23218+24

24

..



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19/333

10 1.2.

.

. 2

8, 12 24 ...

+.

+ .

. . :

.

, = 2 = 2. + . 2 + 2 2 + . . + , . + = 2 . , ,

:

+

= 2

+ 2

= 2 +

= 2 =

4. :

1.2.1. .

;

. -

:

1.2.2. .

, ,

:

+

= 2

+ 1 + 2

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

= 2 = :

1.2.3. .

2)

6. 1.2.3.

4, ( ), ( ), (

( )


..


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20/333

1. 11

1.2.4. 1) .

2) .

. 1) = 2 + 1, = 2+ 1 , . = 2 + 1 2+ 1

= 4+ 2 + 2+ 1 2 4, 2, 2 -

=2

2

+

+

+1

= 2+ +

2 (

), .

.

= 2 + 1 .

2) = 2 .

.

= 2 2 2 :

= 2

=2

. ;

.

:

..



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21/333

12 1.2.

1.2.5.

.

.

1 .

=2

= 2+ 1 . + = 2 + 2+ 1 = 2 + + 1 = 2 + 1

= + . + .2

. , + , . x = + :

1 x

2 x

.

. ;

+ = x x .

= x : .

1.2.3 . (

) ( ). ,

.

. ...

1. ...

(

).

. Q.E.D.

quod erat demonstrandum

( ). 2. 1.2.5 -

( ).

. .

. ( -

)

.


..


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22/333

1. 13

( 1.2.5

).

( ) -

.

:

-

-

7. , . ( ) :

1. + + 20112.

+

1

3. + 4. 2 + 1

8. + 1 . .

9. + 1 .1.2.6 ,

-

2. 0 -

2 - 2 .

2+ 1 . 0 2

3 3

..



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24/333

1. 15

y = xz. y x y x x y.

3. 3 3

1 3

= 1

3

.

12 36 3

3 12 = 36. y = xz y xz x.

z x

:

...5x, 4x, 3x, 2x, x, 0 x, x, 2x, 3x, 4x, 5x ... 12. 12

=1

a,

= 1

a

:

1 a. :

1. 0.

2. 1.3. 1,.

1. 0

2. 13. 1,

2 x 3 x 4 x x 1, 0 2 3 4...

276 .. - 195 .. 0,

1, 1 .

1

.

-

0, 1, 1 2, 4,.... . - 0, 1, 1 2, 4,... .

.

..



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16 1.3.

1.1: 1 400

2 4, 6 ... -

3.

6, 9 ..

1 400.

-

.

.

1.3

1.3.1

( )

1

2,

5

3,

34

,6

3

,

( 0). 5 -

-

.

36

72= 50

100= 14

28= 816 = 12

5 . 1.4.12


..


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1. 17

() . -

(

). (

)

.

4. :12560

= 2512

144

64= 9

4

77121 = 711 13. 122

56, 7080

, 7181

1.3.2

:

=

= =

=

1.2.2

=

= .

=

, , .

.

:

0 0

=0.

14. ( ,, ) =

15. 2358987812345912345923589878

= 7812345923589858987812345923

;

16. x, y x3y

.

x;

,

, >

<

.

>

>.

00 = 0a+ + ... + < 0

=

a ...

> 01 a

0,

=0

1

a

1

a

...

1

a

=1

a

0

1.4.10

, ,

.

: < < < (1.24)

< < (1.25) < < (1.26) (1.27)

(1.24) (1.25), (1.26), (1.27)

.

2 < x x < y 2 < y. a < b, b < c, c < d a < c c < d

a

0

+

>0. :

> 0, > 0 + > 0 (1.33)

.

:

0, > 0 + > 0 (1.34)

0, 0 + 0 (1.35)

.

: > 0 - > 0 > . < 0 > 0. :

> 0 < 0 (1.36) (1.29)

.

. .

< > 0. > 0 ( (1.30) ) > 0. > 0 > 0 < (1.37) < < 0 > 0 < < > . :

< < 0 > (1.38) (1.30)

+ + = +.

.

13.

13

http://www.archive.org/details/diophantialexan01plangoog


..


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http://www.archive.org/details/diophantialexan01plangooghttp://www.archive.org/details/diophantialexan01plangoog


44/333

1. 35

Teubner, 1893.

> 0 < 0 > 0 > 0 (1.30) > 0. = > 0 < 0.

> 0 < 0 < 0 (1.39) + = . < 0 < 0 =

+

+> 0 :

< 0 < 0 > 0 (1.40) = +

:

.

< < 0 > 0 < 0 > . :

< < 0 > (1.41) -

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

< 0

.

> 0 1> 0 (1.42)

0, < 0

1.

+

2.

3. + 1 14.

+

5.+1

6.

+1+1

1.4.12

x

.

x. x

x.

x

O x

x

x

x

x

.

2, 3

=2,

7

= 7,

[1

2] =0,

[2

2] = 2.

:

x = x =

x

(1.48)

..



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47/333

38 1.4.

x x < x + 1 (1.49)

. x ,

, , .

x.

x = x + x (1.50)

x = x x (1.51) 3, 2 3, 2 3 = 0, 2 4, 78 0, 22.

,

.

10:

...

104 = 11000

,

103 = 11000

,

102 = 1100

,

101 = 110

,

100 = 1,101 = 10,102 = 100

...

(1.52)

(1.52) 10 .

10 .

.

x. x 10

: x x

(1.52) x

10 10

.

x = 8243

. x 102 = 100 103

=1000. :

102 < 8243 < 103

102 103

102 100.

100 = 1 100, 2 100, 3 100, 4 100, 5 100, 6 100, 7 100, 8 100, 9 100 = 1000


..


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48/333


49/333

40 1.4.

100 = 102 . 101.

2

102

+1

10, 2

102

+2

10, 2

102

+2

10, ..., 2

102

+10

10

x x

. x

2 102 + 7 10 < 8243< 2 102 + 8 10

10

100

=1

2 1002 + 7 10 + 4 100 < 8243 < 2 1002 + 7 10 + 5 100

101 = 110

. :

2 1002 + 7 10 + 4 100 + 6 101 < 8243< 2 1002 + 7 10 + 4 100 + 7 101

: 102, 103 .

2 102, 7 101, 4 100, 6 101, 6 102, 6 103, 6 104, .. (;;) (

10 ) -

x. (bold)


..


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50/333

1. 41

:

2 102 7 101 4 10 6 101 6 102 6 103, 6 104 ...

