-
(001-008)- 3-11-06 00:30 1
-
... / II / 2.2.1. / 2.2.1.:
. ...,
:
.
. .
75% 25% .
, , /
, , /
, , /
, , /
,
- ,
, , ,
/
, -
,
,
,
(001-008)- 3-11-06 00:30 2
-
(001-008)- 7-11-06 16:06 3
-
(001-008)- 3-11-06 00:30 4
-
, , . . 5 , 2 . .
:1. . , , .
2. . , , . , .
3. . , , , . .
4. - . , , , .
5. . , .
(001-008)- 9-12-06 13:48 5
-
6. . . , , ( - - ...) , .
:- .- . . :- - . : - .
. , .
(001-008)- 27-12-06 16:27 6
-
1 - 1.1
(- ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . .12. . . . . . . . . . . . . . . . .12. . . . . . . . . . . . .
. . . . . . . . . . . . . . . . .17. T . . . . . . . . . . . . . .
. . . . . .20
1.2 - . . . . . . . . . . . . . . . . . . . . . . . . . . . .
25. - . . . . . . . . . . . . . . . . . . . . . . . 25. . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
1.3 - . . . . . . . . . 331.4 . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . .381.5 . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . .421.6 . . . . . . . . . .
. . . . . . .531.7 . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . .631.8 ... ... . . . . . . .681.9
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
.711.10 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . .75
. - . . . . . . . .75. - . . . . . . . . . . . . . . . .78 1 . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . .81 - 1 . . .
. . . . . . . . . . . . . .83
2 - - 2.1 x + = 0 . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . 862.2 . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . 89
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . 90. . . .94
2.3 . . . . . . . . . . . . . . . . . . . .992.4 . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
.1032.5 - . . . . . . . . . . . . . . . . . . . . .110
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .110.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . .111. . . . . . . . . . . . . . .113 2 . . . . . . . . . . . .
. . . . . . . . . . . . . . . . .118 - 2 . . . . . . . . . . . . .
. .120
3 - 3.1 . . . . . . . . . . . . . . . . . . . . . . . . . . .
.1223.2
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . .128
(001-008)- 27-12-06 14:59 7
-
3.3 . . . . . . . . . . . . . . . . . .133
3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . .140 -
3 . . . . . . . . . . . . . . . .141
4 - 4.1 y = x2 0 . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . .1444.2 y = x2 + x + 0 . . . . . . . . . . . . . .
. . . . . . . . . . .150
4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. .156 - 4 . . . . . . . . . . . . . . . . . . . .158
5 - 5.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1605.2 -
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
.1675.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . .174
5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. .180 - 5 . . . . . . . . . . . . . . . . . . . .181
-
1 - 1.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . .1861.2 . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . .1981.3 . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . .2061.4 . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . .2101.5 . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . 215
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . 215. . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
.220
1.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. .225 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . .229 - 1 . . . . . . . . . . . . . . . . . . . .230
2 - 2.1 0 180 . . . . . . . . . . . . . 2322.2 . . . . . . . . .
. . . 2372.3 . . . . . . . . . . . 2402.4 - . . . . . . . . . . . .
. . . . . . 244
2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. 251 - 2 . . . . . . . . . . . . . . . . . . . . 253 . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . 254
- . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . 255 . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . 256
(001-008)- 27-12-06 15:00 8
-
009-010 3-11-06 00:31 9
-
009-010 3-11-06 00:31 10
-
A
1.1 ( )
1.2
1.3
1.4
1.5
1.6
1.7
1.8 ... ...
1.9
1.10
1
1o
011-019 3-11-06 00:33 11
-
12
, .
. .
.
.. , , 7,34, 2, 3, , , 4, 0,5, 1 + 3, 6,1010010001...
.
, , 0.
.
.
, , .
: 2 = 2, 2 = 2, 0 = 0, =
,
, .
34
34
53
52
34
( )11..11
, - = , 7,34 = ,
3 = , 4 = 2 = , -0,5 = .
2, , , 1+ 3, 6,1010010001...53
-510
21
31
734100
-52
52
34
+7 + 5 = +127 5 = 12
00 11 22 33 441122
5522 44,,88
2233,,88
33
44xx xx
011-019 3-11-06 00:33 12
-
, - , .
, -
++
, -
:
: 0 = 0. A = 0, = 0 = 0. , . , .
. ,
.
( :: , 0),
.
1 =
1 : =
= + ( )
(( ++ )) == ++
11 == 11,, 00 ++ (()) == 00
11 == ++ 00 == (()) == (()) ++ (( ++ )) == (( ++ )) ++
== ++ == ++
13
1.1
+5 7 = 25 + 7 = +2
(+5) (+7) = +35(5) (7) = +35
(+5) (7) = 35(5) (+7) = 35
5 7 = 5 + (7) = 2
5 (7) = 5 + (+7) = 12
5 : 15 = 5 = = 13
515
115
3 , 3
, 54
45
011-019 22-12-06 16:33 13
-
NN ::
)) ((33)) ( ) ( ++ 33) (++ ) ( ) ))
)) (3) ( ) ( + 3) (+ ) ( ) = + + 3 ( ) =
= + + 3 + = + + = = 2
)) = = = =
++ == 33 ++ == 55,,
== (( 22)) ++ 22( )..
= ( 2) + 2( ) == + 2 + 2 = ( )= 2 + 2 = ( )= 2( + ) ( + ) = ( )=
2(3) (5) == 6 + 5 == 1
x .
:)) 3 + 7 = ..... )) 6 + 6 = ..... )) 2 9 = .....
)) (2) = ..... )) 0 ( ) = ..... )) ( ) ( ) = .....))(6) :: ( )=
..... )) ( ) :: (+4)= ..... )) ( ) :: (+ )= .....434385125
54
45
27
13
22
22227733,,114411663300,,8800,,3366
112233
11
E
2
22
22
32
1510
5 25 3
6 1 + 2 26 1
3 3
13 + 212 3
126
16
186
26
276
16
13
92
16
13
92
12
13
13
32
1133 ++ 221122 33
1122
1133
1133
3322
11
14
M - 1
011-019 3-11-06 00:33 14
-
:)) (3 2 5)x = ........ )) 3(2 5x) = ........ )) 3(2 5)x
=........)) 2(x ... .....) = ..... + 6 )) (3 + x)(2 + y) = ........
)) 4(... + ...) = 12x + 8
:ii)) A , :
)) )) )) )) .
iiii)) A , :)) )) )) )) .
(), (), :
)) .
)) .
)) .
)) .
N :)) 2 + 3 4 12 :: (4) + 1 )) 2 + 3 (4 12) :: (4 + 1))) 3 (2) 5
+ 4 :: (2) 6 )) 8 :: (3 + 5) 4 (2 + 6)
.
(5 4) (+2) + (6 + 4) (7) =4 ( 2 + 6 3) + (9 + 6) =
14 + (6 + 5 3) ( 4 1) (2) =(3) (2) + 4 (+5) (1) :: (1) =
, xx . = 5km, .
33
22
11
55
44
33
15
1.1
OO AA BB
112233445566 00 11 22 33 44 55 66xx xx
011-019 3-11-06 00:33 15
-
:
)) ( ) + ( ) (+ ) )) ( + ) + ( + )
)) 5 5 ( ) )) (1 ) ( ) :: ( + )
:
)) )) )) 7 +
:1, 3, 0, 2, 1, 2, 5, 0, 3, 1.
.
(+ ).)) 12 ... 5 ... 20 = 3 )) 8 ... 9 ... 1 = 0
)) ... ... = 3 )) 0,35 ... 6,15 ... 8,50 = 2
:)) 8 ( ) + ( 5 ) = 3)) 2 ( + ) (4 + ) (2 ) = 0)) 2 ( 3) + (7 +
9) 3 (+2) = 0
x + y = 5 + = 7, : = 4 (x ) (y ) = (5 x + ) + (8 + y) ( 4)
, , 56 , , 32, = (9 2) (15 2).
7, 6, 5, 3, 1, 2, 4, 5, 9
, .
1111
1100
99
88
104
34
54
77
66
1 3 31 2 + 3
1 2 3 41 2 (3 )4
1 2 + 12 31 13 + 6 2
55
23
25
35
45
12
72
23
12
23
12
116
53
12
56
32
13
112
12
14
23
44
16
M - 1
+ + =
+ + =
+ + =
011-019 3-11-06 00:33 16
-
2 . == ......
-
: 11 == 00 == 11 00
== 00
, :
NN ::
)) )) xx22 ((xx yy22))33 :: ((xx22 yy33))22
)) = = = =
)) x2(xy2)3 :: (x2y3)2 = = = = x
AA xx33 yy22 == 33,, == xx22 ((xx22 yy33))22 ((xx11))33..
AA = x2 (x2 y3)2 (x1)3 = x2 x4 y6 x3 = x2 x4 x3 y6 = x9 y6 =
= (x3 y2)3 = (3)3 = 27.
22
x5y6
x4y6x2x3(y2)3
(x2)2(y3)2x2(xy2)3
(x2y3)2
34
322
22 33
32 2422 (33)32 (22)2
(2)2 (3)3
(3 22)2
((22))22 ((33))33
((33 2222))22
11
2 4 3 4() = ()3 2 ( ) == ( )
(23)2 = 26 = 64(( )) ==
2 3 23 8( ) = = 3 33 27 ( ) ==
(2) 2 = 222 = 42(()) == 35 : 33 = 353 = 32 :: == 23 24 = 23+4 =
27 == ++
11
BB
17
1.1
23 = 2 2 2 = 8
(3)2 = (3) (3) = 9
011-019 3-11-06 00:33 17
-
NN ::AA == ((22))22 ((33)) ++ 22 3322 5522 ((22)) :: 55 66 BB ==
((22 55 3322)) ++ 22 ((2233 44)) 1122 :: ((33))
= (2)2 (3) + 2 32 52 (2) :: 5 6 == 4 (3) + 2 9 25 (2) :: 5 6 ==
12 + 18 + 50 :: 5 6 == 12 + 18 + 10 6 = = 10
= (2 5 32) + 2 (23 4) 12 :: (3) == (2 5 9) + 2 (8 4) 12 : (3) ==
10 9 + 2 4 12 : (3) == 1 + 8 + 4 == 9 + 4 = = 13
(), (), :
)) + + + = 4.