2 7 4 6 6 6 6 ...

2 7 4 6 6 6 6 ...

.

10 .

.

,:

2 7 4 , 6 6 6 6 ...

:

8 2 4 3

- 6 2 7 4, 6 6 6

2 2

- 2 1

1 4

- 1 1 2

2 0

- 1 8

2 0 0

- 1 8

2 0

x.

x = 8243

.

10 -

0, 1, 2, 3, 4, 5, 6, 7, 8, 9

( 0 )

.

.

..



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51/333

42 1.4.

-

( )

( ).

-

.

.

.

:

: .

: .

0.

-

. , ,

.

9. -

. 0, 999...

1. 0, 9999

a

=0, 9999 10a

=9, 9999... 10a

=9

+a 9a

=9 a

=1.

9 -

1 0, 9999.

mm1...0

, 122...

64. 197

. .

65. 3 0, 23

66. a2a1a0 b2b1b0

a2 + b2 = a1 + b1 = a0 + b0 = 11. a2a1a0 + b2b1b0. 67. 0, 3 ( )

68. ( ):

0, 1 01

1 0 02

1 0 0 03

1 0 0 0 01

. . .


..


taexei


52/333


53/333


54/333

1. 45

.

. ( ) . 60 .

1. , ,

.16

2.

.17

3. , ,

, .18

4. .

5. AB ,

.

6. .

7. ....

8. .

16, , . . .

Legendre. 19 .17

: , 20 .18

: , 20 .

..



taexei


55/333

46 1.5.

9.

10. : ...

-

.

.

-

.

. -

.

: .

384 .. - 322 ..

; 1,

2, 4, 6, 7, 9. 5 9

. .

. 8

.

. , , -

. .

, , .

19

.

.

.Augustus De Morgan

1806 - 1871

George Boole

1815 - 1864

. ( )

. - De Morgan Boole.

(

) .

- -


..


taexei


56/333

1. 47

.

1.5.2

.

7.

. -

.

11. 1. 3 > 5+ > .

2. 11

.

3. 3 > 5 4. 33 > 5 .5. x + 2 = 3

x .

. x 3

3 + 2 = 3 . x 1 1

+2

=3.

7 .

. :

.

, ,

:

.19.

.

.

19 ( ) . -

. .

Goldbach:

2 .

428. 1742!

..



taexei


57/333

48 1.5.

1.5.3

x + 2 = 3 - (

x) .

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

.

-

.

-

20.

1.5.4

:

P

Q

.

P Q ()

().

.

.

. .

.

:

= + = +

. ,

=

+

=

+.

:

= + = + 20

:

.


..


taexei


58/333


59/333

50 1.5.

1.5.5

( - ):

.

AB :

= B =

B

=

=

.

:

= B = .

P Q

P Q -

.

.

76. ;

1. > 5 2 > 102. 2 < 0 4 < 03. > 0 > 4. > 0 + > 05. > 2 > 7

77. ;

1. x

2

x

+2

4

2. x 2 x > 23. x y x2 y24. x = y + 1 x y5. x = 1 x2 1 = 0


..


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60/333

1. 51

1.5.6

1 < 2 < 3 : 1 < 2 2 < 3. 1 < 2 < 3 1

32. 2 33. x < x + 14. 2 75.

a

+

=

1.5.9

. -

-

.

.

P1

P2 ... P

P1 P2, P2 P3,P1 P . : P1

P1 P2 P2 ( ). P2 P2 P3 P3 .

P .

12. x < y 2x3

4< 2y3

4. :

x < y 2x < 2y 2x 3 < 2y 32x

3

4 0

=0 .

#

## # . + < + + = + . :

+ +

##

:

+ + + + + + + + . :

+ + (3.37)

i, ii,

3.37

. (3.36)

1 12 2...

1 + 2 + ... + 1 + 2 + ... + (3.38)


..


taexei


116/333

3. , , , 107

:

.

201. x

0

.

3.2.6

< 1 > . > < 1. :

< > (3.50) 3.50 : -

0

:

.

. :

< 1

, 1

; :

2 < 3. 12> 1

3

2

1

.

: 1> 1

1

1

> 0

> 0.

> 0, > 0 . ,

< . > 0 <

1

< 1

1

> 1

.

:

, < 1> 1

(3.51)

227. (3.51)

.

; 228. , < 1

> 2

+ > 1 . 229. A = ,1, B = 1, 0, C =0, 1 D = 1,+. x (4 ) A,B,C,D 1

x;

230. p,q,r p < qr< 1

p.

1. p < 1.2. p < r

q< 1

p.

3.2.7

. .

+ + +4 9 13 2 3

13 5

16 9 25 4 3 5 7

0 16 16 0 4 4 4

144 25 169 12 5 13 17

+ + , .

+ = +.

+ = + , :


..


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126/333

3. , , , 117

231. , + = + = 0 = 0

232. ,

=

;

. - :

4 9 36 2 3 6 6

16 9 144 4 3 12 12

0 16 0 0 4 0 0

4 25 100 2 5 10 10

-

=. 3.2.5: =

2 = 2 2 = ()2 2

2 = 22

=

. , :

= (3.52) (3.52)

. ..: = =

(3.52)

= (3.52)

:

4 9 49 2 3 23 23

16 9 169

4 3 43

4

3

0 16 0 0 4 0 0

4 25 425

2 5 25

2

5

..



taexei


127/333

118 3.2.

=

. -

(3.52):

=

2 =

2

2 = ()2()2

2

=

2

2

=

, 0 > 0

=

(3.53)

(3.53)

(3.52):

=

= (3.52)

=

=

. 2 = . - .

2

2.

2.


..


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128/333

3. , , , 119

0. < 0 ( 2 = 2) . :

0

2

=

2 = { 0 < 0(3.54)

233.

:

1.

22 34 58 = 2 32 542.

311 519 114 = 35 59 112153.

35 73 117 = 32 7 113231 234.

,, . :

1. A = 242. B = 824

3. =2484. = 577

235. ,, . :

1. A = 446

2. B = 3283.

=

6

4

84

4. = 444926 236.

2 3 = 1

2

2

3 12

2

237. t1 = +, t2 = :1. t1 + t2 = 22. t1t2 = 2 3. t

2

1+ t2

9= 22 + 2

4. t31+ t3

2= 23 + 6

238. :

1. 322.

323. 32

4. 325.