)) = 4.
)) (5)6 56 .
)) ( )8 ( )8 . )) (3)2 = 92.
)) (5)2 .
)) 32 .
(= ).
)) (1)6 ... 1 )) 32 ... 9 )) 42 ... 16 )) ( )1... )) 52 ... ))(
)0 ... 0 )) ( )5 ... )) (7 + 2)2 ... 72 + 22
.
ii)) H ( )2 :)) )) )) )) 49
94
94
49
23
33
132
12
25
125
25
52
22
32
23
11
E
33
18
M - 1
HH .
.
, .
, .
011-019 3-11-06 00:33 18
-
iiii)) H (2)03 :)) 23 )) 6 )) 23 )) 1
iiiiii)) H 23 + 32 :)) 55 )) 17 )) 56 )) 65
, .
N :
)) 25 28 )) 34 :: 32 )) 23 53 )) (52)4
)) 32 (3)4 )) )) 42 :: 34 )) 27 34
N :
)) (22)3 28 )) (3)2 (3)4 )) (0,75)2 ( )2 )) 363 :: (12)3
)) (2,5)4 (4)4 )) 412 :: 220 )) ( )12 ( )14 )) (0,01)3 105
N :)) (x2)3 5x4 )) (xy3)2 x3y )) (2x)2 (2x2)
)) ( x)3 :: x2 )) (3x2)3 (2x3)2 )) x3 : ( x)2
N : = 3 (2)2 + 4 (7)0 2 8 (21 1) 2 32 = (4)2 :: 2 5 (3) 22
(2)4
= (2,5)2 (1,25)3 (4)2 (8)3 = (257 84) :: (57 404)
, ;
55
44
32
32
23
33
23
23
34
22
135
(6)6
26
11
11.. 14
22.. 24
33.. 444.. 23
55.. 24
66.. 1
.. (24)1
.. (25)2 210
.. (2)2
.. (24 :: 23) 22
44
19
1.1
011-019 3-11-06 00:33 19
-
20
M - 1
x xx - x. .. 25 = 5, 52 = 25 0 = 0. (5)2 = 25, (5)2 = 25 = 5 = 5
.
, xx :
, . : (9 )2 = 32 = 9, (9 )2 = 9.
II
1. (= )4 100 ... 4 100 ...
2. :2 5 ... 2 5 ...
4 100 :
4 100 = 4 100 = - 2 5. , , . .
, :
.
.
4100
4100
25
25
4100
4100
x 0, (x )2 = x
x2 = xx
(7)2 = 7 = 7, 72 = 7
=
== >> 00
020-024 3-11-06 00:35 20
-
, .
( )2 = ()2 ()2 = ( )2 =
, , .
= . .
16 + 9 = 4 + 3 = 7, 16 + 9 = 25 = 5 16 + 9 16 + 9.:: ,, ,, +
+
NN 2200 == 2255 ,, 22 == ..
20 = 4 5 = 22 5 20 = 22 5 = 22 5 = 25. 2 = 2 = . 20 -
.. 20 = 2 10,
.
NN ::)) 3333 ++ 2233 == 5533 )) 33 2244 == 6622 )) 5500 1188 ==
2222
)) :
33 + 23 = (3 + 2)3 = 53)) 3 24 = 3 24 = 72 = 36 2 = 36 2 = 62))
50 18 = 25 2 9 2 = 25 2 9 2 = 52 32 = 22
NN ,, ,,
..
= = = 533
53(3 )2
5 33 3
53
5533
33
22
11
21
1.1
o .
020-024 3-11-06 00:35 21
-
1122 mm22 33 mm22 .. ..
2 = 12 m2, = 12 m. T 2 = 3 m2, = 3 m. : = = 12 3 = 12 3 = 36 = 6
m2. : = + = + = 12 + 3 =4 3 + 3 = 23 + 3 = 33 m.
:)) 33 + 3 = ..... )) 52 32 = ..... )) 5 + 45 55 = .....)) 12 3
= ..... )) 18 :: 2 = ..... )) 32 8 = .....
.
:
366416914
++ ++
33
11.. 5
22..
33.. 5
.. 25
.. 25
.. 25
.. 52
.. (5)2
.. 52
BB
22
11
E
44
22
M - 1
12 m2
3 m2
020-024 3-11-06 00:35 22
-
(), (), .)) 2 3 = 6
)) 2 + 3 = 5
)) = )) (3)2 = 3
)) ( 1)2 = 1)) 5 10.
)) 12 3.
50 m2. 52 m;
N :
)) 35 75 + 25 )) 57 83 27 + 43
)) )) + N :
)) 32 50 + 32 68 = 102 )) 27 20 + 12 5 = 53 35
)) 3 18 2 48 + = 6 )) 3,6 4,9 0,8 0,2 = 3,8
N :
)) 12+16 )) 86+2529 )) 6 1239
N , . ;
32322242E252
44
33
1205
22
143
212
107
145
127
37
58
52
11
55
12
12
32
94
44
23
1.1
020-024 7-11-06 21:26 23
-
N :
)) 2 (18 + 8 ) )) 6 (27 3 ))) (75 + 45 300 ) :: 15 )) (7 5)(7 +
5)
N , , .
)) )) )) ))
N :
)) 5 + x = 35 x )) 6 x = 24
)) = 32 )) 33 x = 27
N (3 1)(3 + 1) = 2. - , , -
.
, E 50 m2 8 m2 , - 98 m2.
= 3 cm = 6 cm , , , = 2 cm AE = 1 cm. = 3.
( = ), = 4 cm = 4 cm.)) -
4 + 45 cm.
)) 4 :4 + 20, 4 + 220, 85 , 2(2 + 20). ;
1111
1100
99
13 1
88
x2
77
23 + 6 3
525
46
12
66
55
24
M - 1
50 m2
8 m2
BB
4 cm
4 cm
020-024 3-11-06 00:35 24
-
25
M .
.
.
--
1. .
2. . 2x2 + xy. x = 5 y = 8, ;
AA , - ..
- , , . .
, - .
A , . , 2x2 + xy x = 5 y = 8, 2 52 + 5 8 = 90.
M 11..22
x
x
x
5
5
5
8
x
x
y
6 5 + 2 8, 2 52 + 5 8
4x, 2 + 2, x2, ,
, 2x2 + xy2xy 3
2x + 32, + 2, R343
12
025-029 3-11-06 00:38 25
-
, , .
, .
, .
.
x3y2, 5x3y2, x3y2, .
,
, .
2x3y 2x3y .
. , 0 , . , 5 .
NN )) 33xx22yy33,, xx == 22 yy == 11 )) 2222 33 ++ 66 == 33 ==
88..
)) H 3x2y3 x = 2 y = 1 :
3 (2)2 (1)3 = 3 (+4) (1) = 12.
)) 22 3 + 6 = 3 = 8 :2 (3)2 3 8 + 6 = 2 (+9) 24 + 6 = 18 24 + 6
= 0.
11
25
26
M - 1
4x, x2, , 2 x 4y2323
22xx33yy ::33 xx11 yy44 xx yy
2x3y
025-029 1-12-06 21:36 26
-
TT (( )) ,, (( ccmm))
== ( 110000 ++ ),, tt (( )) (( == 00,,99 == 00,,88)).. ,, 3300
11,,7777 mm..
T ( ) 30 1,77 m = 177 cm,
= 0,9 (177 100 + ) = 0,9 (177 100 + 3) = 0,9 80 = 72 .
T ( ) 30 1,77 m = 177 cm,
= 0,8 (177 100 + ) = 0,8 (177 100 + 3) = 0,8 80 = 64 .
,, .. -- ,, .. == 1100 ccmm..
T 2, (2)2 = 42. 2, 42 2. 42 2
42 2 = (4 )2 = (4 3,14)2 = 0,862
0,86 2. = 10 cm 0,86 102 = 0,86 100 = 86 cm2.
;
)) 3x2y )) 3 + x2y )) )) 2x2y3 )) (3 2)3 )) 3
:
)) 6x2y2 )) xy3 )) x3y )) 5y3x )) )) y2x2
)) )) x2y2 )) yx3 )) 2xy3xy3
7
52
yx3
435
22
23
x3y2
11
E
33
3010
3010
tt1100
22
27
1.2 MM
025-029 3-11-06 00:38 27
-
N :
xy23.
.
.
OO 11.. 11.. .
22.. 7x4y5 x ( ). 4.
22.. 8, 5, 0, 3. 33.. -33..
3x2 y. 44.. 2x2y 2. .
55.. x3y, 3x3y. 44.. 5xy2, 25xy2.
66.. O xy. 55.. E 425, 25.77.. xy2 4xy2 66.. H .........
2x2y
( ). x = 2 y = 1 8.88.. . 77.. -
6, 3, 7.88.. -
.
62
55
1344
33xx44
11 xx22yy5577
xxyy 2255xxyy 44
xx yy
yy
xx
33
28
M - 1
025-029 3-11-06 00:38 28
-
N :)) 2xy3 + x2y 4 x = 2 y = 1
)) x2 + 3 x = 3 = 2
, .
, 2 5.
, 3xy2
)) x)) x y)) 48, x = 2 y = 1.
. , , 4x3y, xy2 :)) )) ))
. , . , = 10;
9 . , 2 1 .
. , x = 12;
77
66
55
44
33
5722
12
23
11
29
1.2 MM
xx
55
025-029 3-11-06 00:38 29
-
30
M - 1
.
.. 5x3 + 2x3
5x3 + 2x3 = (5 + 2)x3 = 3x3
: ..
, 12x2y 3x2y = 15x2y.A , 3x 5y, 3x + 5y .
.. (2x)(3x2y) -
(2x)(3x2y) = (2)x 3x2y = (2) 3(xx2)y = 6x3y.
: :
.
(3x4y3)( x3) = x5y34.
, , . ,(12x4y2) : (4x 2y) = 12x4y2 = = = 3x2.
: (7xy4) : (x3y) = =
, .