(3 3)26.

(1 3)2..



taexei


129/333

120 3.3.

3.3

() . .

.

.

.

x + 12 = x2 + 2x + 1 x .

x .

x -

.

2 3

x

+1

2

=x2

+3x

+1

x = 0. 0.

1 2

x + 12 = x2 + 2x + 2 x. .

x 3 = 0 x y = 4 . x 3 = 0 x = 3 . x y = 4 : x = 4, y = 0 x

=5, y

=1 ... 4 0

x 4 y 0 . y 4

x 0 0 4 = 4. .

:

1 , 2 , ..

. -

, ..

.

x

y

=4

x, y

-

x, y, 5, 1, 11, 7, 11,15 ... x 2y + 4z = 7 - x, y, z. 1, 1, 8 x = 1, y = 1, z = 8 1, 8, 1 x = 1, y = 8, z = 1 .


..


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130/333

3. , , , 121

239. x = 3 3x = 9; x3= 9;

27

x= 9; x3 = 27; xx = 27;

240. 2, 3, 6 x + y + z = 11. .

3.3.1 x + = 0 x + = 0 ( -

) . .

.

-

:

23. : 2x + 8 = 0.: 2x

+8

=0

2x

= 8

2x2

=82

x

=82

x

= 4

24. : 0x + 8 = 0.: 0x + 8 = 0 0x = 8.

x 0 = 8. .

25. : 4 x + 5 = 2.: : 4 x + 5 = 2

4x + 20 = 2 4x = 2 20 4x = 18 x = 184 x = 9

2

26. : 4

x

+5

=4x

+20.

: 4 x + 5 = 4x+20 4x+20 = 4x+20 4x4x = 2020 0x = 0. 0x = 0 ( ) x .

. -

: .

x + = 0 x = x =

0.

=0,

0.

= = 0. 241. ( x):

..



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131/333

122 3.3.

1. 2x + 8 = 02. 3x 8 = 133. x + t = 0, ( 0)

4. 1

x + t = 0, ( 0)5. + 1x = 0, ( 1)6. x + 4 = 1, ( 0)

242. :

1. 6 x + 3 5x = 262. 6 x + 3 5x = 26 + x3. 4 4 + 2x = 60 3x

4. 60x + 1 = 3 3 + 4x5. 6 x x + 4 = 8 x26. x 32 5 10 + x = x2 8

243. :

1. 3x + 100 = x3+ x

2 4

2. 5x114 x1

10= 11x1

12

3. 4x+2

+7

x+3

=27

x+2x+3

4. 30x+5 153 5+4xx+5 = 05. 3 x 1 x 2 x 3 x 4 = 2x + 7 x+ 46. 2x 2x

9= 1

9(16x 3

2)

3.3.2 x + = 0 243

x+= 0 . :

27. :

x3 + x + 14 = 8

. :

. x3+ x+1

4= 8

12( x3+ x+1

4) = 12 8 4x + 3 x + 1 = 96 4x + 3x + 3 = 96 x = 93

7

28. :

2

x

+1

=3

4

. : x + 1.

x + 1 0 x 1. .


..


taexei


132/333

3. , , , 123

x 1. : 2x+1 = 34

2 4 = 3 x + 1 8 = 3x + 3 x = 53

1. 29. :

3x + 2 + 6 + xx + 2 x + 1 = 5x + 1

. x + 2, x + 1 x 2,1.

x + 2 x + 1. :3

x + 2 + 6 + xx + 2 x + 1 = 5x + 1

x + 2 x + 1 3

x + 2 +6

+x

x + 2 x + 1 = x + 2 x + 15

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

.

.

30. :

4

3

x

2

2x

1

x

1

=0

.

:

4 3 x 2 2x 1 x 1 = 0

4 3 x 2 = 0

2x 1 = 0

x

1

=0

x =10

3 x =1

2 x = 1 -

x = 103

, x = 22

, x = 1. 244. :

..



taexei


133/333

124 3.3.

1. 2x 6 x+ 35x 6 = 02. 3 2x 4x 20 x + 2 4 + 2x = 0

245. :

1. x2 4 = 02. 4x2 9 = 0

3. x2 2x + x 2 = 03.3.3

31.

v0 = 3msec a = 2msec2 . v = 6m

sec;

a

=1m

sec2; a

=5m

sec2;

. .

:

a = 2msec2

. v = v0 + at 6 = 3 + 2t 2t = 3 t = 3

2

t = 32

sec

a = 1msec2

. 6 = 3 + 1 t t = 3 t = 3sec

a = 5msec2

. 6 = 3 + 5t 5t =3

t

=3

5 t

=3

5sec

: .

a

. :

6 = 3 + at at = 3 t = 3a

-

a.

t = 3a

a

.

6 = 3 + at t a . . -

.

.


..


taexei


134/333

3. , , , 125

32. x 2x + 2 = + 4x.. .

:

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

2 4 = 0 2 + 2 = 0 = 2 = 2 :

=2 0x

=0 .

= 2 0x = 4 . 2 4

2 x x = 2

2 4 x = 2 2 + 2 x = 1 + 2 246. :

x + 3 = 5 x + 1 247.

1x 2 + 1 = 0 x ;

248. t

t

x

+1

3t

=x

t

x = 5; 249.

x, y xy= .

.

..



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135/333

126 3.3.

3.3.4 x2 + x + = 0 x2 + x + = 0.

= 0 x + = 0 ;;.

0 x2

+ x + = 0 x2 ( 0x2 +x + = 0 ). .

.

.

-

.

,

x2 + x 2 = 0

, -

: ,

. -

AB (A )

, 2

.

A= 2

. E, Z

B. AE. ZAE . , ZA .

BAE = 90 A = A =ZA AEB, ABZ B AEB =BZA.

AB

BZ= BE

AB= AE

AZ


..


taexei


136/333

3. , , , 127

x + 2+

2

= x

2 = x x + x x2 + x 2 = 0.

19.

x2 + x + = x + 22 2 4

42

:

x + 22 2 4

42 =

x2 + x+ 2

42 2 4

42 =

x2 + x+ 2

42 2

42 4

42 =

x2 + x+

=

x

2

+ x+ 0 x2+x+= 0 x +

22 2 4

42 = 0 x +

222 4

42= 0

2 4 - . x +

22 24

42= 0

x +

22

42 = 0 < 0

42 .

x + 22

.

..



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137/333

128 3.3.