7y 3
x27y 4
x3y
2
yy
4
2124
12 4y2
4x2y1
4x2y
65
25
BB
030-032 3-11-06 00:45 30
-
NN ::
)) 77xx22 xx22 ++ 44xx22 )) ( xxyy22) ( xx33yy22) )) ( 33) :: (
33)
)) 7x2 x2 + 4x2 = (7 + 4) x2 = ( + ) x2 = x2
)) ( xy2) ( x3y2) = x4y4 = x4y4
)) ( 3) :: ( 3) = : ( ) = ( ) = =
.. tt sseecc ,, ,, ..
A = gt2, g = 10 m/sec2 . = 5t2.
A , sec
= g ( )2= 10 = t2. H = = 5t2 t2 = t2 t2 = t2.
MM ,, ,, .. .. ;;
V1 = 2. , V2 = (2)2 = (42) = 42. - V2 V1 = 42 2 = 32, 2 . .
33
154
54
204
54
54
t24
12
t2
12
t2
12
22
3222
6343
23
334
32
334
12
34
16
212
14
23
72
82
12
142
12
12
1122
3344
1144
2233
1122
11
31
1.2 MM
030-032 3-11-06 00:45 31
-
(), () .)) .)) .)) .)) .
:)) 5x2 + 2x2 = .......... )) 5x2 2x3 = .......... )) 3x 2y + 2x
= ..........)) 4x2y yx2 = .......... )) 2xy y2 = .......... )) 6x3y
:: 3xy = ..........
)) 5x43 (.....) = 10x64 )) = )) 3x2y ..... = 4x2y
N :)) 7x2y + 4x2y )) 4x2 6x2 + x2 )) 6x3 x3
)) 0,25 0,35 + 0,5 )) xy24 1,2xy24 )) 32x2 + 42x2 2x2
N :)) 3x 5x2 )) 6x2 x3 )) 2xy3 (3x2y) )) 3x2y (2xy4)
)) 3 43 )) x32 ( x3) )) ( xy3) (3x2) ( y3)N :
)) 123 :: (3) )) 8x2y :: (2xy2) )) ( 35) :: ( 22))) (0,84x25) ::
(0,12x3) )) (x34) :: ( x2) )) (0,537) :: ( 22) :
)) ( x2y)2 (6xy3) )) (2x2y3)3 :: (8x3y4) )) (2xy43)2 (x2y)3 .
;
.
66
55
13
44
710
14
65
13
33
56
25
14
43
13
34
22
25
92
11
4x2
y12x3y........
22
11E
32
M - 1
BB
xx
yy
xx
xx
xxxx
))
yy
))yy
xx
xx
xx xxxx
yy
))
22xx
22xx
))
22xx
yy
))
030-032 3-11-06 00:45 32
-
33
M , .
M .
1. .
2. . ;
3. , .
, . , , . .. 3x2y + 2xy4 5x3y3
K .
E, , , .
, - .
, , , . , , .
A
11.. 33
32 222 3 + 4
3x2y + 2xy4 5x3y3
3x2y, 2xy4, 5x3y3.
3x2y + 2xy4 5x3y3 3 x,4 y,
6 x y.
033-037 7-11-06 16:09 33
-
3x + 2x2 + 5 x (x) Q(x) A(x) ... (x) = 3x + 2x2 + 5 , .
, (x) = 2x2 3x + 5.T, , xx.H (x) x = 5, (5) :
(5) = 2 52 3 5 + 5 = 50 15 + 5 = 40.
, .
, , - . .
. , (x) = 3x3 2x2 7x 5 (x) = 2x3 x2 + x :((xx)) ++ BB((xx))=
(3x3 2x2 7x 5) + (2x3 x2 + x) = ( )
= 3x3 2x2 7x 5 + 2x3 x2 + x = ( )= 5x3 3x2 6x 5.
, :((xx)) BB((xx)) = (3x3 2x2 7x 5) (2x3 x2 + x) =
= 3x3 2x2 7x 5 2x3 + x2 x == x3 x2 8x 5.
)) ((xx)) == 44xx22 88xx ++ xx33 55 xx ..
)) ((xx)) QQ((xx)) == xx22 ++ xx ++ ,, ,, ,, ,, ;;
11
34
M - 1
3x2 5x + 1 x2 + x + 1 , = 3 = 5.
22 3 + 42 5 =22 + 42 3 5 = 62 8
, , .
033-037 3-11-06 00:50 34
-
)) (x) = 4x2 8x + x3 5, x
(x) = x3 + 4x2 8x 5. To P(x) , 0 , = 0.
)) (x) = x3 + 4x2 8x 5 Q(x) = x2 + x + , = 0, = 4, = 8 = 5.
xx
550000 C ,, 1100 C
((,, ,, ......)) xx22 C ((,,
......)).. xx ;; ,, 5500 ;;
10 C, x 10x C. C, (x) = x2 + 10x + 500
50 :(50) = 502 + 10 50 + 500 = 2500 + 500 + 500 = 1250 C
((xx)) == xx22 33xx ++ 44,, QQ((xx)) == PP((22xx))
PP((xx))..
o (2x) , (x) , x 2x, :(2x) = (2x)2 3(2x) + 4 = 4x2 6x + 4 (x) ,
(x) , x x, :(x) = (x)2 3(x) + 4 = x2 + 3x + 4. :Q(x) = P(2x) P(x) =
(4x2 6x + 4) (x2 + 3x + 4) =
= 4x2 6x + 4 x2 3x 4 = 3x2 9x
;
)) 4x3 5x2 + 2x )) 3x4 7x2 12
)) 2 x2y 5xy + y2 + )) x3 + 2x2y x y2 + 3y3
2 x;)) 7 3x 2x2 )) 3x2 5x 3x2 + 10)) 4x3 + x2 3x3 + 2x x3 + 6 ))
2xy 3y + 9
22
13
1x
11
E
33
110
110
110
111100
22
35
1.3 oo
033-037 7-11-06 16:10 35
-
- 4x3 8x2 + x + 7 x3 6x + 2
4x3 8x2 + x + 7 4x3 8x2 + x + 7
+ x3 6x + 2 + x3 + 6x 25x3 8x2 5x + 9 3x3 8x2 + 7x + 5
E ; .
. 2x2 + 5x + 7 8x2 + 4x 5 :)) 6x2 + x 2 )) 10x2 + 9x + 2 )) 6x2
x 12 )) 6x2 + x + 12.
T (x), B(x) (x) 2, 3 2 .)) (x) + B(x).)) A (x) + (x) , ;
N x.)) P(x) = 3x 5x2 + x4 + 10 + 2x3 )) Q(x) = 6x + 2x3 + 1))
A(x) = 3x2 + 7 + 2x3 + 7x )) (x) = x x4 5
= 2xy2 + y3 + 2x3 xy2.)) x = 2 y = 1.)) y.
x y;
A (x) = 2x2 + 2x 9, :)) (3) = (2) )) 3(1) + (3) = 0
-, 100 2x .)) .))
, 60 .
44
33
22
11
55
44
33
36
M - 1
110000 mm
22xx
033-037 3-11-06 00:50 36
-
N :)) (2x2 x) (x3 5x2 + x 1) )) 3x2y (2xy yx2) + (3xy y3))) (22
3) (2 + 4) (2 + 2) )) 22 4 3 (2 + 5))) ( x2 x + 1) ( x + x2 ) ))
(0,4x3 + 2,3x2) + (3,6x3 0,3x2 + 4)
A (x) = 2x3 x2 + x 4, B(x) = 3x3 + 5x 2 (x) = 4x2 3x + 8, :))
(x) B(x) )) (x) + (x) )) (x) A(x) + B(x)
N :)) (... ..... 4x ... .....) + (x2 ... ..... + 4) = 6x2 8x +
7)) (x3 ... ..... + 8) (... ..... + x2 ... .....) = x3 x2 + 5x +
9
. ( - , ).
A P(x) = (5x2 + 4x 3) (x2 2x + 1) + (3x2 + x) Q(x) = x2 + x + ,
, , , (x) Q(x) .
t sec = 2 m/sec2. , 10 sec . . , t = 5 sec;
1100
99
88
77
66
13
16
34
12
55
37
1.3 oo
2x2+2x3
9x23x+2
4x2+4x5
7x2+3x4
033-037 3-11-06 00:50 37
-
38
:
1. ( + ) 3x2(2x3 + 6x).
2. ( + )( + ) (3x2y + 2y)(2x2 + 5).
3x2(2x3 + 6x) 3x2 2x3 + 6x, ,
3x2(2x3 + 6x) = 3x2 2x3 + 3x2 6x = 6x5 + 18x3
: , - .
(3x2y + 2y)(2x2 + 5) 3x2y + 2y 2x2 + 5, , (3x2y + 2y)(2x2 + 5) =
3x2y 2x2 + 3x2y 5 + 2y 2x2 + 2y 5 =
= 6x4y + 15x2y + 4x2y + 10y = 6x4y + 19x2y + 10y
: , .
, ..
11..44
038-041 7-11-06 16:11 38
-
NN ::
)) xx22yy(xx yy 33) )) ((22xx22 55xx ++ 66))((xx 22))))
44xx((22xx22 ++ 33xx 11)) 33xx22((22xx ++ 55)) )) 22xx22 ((xx ++
44))((xx 11))
)) x2y(x y 3) = x3y + x2y2 + 2x2y
)) (2x2 5x + 6)(x 2) = 2x3 4x2 5x2 + 10x + 6x 12 = 2x3 9x2 + 16x
12
)) 4x(2x2 + 3x 1) 3x2(2x + 5) = 8x3 + 12x2 4x + 6x3 15x2 = 2x3
3x2 4x
)) 2x2 (x + 4)(x 1) = 2x2(x2 x + 4x 4) = 2x2(x2 + 3x 4) = 2x4
6x3 + 8x2
O - - . -, () (2x2 5x + 6)(x 2) :
,, .. ,, ,, ..