. 0 =

2

x

+

2

2

2

2

=0

x + 2

2x +

2+

2 = 0 (3.55)

x + 2

2= 0 x +

2+

2= 0

x = 2+

2 x =

2

2

= 0 x

=

2. :

x2 + x + = 0 0 = 2 4: < 0 0. > 0

() x1,2=

2. = 0

x = 2

250.

1. 4x2

+92x

+333

=0

2. x2 + 23x + 76 = 03. x2 + 23x + 102 = 0

4.

2x2

+3x

+20

=0

5. 2x2 + 3x + 77 = 06. 64x2 64x 273 = 0

251. - :

1. x2 +x = 02. 2x2 + 4x = 03. x2 + 2x 2= 0

4. 2x2 + 5x 122 = 05. x2 + x + = 06. x2 + x + = 0

252. :

x2 + 2xy y2 = 0 ():

1. x ( x).


..


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138/333

3. , , , 129

2. y.

253. :

1. x2

+3x

+2

=0 x

= 1

2.

2

x

2

3x + = 0 x = 13.3.5 x2 + x + = 0

x2 + x + = 0 . -

:

33. :

2

2 x +1

2 =4

2x x

2

. 2 x 2x x2 = x 2 x .

x 0 x 2. :

2

2 x + 12 = 42x x2 2

2 x + 12 = 4x 2 x 2x2x2x 2 x

2

2 x +1

2 = 2x 2 x4

x 2 x 4x + x 2 x = 8

6x x2 = 8x2 + 6x 8 = 0

x2 6x + 8 = 0 .

x2

6x

+8

=0 x

=2

x

=4. x

0 x

2

2 4.

34. :

2x 12 3 2x 1 + 2 = 0..



taexei


139/333

130 3.3.

.

.

x -

2x

1.

2x

1

=u

u.

x.

2x1 u u23u+2 = 0 u1 = 1 u2 = 2. 2x 1 = u 2x 1 = 1 2x 1 = 2 x1 = 1 x2 = 32 . 35. :

x4 5x2 36 = 0. x4 = (x2)2

(x2)2 5x2 36 = 0 (x2)2 . x2 = u u x.

u2 5u 36 = 0 u1 = 4 u2 = 9. u x2 = u :

x2 = 4 x2 = 9 x1 = 3 x2 = 3. 254.

1. 2x + 21x+4 = 9

2. 23x1

+3

+70

x24x+3

=0

3. 3x + 5x2 = 2

4. 1x+ 3

2= 1

x+35.

3x3+4+x33x2+4+x2

=7

6. x2+x+1

x2x+1 = 32+22+32 255.


..


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140/333

3. , , , 131

1. x4 5x2 + 4 = 02. 2x+ 12 3 2x + 1 1 = 03. (2x+3

x1 )2 3 (2x+3x1 ) + 2 = 04. 4

x2+4 + 5x2+5 = 25. (x2 + x + 1)(x2 + x + 2) 12 = 06. 7 (x + 1

x) 2(x2 + 1

x2) = 9

3.3.6 Vieta

x2 + x + = 0 . Francois Viete

(Franciscus Vieta)

1540 - 1603

.

x1 = 2 x2 = +2 . :x1 + x2 =

2+ +

2= () + (+)

2= 2

2=

x1 x2 = 2 +2 = () (+)42 =2 242

= 2 42

= 2 (2 4)42

= 442

=

3 S P.

S= x1 + x2 =

P = x1 x2 =

Vieta (3.56)

36. -

2x2

4x

1

=0 .

. (;). S = 42 = 2 P = 1

2= 1

2.

.

37. x2+x+= 0 x1 x2. x21 + x22

. x21 + x22 : x1 x2 x1 x2

. x1 x2

x2 x1 :

x21 + x22 x22 + x21

3 summa, productum.

..



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141/333

132 3.3.

x1, x2 (

4 ) ,

, S = x1 + x2 P

=x1

x2. :

x2

1 + x22 = x1 + x22 2x1x2 = S2 2P ,,

:

S2 2P = 2 2

= 2

2 2

= 2 2

2

x21 + x22 = 2 22

Vieta

S P.

= S

= P = S = P

x2 Sx + P = 0 (x2 Sx + P) = 0 x2 Sx +P = 0. :

S P

x2 Sx +P = 0 (3.57) 38. 4 2.

. S

=4

2. x2 4x + 2 = 0 . . :

x1 = 2 +2 x2 = 2 2. 256. .

1. 2x2 58x+ 12 = 02. t2 + 1500t 2 = 0 3. 2x

2 2x + 6 = 04. tx2 2tx+ t2 = 0

257. x2

+x

+

=0 1, 2.

,, :

1. 1 2 + 1 + 2 1 + 1 2. 11+1 + 12+1 258.


..


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142/333

3. , , , 133

1. 1 42. 5

8 1

3

3. 16+ 1

6

13 1

6 1

6

13

4.

3 13

259. 1. .

1. 2x2 + 928x 930 = 0 2. x2 8193x+ 8192 = 0 260. S P . .

1. S= 15, P = 142. S= 19, P = 84 3. S= 12, P = 34. S= 2, P = 2 2

261. .

1. 7x2 + 14x 1 = 02. 3x2 9x + 2 = 0 3. 6x

2 14x + 1 = 04. 3x2 + 5x + 1 = 03.4

()

. -

(. 3.3.1). -

. x

2

>0

2

2,+. x + 1 (y2 + 1) > 0 x, y x, y x > 1, y R.3.4.1 x + > 0

2x + 4 > 0 -:

2x + 4 > 0 2x > 4 2x2> 4

2 x > 2

x

x

> 2

2,

+.

:

:

..



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143/333

134 3.4.

2x + 4 > 0 :2x + 4 > 0 2x > 4 2x

2 < 42 x < 2 2 .

:

:

- x

+

>0

x + > 0 x >

x

>

>0

x <

< 0

0 > = 0

> 0 < 0 : -

,+

, . = 0

. x + > 0 0 > . x

+

>0 x 0

>

>0.

>0 x

+

>0

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

:


..


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144/333

3. , , , 135

x + > 0:

,+ > 0.

,

< 0. R = 0 > 0. = 0 0.

4. x + 0, x + < 0, x + 0 x + > 0. 39. 3x + 2 0..

3x + 2 03x 2 $$$3x3 23 x 23

40. x

+1

x.

. : x + 1 x x x 1 1x 1. :

1 > 0 > 1. x x 11 .