H 2x + y 4x + 10, (2x + y)(4x + 10). y x + 10, y(x + 10)., -
:(2x + y)(4x + 10) y(x + 10) = 8x2 + 20x + 4xy + 10y xy 10y =
= 8x2 + 20x + 3xy
22
29
23
13
23
1133
2233
11
39
1.4
xx xxxx
xx++1100
22xx
yy
2x2 5x + 6x 2
4x2 + 10x 12+ 2x3 5x2 + 6x
2x3 9x2 + 16x 12
22 ((22xx22 55xx ++ 66)) xx ((22xx22 55xx ++ 66))
038-041 7-11-06 16:12 39
-
, .
(), () .)) (x) 3 Q(x)
2, (x) Q(x) 6.)) (x) Q(x) 7
(x) 3, Q(x) 4.
:)) x(2x + .....) = ..... + 4x )) 3x2(..... 2) = 3x3y .....)) (x
+ 5)(..... + 3) = 2x2 + ..... + 10x + ..... )) (x2 + y)(x .....) =
..... x2y2 + ..... y3
.ii)) O
:)) 3x + 1 )) x3 + 1 )) x3 + x2 )) x3 + x
iiii)) T - :)) 6x2 + 4x + 1 )) 4x2 + 6x)) 6x2 + 4x + 2 )) 6x2 +
4x
O - - :)) (x + 2)(x + 3) )) 2x 3x)) x2 + 6 )) x2 + 5x + 6
;
55
44
33
22
11.. x2 x22.. x2 + 133.. x2 + 2x + 144.. x2 + 2x + 355.. x2 +
x66.. x2 + 3x + 277.. x2 1
.. x(x + 1)
.. (x + 1)(x 1)
.. x(x 1)
.. (x + 1)(1 + x)
.. (x + 1)(x + 2)
BB
11
E
40
M - 1
22
33
xx
xx
xx
xx
xx++11
038-041 3-11-06 00:54 40
-
N :)) 3x2y(5x + 2y) )) 4x(2x2 x + 2) 8x)) 5x(2x 3) 3x(2 3x) ))
2xy(x2 3y2) 4x(x2y 2y3)
N :)) (2 3)(4 + 2) )) (x2 2x + 4)(x + 2) 8)) 3x2(2x + 3)(5 x) ))
(4 3x)(5 2x) 6x(x 4))) (2x2 3x 4)(3x2 + x) )) (3x2 2xy 5y2)(4y
x)
N :)) (3x 2)(x2 x)(4x 3) )) 2x(x2 x + 1)(x 2) (x 1)(2x 3)(x +
2))) (2x + y)(x2 3xy) (3x y)(4x + y)(2x 3y)
N :)) (x2 4x + 4)(x2 + 4x + 4) x2(x2 8) 16 = 0)) (3 + 8)( ) ( +
2)( 3) = 62
(x) = 2x2 + 5x 3 Q(x) = 4x 5, :)) (x) Q(x) )) P(x) 3Q(x) + 11x
12 )) P(x) 2 Q(x) + 3
P(x) = 3x(2x + 4)(x 1) Q(x) = x3 + x2 + x + , , , , , (x) Q(x)
.
.
x 5 . 3 1 , 4x + 2 .
88
77
66
55
44
33
22
11
41
1.4
xx
22xx
xx
yyyyyy
038-041 3-11-06 00:54 41
-
42
. . .
1. 3x = 12, x + y = 7, 4 = 3 + , x(x + 2) = x2 + 2x, ;
2. )) - .
)) -;
i) x2 + 9 ii) (x + 3)2
iii) x2 + 6x iv) 6x + 9)) N
.
Y . , 3x = 12, x = 4 x. O, x + y = 7, x = 1 y = 6, x = 3 y = 4,
x = 4 y = 5.
Y , : + = + , 4 = 3 + , x(x + 2) = x2 + 2x. .
.
T , . :
A
11.. 55
xx
xx
xx
xx
xx
xx
33
33
33
33
33
33
042-052 3-11-06 00:58 42
-
)) ( + )2 ( + )( + ) , :
( + )2 = ( + )( + ) == 2 + + + 2 == 2 + 2 + 2
( + )2. , (y + 4)2 , y 4, :
(y + 4)2 = y2 + 2 y 4 + 42 = y2 + 8y + 16.
H , , , , ..
(2x + 3y)2 = (2x)2 + 2 2x 3y + (3y)2 = 4x2 + 12xy + 9y2
( + )2 = 2 + 2 + 2
)) ( )2 ( )( ) ,
:( )2 = ( )( ) = 2 + 2 = 2 2 + 2
, (y 4)2 , y 4, :
(y 4)2 = y2 2 y 4 + 42 = y2 8y + 16O, (3x 4y)2
:(3x 4y)2 = (3x)2 2 (3x) (4y) + (4y)2 = 9x2 24xy + 16y2
( )2 = 2 2 + 2
( )2 = 2 2 + 2
( + )2 = 2 + 2 + 2
43
1.5 AA
( + )2 = 2 + 2 + 2
. + ,
: = ( + )2 (1)
.
= 2 + + + 2
= 2 + 2 + 2 (2)
(1) (2)
(( ++ ))22 == 22 ++ 22 ++ 22..