1 > 0 < 1. x x 11 .

1 = 0 = 1. 0 1 x .

262. :

1. 2x > 32. 2x > 33. 2x < 3

4. 2x < 35. 4x + 2 < 16. 8 x + 2 > 4

263. :

..



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145/333

136 3.4.

1. x > 4 > 0.2. 2x + 2 0 > 2

3. 2 1x < + 1 > 12

4. (2 + 2 + 1)x < 3 + 2 264. :

1. x+13 x14 > 1 2. 2 x 1 + 2x14 < 3

265.

3x + 4 < 5 t 2t 1 ; 266. 2x 3y > 1.

1. x.

2. y.

267. m 2x2 3x+m+ 1 = 0 x ;

3.4.2 ()

-

.

.

.

41. 4 3x > 0 5 6x > 0.

(, 43) (, 5

6).

: (, 43) (, 5

6) = (, 5

6).

42. 4 < 3x 1 < 5.


..


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146/333

3. , , , 137

. x

: 4 < 3x 1 3x 1 < 5. (5

3,+)

, 2

.

:

(5

3,

+), 2

=(53 , 2).

43. m -

m

2 5 6, 6.. 6 < m2 5 < 6

.

.

6 < m2 5, m2 5 < 6 12

2 0

269. 3 5 6x < 58

.

270. x2 4x + 5 1 = 0 .

3.4.3 x + , 0. ,

0

x x+= 0 ( ) x x + > 0 x + < 0 ( ). 4 x +

4

..



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147/333


148/333

3. , , , 139

x

3x + 4 4

3+

+ 0 45. x (3 1)x + (3 + 1).. = 3 1 > 0 (3 1)x + (3 + 1)

x = 3+131 = (3+1)(3+1)(31)(3+1) = 4+232 = 2 3. x

x

.

x

(3

1

)x

+ (3

+1

)

2 3 +

0

+3.4.4 -.

x + , 0 .

46. x 2x 1 4 3x.. -

.

:

x

2x 1 1

2+

0 +x

4 3x 4

3+

+ 0 -

-. (, 12) ,( 1

2, 4

3) , (4

3,+).

x

:

..



taexei


149/333


150/333

3. , , , 141

x2x 1

5x + 10

2x15x+10

2 22

+ 0 + + ++ 0 +

271. 46 x = 2012 .

272. 47 12012

.

273. 3 x 5 + 4x 3x 1.

x

3 x5 + 4x3x

1

3 x 5 + 4x 3x 1

54

1

33 +0

0

0

0 0 0

274. x :

1. 2 4x x + 52. (x 1

2) 2x 1

3.

1

x

2

x

3

x

275. x :2x 1 x + 35x + 7

.

..



taexei


151/333

142 3.4.

3.4.5 x2 + x + > 0, 0 x2+x +> 0, 0

<

>,

,

. x2

+x

+

x + - x2 +x + -

. x2 + x + = 0 .

3.4.6 x2 + x + > 0, 0 0 :

x2

+x

+

=

x

+

2

2

2 4

42

(3.58)

0 (. 128, (3.55) )

x2 + x + = x + 2

2x +

2+

2

2

2,

2+

2

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

+x

+.

x2 + x + 0 x2 + x + = x x1 x x2 x1,2 = 2 (3.59)

-

. -

px + q ( 278). = 0

(3.58).

x2

+x

+

=

x

+

2

2

(3.55) ): -

x1 = x2

x2 + x + = x x1 x x1 = ax x12


..


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152/333

3. , , , 143

x1 . x1 = 2

x2+x+= ax 22 = ax +

22.

48. 3x2

+2x

1.

.

x1 = 1 x2 = 13 3x2 + 2x 1 = 3 x 1 (x 13) 3x2 + 2x 1 = 3 x + 1 x 1

3

3 (x 13)

3x2 + 2x 1 = x + 1 3x 1. 49. 3x2+x

+1.

. x1 = 1+136 x2 = 1136 . 3x2 + x + 1 = 3x 1 +136

x 1 136

50. 9x2 12x + 4.

. 23

9x2

12x

+4

=9

x

2

3

x

2

3

=9

x

2

3

2

9x2 12x + 4 = 3x 22. 9 (x 2

3)2 = 32 (x 2

3)2 = (3(x 2

3))2 = 3x 22.

51. 22+11+ 152.

.

.

, , :

2

2

+11

+152

112 4 2 152 = 2 1122 5

2 3.

22 + 11+ 152 = 2 52

3 = 2 + 52

+ 3..



taexei


153/333

144 3.4.

2 + 52

22 + 11+ 152 = 2 + 5 + 3 276. :

1. 2x2 5x + 22. 4x2 + 4x + 13. x2 8x 33

277.

1. x2+6x91

x2+8x105

2. 2a2+8a90

3a236a+105

3. b2

9bx+14x2

b2bx2x2

278. x2 +x + - x2 + x + = px + q px + q p, q, p, q .

3.4.7 x2 + x + , 0 -

x2 + x + . :

x2 + x + = x + 22 42 x2 + x +

x + 22

42

-

x

x + 22 42 x -

,

.


..


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154/333

3. , , , 145

< 0 x + 22

42 :

x

+

2

2

42

= x

+

2

2

0 +42

>0>0 >0

x2 +x + x.

= 0 x + 22

42= x +

22.

.

x 2

x = 2

.

x

2 .

> 0 x1 < x2 x + 22

42= x x1 x x2.

:

x

x x1x

x2

x x1 x x2

x1 x2 + 0 + +

0

++ 0 0 + x

, x (

)

.

:

x2 + x + x < 0. x = 0. x > 0. (3.60)

..



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155/333

146 3.4.

:

+

+

0

52. 2x2 + 4x + 1. 1 1

2

6

1+ 12

6. = 2

. x

.

53. (7 3) x 2 x + 3.. ( )

x2

7 3 3 2. 3 2 .

279. :

1. 3x2 + 18x 212. 9x2

+3x

2

3. 5x2

35x + 404. x2 + x + 15. 3x

1

4x2

6. 1 6x+ 9x2

280. ,

1. 2 + + 2 02. 2 + 2 0


..


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156/333

3. , , , 147

281. x, y :

x2 2x + 6 + y2 4y > 0 282. x

=1 x

=2

x

=3. x

=4;

3.4.8 x2 + x + > 0, 0 x2 + x + > 0

x

.

54. 2x2 2x 24 > 0.. 2x2 2x 24 = 2 > 0 3 4.

x < 3 4 < x.

,

3

4,

+.

55. 3x2 2x 1 < 0.. 3x2 2x 1 = 3 > 0 1

3 1.

.

x 13

1. x 13< x < 1.

(13

, 1). 56. 3x2 11x 4 0.

.

1

3 4.

. (,13] 4,+. 57. 2x2 x + 1.

. -

= 2 . x .

.

58. x22x+43 = 0 .

.

2

2

4

1

4

3

42 16 + 12 > 0. , 1 3,+.

,

:

..



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157/333

148 3.4.

x

2x2

7x

4

3x2 16x + 5(2x2 7x 4) (3x2 16x + 5)

12

1

34 5 +

+0

0

+ ++ + 0 0 ++ 0 0 + 0 0 +

(2x2 7x 4) (3x2 16x + 5) - 0 1

2 x 1

3 4 x 5.

[1

2, 1

3

] 4, 5

.

59. x2x+2 2x34x1

. x 2, 14

.

.

. x 2, 14

:

x

2

x

+2

2x

3

4x

1

x 2x + 2 2x 34x 1 0

x 2 4x 1 x + 2 2x 3x + 2 4x 1 02x2 10x + 8x + 2 4x 1 0

(2x2 10x + 8) x + 2 4x 1 0 :


..


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3. , , , 149

x

2x2 10x + 8x + 2

4x 1(2x2 10x + 8) x + 2 4x 1

2 14

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

,2 (14

, 1]

4,

+ 283. , , 1. x + 2 x 2 < 02. x + 3 x+ 7 03. (x + 1

2) (x 3

4) < 0

4. 2x + 3 3x 4 05. x + 5 2x + 5 > 06. 3 2x 3x + 2 > 0

284. 0 < 0 285.

1. x2 x 12 > 02. 3x2 + 2x + 1 > 03. 4x2 4x+ 1 > 04. 3x2 4x+ 1 < 0

5. 4x2 + x 1 > 06. x2 + 4x + 6 < 07. 4x2 + 4x 1 08. 4x2 + 4x + 9 > 0

286.

1. 1 < x2 x < 22. 1 < x

x2+1 23. 1

x< x

x+1 < 4..



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160/333

4

4.1

4.1.1 .

-

.

( )

.

O B 0 1 -

12

A 2 .

1.

x

x.

,

.

1, 2, 3, 1,2, 3 1, 2, 3.

1, 2, 3 1

-

151

taexei


161/333


162/333

4. 153

... - -

... .... 2 3 =

2

+3

2

3

= 2

+3.

, . .

x 0 x x = 0. (4.6)

( ):

x = x x. (4.7) x

x.

x

= x

.

x

x

x

:

x x x x. (4.8) 291. :

1. 7 =2. 1

3 =

3. 32 11 =4. 5 32000 =5. 1

4111 1

8111 =

6. x2 + 3 = 292. :

1. A = 1 2 32. B = 1

3 4 2

3 1

2

3. = (5 252) (3

4 1)

4. = 3567

5. E= 4 8 7 9 116. Z= 1 334 7 334

293. > :

1. A = + 12. B

=33

3. = 3 3 + 4.

= 22+

+

294.

+ + = + + 295. x > 1. x31

x2+x+1

..



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163/333

154 4.1.

296. x > 5. x 2 3 x 5

297. 1

0 2x 1 = 2x 1.

:

x

2x 12x 1

1

2 + 0 +1 2x 0 2x 1


..


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164/333

4. 155

2x 1 =

1 2x x < 12

0 x = 12

2x

1 x

>1

2

. x = 1

2

. . :

2x 1 = { 1 2x x < 122x 1 x 1

2

x = 12

-

: 2x 1 x = 12

0 .

61. x 2x 1 + x 1.. .

.

:

x

2x

1

2x 1x + 1x + 1

2x 1 + x 1

1 12

+

0

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

x 1 x + 1 x + 13x 2 x 3x

. :2x 1 + x + 1 =

3x x < 12 x 1 x < 1

2

3x x 12

..



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165/333

156 4.1.

62. x

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

x

2x 1x + 1

2x 1x + 12x 1x + 12x 1x + 1 x

1 12 + 0 + 0 + ++ 0 0 +

2x 1 x + 1 2x 1x + 1 2x 1 x + 12x

2 1 2x2 2x + 1 2x2 1:

2x 1 x + 1 x = 2x2 1 x < 12x2 2x + 1 1 x < 1

2

2x2 1 x 12

299. x

2 x + 4 + x. 300. x

x 2 x + 2 x.4.1.3

.

.

x = . x .

0

.

x = < 0. 0 = 0. > 0. (4.9)


..


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167/333


168/333


169/333

160 4.1.

. :

x <

x < x 0

x

< x

2 x < .. ... < x

306. :

1. x < 7 ... < x < ...2. x 12 ... x ...3. x > 11 x < ... x > ...

4. x 3 x ... x ...5. 1

2< x < 1

2 x < ...

6. x < 8 x > 8 ... > .....



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171/333


172/333


173/333

164 4.1.

2 = 2

2

=2

2

2 = 2

2

2

2= 2

2()

:

. 4.1.2. -

:

=

=

=

=

71. = 3, = 5 = = 3 5 = 15

= = 35

4.1.2 :

4.1.4. 3 1

2

...

1

= 1

2

...

1

(4.13)

.

1 2 ... 1 = 1 2 ... 1 =1 2 ... 1 = 1 2 ... 1 == 1 2 ... 1 = ... =1 2 ... 1

4.1.5.

= (4.14)


..


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4. 165

. = ...

= ...

=

, , :

72. xyz3 = xyz3 = x yz3

309. ,, = 2, = 4, = 12

.

:

1.

2. 3. 2

4. 2

5. + 6. 3 + 3 2

310. xx (x 0) 1. 311. :

x1+x2 12 , x R.

.

-

.

:

x y x + y x y x + y x + y2 3 5 2 3 5 5

2 3 1 2 3 1 52 3 1 2 3 1 5

2

3

5 2 3 5 5

2 0 2 2 0 2 2

0 3 3 0 3 3 32 0 2 2 0 2 20 3 3 0 3 3 3

..



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166 4.1.

x + y x + y. x, y

.

x

+y

x

+ y

.

4.1.6. ,

+ + (4.15). : (4.8)

:

+ + + A = + X= + . A 0.

A X A A > 0 X A. A = 0 X= 0 X A.

X

A

X, A X, A :

+ + : + + + 2 + 2 + 2 2 + 2 + 2

+

2

2

+2

+2

2 + 2+ 2 2 + 2 + 2 2 2 (4.8)


..


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4. 167

: + + ; 0.

=

+ = + =

= 0. : + = + 0 (4.16)

: = a + a + = + :

+

(4.17)

73. x 1 y 9 x + y 10

. x + y x + y 1 + 9 = 10. 74.

+ + .

=

+

+

=

+

+

+

.

75. x = y = 2 2x 3y 10. 2x 3y 2x + 3y = 2 x + 3 y = 2 2 + 3 2 = 10. 312. ; 313. x, y x < 2, y < 3. :

1. x + y < 52.

x

+2y

1, y > 2 x1y2 > 0


..


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189/333

180 4.1.

KM =KT M, T -

. M x 2y+ 3 = 0 .

(

340) :

Kx1, y1 x2, y2 :x2 x1 y y1 = y2 y1 x x1 (4.25) (4.25) xx y1 = y2 x1 x2 (4.25) y = y1

yy x1 = x2 y1

y2 (4.25) x

=x1.

337. A2, 4, B 1, 3. 338. :

339. 2x 3y = 1,x + y = 2. 340. (*) -

Kx1, y1 x2, y2 x2 x1 y y1 = y2 y1 x x1.


..


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191/333


192/333

4. 183

f,g,h,, ..

f

A B f A B ( f A B). f x

A y

B

f

x

y y

=f

x

. x

A

y = fx f x. x y

.

x y x

y.

x1 = x2 fx1 = fx2 (4.26) f

f

2

=f

(4

) f

(x2

+2x

+1

) =f

(x

+1

2

) 2

= 4 x2

+2x

+1

=x + 12

. fx1 = fx2 x1 = x2. fx = x2 f2 = f2 2 = 2!

A B -

. -

.

x fx ().

83. A = R, B = R f A B f

x

= x

+1

x

+2

+1.

f1 = 1 + 1 1 + 2 + 1 = 7f3 = 3 + 1 3 + 2 + 1 = 3

f2 = 2 + 12 + 2 + 1 = 32 + 5 84. A = 0,+, B = R q A B g x = x + x2.

g 0 = 0+ 02 = 0g 4 = 4 + 42 = 18

85. A = N A = 1, 2, 3, 4,... B = R A R = 2 + 3. 1 = 2 + 3 1 = 5 2 = 2 + 3 2 = 8

..



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194/333

4. 185

:

-

.

.

.

: :

-

.

87. f R R, g R R fx =x2, g x = 2x 1 :

88. -

:

..



taexei


195/333

186 4.2.

x g 2x1 f. f . 2x 1 2x 12.

.

-

( x) ( y).

.

f

x

=x3.

x -3 -2 -1 0 1 2 3 4 5

fx -27 -8 -1 0 1 8 27 64 125 . -

. f A R A ( ) R f

x A y = fx Mx, fx.

x Mx, fx . A

.

:


..


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196/333


197/333

188 4.2.

f.

y f.

1. x fx > 0: xx.

2. x f

x

x. : x > x

x>0 1 > xx 1 > x 1 >


..


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202/333

4. 193

x 1 > x . : 0 < x < 1

,

< x

>x

(4.28)

x 1 :

:

x > 1 , < x < x (4.29)

: x < x x>0 1 < xx 1 < x

1

x ()

351. :

1. 23...24

2. (23)5 ...( 2

3)9

3. (2 1)33 ...(2 1)444. 2+1

3+14 ...2+13+175. 1 + x2012 ...1 + x2000 x 06. 1 + (2

3)11 ...1 + (2

3)15

352. 0 < x < 1.1. 0 < x < 1 x.

;

2. (4.28)

4.2.5 - .

x

g

x

=x2.

0 yy

x

0 g

x

=.

: yy ,

, xx: x gx = x2 = . x

.

..



taexei


203/333

194 4.2.

hx = x3. 0 x 0 hx = x3 = . x

3

.

x (=

) .


..


taexei


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205/333

196 4.2.

< 1 :

.

. :

x = 0x 0x =

(4.31)

= (4.32) :

1

= , 1 = 1, 0 = 0 (4.33)


..


taexei


206/333

4. 197

, ,

:

95. ) 4

81 = 3 34 = 81. ) 4625 4

4

625. (

) 4

625 = 5. 96. m.

3

m3

=m

m3

m. 3m6 = m2 m6 = (m2)3.

5

m15 = m3 (m3)5 =m15. 353. :

1.4

1296 =2.

5

a10 = a > 03. 4

16

81=

4.r

art

=, a

>0

5.33a36 = , a > 0

6.4

83

8 = 354. 4

= 3 4= 5 + .

355. :

1. (1 + 32)3 = 3 + 3 32 + 3 3222. ( 33 32)3 = 1 3 332 32 + 3 33 3223.

3

3

3

2

=1

3

32

+3

33

2+3

22

356. :

1. ( 3x + 3y)3 = x + 3 3x2 3y + 3 3x 3y2 + y2. ( 3x +y)6 = x2 + 2xy + y

..



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207/333

198 4.2.

3. ( 3x +y)6 ( 3xy)6 = 4xy 357. :

1.

(3

x

+3

y

)3

=x

+3 3

x23

y

+3 3

x 3

y2

+y

2. ( 3x +y)6

= x2

+ 2xy + y3. ( 3x +y)6 ( 3xy)6 = 4xy

358. :3

5 + 6 32 + 3 322 = 1 + 34

359. 3

= x 3 = y. x+ y3 3xy x + y = + 360. 3x + 13x3 = x + 3 3x + 33x + 1x .

4.2.6 - .

(3.52), (3.53) . , :

= (4.34)

=

(4.35)

:

=

=

=

=

=

=

=

=

= =

.

(4.34) .

1, 2,..., :

12 ... = 1 2... (4.36) 1, 2,...,

:

=

(4.37)

= = (4.32) 0 :

= = (4.38)


..


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4. 199

4.38 -

:

- .

(

).

.

. (

a). x 0 . x ( x

a). y 0 x. :

y

y x x

. y = y = . y . y = y = . y = a. :

a

=

(4.39)

(4.39) :

a = a = ( ) a = ( a) = a =

.

.

- : = .

= . :

= (4.40)..



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210/333


211/333

202 4.2.

4.2.7 - .

x

x 0. y

> > . - .

. ,

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

x = 0 :

x

=

x

=

x

=

371. :

1. x3 = 82. x3 = 8 3. x

3 = 1 24. x3 = 8

27

372. :

1. x12 = 2122. x13 = 213

4.2.9

. . ,

,

:


..


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214/333

4. 205

1 = 1 = 1

=1

1

=1

0

=0

0

=0

= = =

= = =

= + =

= + =;

= =;

.

!

. = + .

.

.

= . > 0 . ; .

= (

)

= (

)

( ) = ( )

=

( )( ) = ( )( )( ) = ( )

= +

=

+ = + = =

+. (.. = 3, = 3 = 12

7).

..



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215/333

206 4.2.

= . :

+ = + =

= +

+ = +

.

:

>0 t, s

t

s = st = st(4.44)

75

3 = 375 = 316807 25, 6 3 53 = 335 = 3243 = 6, 2. : 5x6 = x 54 3x4 = x 43 . :

1

s = s (4.45) :

= 12 , 3 = 13 , 4 = 14 , 4 = 14 , ... > 0 t, s

ts

ts = 1

ts= 1s

t(4.46)

. (4.44)


..


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4. 207

(4.46) 0 = 1. x

53 = x 53 = 1

x53

13

x5 x

54 = x 54 = 1x54

= 14x5

.

:

pq = p+q (4.47)p

q= pq (4.48)

pq = pq (4.49)p = pp (4.50)

p

=p

p(4.51)

. (4.47)

p =

q=

:

pq = = = = = + =

+ = + = p+q

( ) :

> 1 :p

0, < 0 = 0)

y = x+ xx y x 0.

= 0. 0 xx xx. x y .

.

y = x + : yy 0, xx 0

, 0 (4.54) 105. y = x + 3 = 1. 1 - 45 xx 45. 106. y = x 2 = 1. 1 = 135 xx 135. 107. y

=2x

+1.

=2

=1.

=

1

2. x

x y

y

(12 , 0) 0, 3 . 382. y = x+ A1, 2 xx. 60 383. y = 3x+2 . 384. P1,2 . ;

4.2.12 y

=1x

+1 , y

=2x

+2.

y = 1x + 1 y = 2x+2 . .

. m

m, 1m + 1


..


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222/333

4. 213

.

.

1m

+1

=2m

+2

1 2m + 1 2 = 0 m

Am +B = 0 A = 1 2 B = 1 2.

. m

R m

. A = B = 0 1 2 = 0 1 2 = 0.

y = 1x + 1 y = 2x + 2 1 = 2 1 = 2 (4.55)

.

. m

R

. A

=0 B

0 1

2

=0

1 2 0. : y = 1x + 1 y = 2x + 2 1 = 2 1 2 (4.56)

. A 0. :

y

=1x

+1 y

=2x

+2

1

2

(4.57)

-

: y = 1x + 1. y = 2x + 2

xx:..



taexei


223/333

214 4.2.

xx 1

=1.

1

=1 .

.

yy 1 = 2. 1 2 :

. y = 1x+1, y = 2x+2 . . y =1x+, y = 2x+ .


..


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4. 215

y

=1x, y

=2x 1

A1

1, 1

, A2

1, 2

y

=1x, y

=2x.

A1A2O A1A2 :1 02 + 1 022 +1 02 + 2 022 = 1 12 + 1 222

1 02 + 1 02 + 1 02 + 2 02 = 1 12 + 1 22 1 + 21 + 1 + 22 = 1 22

1 + 21 + 1 +22 = 21 212 + 22 2

= 212

12 = 1 12 = 1 ( 389) .

y = 1x + 1 y = 2x + 2 12 = 1 (4.58)

108. y = 2x + 3 y

= 2

2

x

+

+2

.

2 2 = 2 + 2 3. = 2 2 + 2 3. 109. y = (1 2)x + 1 y =(2 1)x + 1 ...



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216 4.2.

1. (1 2)(2 1) = (2 + 1) (2 1) = 22 1 = 1. 385. t y

= 4

2t

x

+1

y

=3x

2.

386. t y = 2t + 4x + 4, y = 3t 1 .

387. y = x + : P1 3, y = 3x + 4

388. y = x + : P

2, 4

,

y = 8x + 3 389. y = 1x+1 y = 2x+2 12 < 0. .

4.2.13 x + = 0 x + > 0 fx = x+

x + = 0 x + > 0 . 2x

1

=0. x

f

x

=2x

1

xx. 2x 1 > 0 x fx = 2x 1 xx.


..


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4. 217

2x 1 = 0 12

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

x

+

>0.

=0.

= 0 fx = x + xx.

xx = = 0 xx > 0 = .

0 fx = x+ -.

.

:

f

.

f x

fx. x1 x2 f

x1

fx2. fx = x + > 0 :

x1 < x2 >0 x1 < x2 x1 + < x2 + fx1 < fx2

..



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227/333

218 4.2.

fx = x + < 0 :x1 < x2

x2 x1 + > x2 + fx1 > fx2 - .

x = x > fx > 0 x < fx < 0. :

x > fx > f fx > 0x < fx < f fx < 0

: f = 0 x > fx < f fx < 0x < fx > f fx > 0

110.

fx = x+. 2. x > 2 fx > 0 x < 2 fx < 0.

111.

fx = x+ . fx < 0 fx > 1.


..


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228/333

4. 219

. f(52) = 0 f3 = 1.

fx < 0 x > 52

fx > 1 x < 3. 390. f

x

=x

+.

1. fx = 32. fx = 33. fx > 14. 5 < fx < 3

4.2.14 fx = x f

x

= x

x

:

fx = { x , x < 0x , x 0

y = x y = x. ..



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229/333

220 4.2.

0 -

. Geogebra

: f(x)=abs(x).

fx =x x = x <

x >

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

112.

2 < x < 3. yy 2 3 xx . 2 3

3

2 2 3.

3,2 2, 3.

391. Geogebra

g x = x 1 x 1 < 3...



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232/333

4. 223

113. , ,

fx = x2 + x + A 0,1, B 1, 0, 1, 2.. :

f0 = 1,

Shmeiwseis Algevra a Lykeiou Taexeiola.gr ΜΥ-ΝΕΤ-LAND.BLOGSPOT.GR

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