2
2
BB
042-052 3-11-06 00:58 43
-
)) ( + )3 ( + )( + )2 - + ( + )2, :( + )3 = ( + )( + )2 =
= ( + )(2 + 2 + 2) == 3 + 22 + 2 + 2 + 22 + 3 == 3 + 32 + 32 +
3
: (x + 2)3 = x3 + 3 x2 2 + 3 x 22 + 23 = x3 + 6x2 + 12x + 8 (2x
5)3 = (2x)3 3 (2x)2 5 + 3 (2x) 52 53 = 8x3 60x2 + 150x 125.
)) ( + )( ) :( + )( ) = 2 + 2 = 2 2
. , :
(x + 2)(x 2) = x2 22 = x2 4 (3 + 2)(3 2) = (3)2 (2)2 = 92 42
)) ( )(2 + + 2) :( )(2 + + 2) = 3 + 2 + 2 2 2 3 = 3 3
. :
(x 2)(x2 + 2x + 4) = (x 2)(x2 + 2x + 22) = x3 23 = x3 8 (x +
3)(x2 3x + 9) = (x + 3)(x2 3x + 32) = x3 + 33 = x3 + 27
( + )(2 + 2) = 3 + 3
( )(2 + + 2) = 3 3
( + )( ) = 2 2
( )3 = 3 32 + 32 3
( + )3 = 3 + 32 + 32 + 3
44
M - 1
042-052 3-11-06 00:58 44
-
)) (( ++ ++ ))22 == 22 ++ 22 ++ 22 ++ 22 ++ 22 ++ 22..)) ((33xx
++ 22yy ++ 44))22..
)) ( + + )2 = ( + + )( + + ) =
= 2 + + + + 2 + + + + 2 == 2 + 2 + 2 + 2 + 2 + 2.
)) , (3x + 2y + 4)2 :(3x + 2y + 4)2 == (3x)2+(2y)2+42+2 3
x2y+22y4+23x4 == 9x2 + 4y2 + 16 + 12xy + 16y + 24x.
)) 22 ++ 22 == (( ++ ))22 22 33 ++ 33 == (( ++ ))33 33(( ++
))..
)) xx ++ == 33,, xx22 ++ xx33 ++ ..
)) :
( + )2 2 = 2 + 2 + 2 2 = 2 + 2
( + )3 3( + ) = 3 + 32 + 32 + 3 32 32 = 3 + 3.
)) x2 + x2 + ( )2
2 + 2 = ( + )2 2 :
x2 + = x2 + ( )2 = (x + )2 2x = 32 2 2 = 9 4 = 5.H x3 + x3 + (
)3 3 + 3 = ( + )3 3( + ) :
x3 + = x3 + ( )3 = (x + )3 3x (x + ) = 33 3 2 3 = 27 18 = 9
,, ,, ;;
To 2. ,
33
2x
2x
2x
2x
8x3
2x
8x3
2x
2x
2x
4x2
2x
4x2
88xx33
44xx22
22xx
22
11
45
1.5 AA
2
2
2
((++++))22 == 22++22++22++22++22++22
042-052 3-11-06 00:58 45
-
+ , ( )( + ) = 2 2., 2 2 2, 2.
..
3 + 5 , (3 5)(3 + 5) = 32 (5)2 = 9 5 = 4.
= = = .
)) (( 11))(( ++ 11)) ++ 11 == 22..)) 22000077 22000099 ++ 11
,, ..
)) ( 1)( + 1) + 1 = (2 12) + 1 = 2 1 + 1 = 2.
)) = 2008, 1 = 2007 + 1 = 2009. :2007 2009 + 1 = ( 1)( + 1) + 1
= 2 = 20082., 2007 2009 + 1 2008.
..99 101 = (100 1)(100 + 1) = 1002 12 = 10000 1 = 99991032 =
(100 + 3) 2 = 100 2 + 2 100 3 + 32 = 10609
::)) ((22xx 33))22 22((33xx 11))((33xx ++ 11)) )) ((xx 22yy))33
((xx yy))((xx22 ++ xxyy ++ yy22))..
)) (2x 3)2 2(3x 1)(3x + 1) = (2x)2 2 2x 3 + 32 2(3x)2 12 =
= (4x2 12x + 9) 2(9x2 1) == 4x2 12x + 9 18x2 + 2 = 14x2 12x +
11
)) (x 2y)3 (x y)(x2 + xy + y2) = x3 3 x2 2y + 3 x (2y)2 (2y)3
(x3 y3)== x3 6x2y + 12xy2 8y3 x3 + y3 == 6x2y + 12xy2 7y3
66
55
3 +52
2(3 +5)4
2(3 +5)(3 5)(3 + 5)
23 5
2233 55
44
46
M - 1
042-052 3-11-06 00:58 46
-
((22 ++ 22))((xx22 ++ yy22)) == ((xx ++ yy))22 ++ ((yy
xx))22
((TT LLaaggrraannggee)).
1 :
(2 + 2)(x2 + y2) = 22xx22 ++ 22yy22 ++ 22xx22 ++ 22yy22
2 :(x + y)2 + (y x)2 = (x)2 + 2(x)(y) + (y)2 + (y)2 2(y)(x) +
(x)2 =
= 2x2 + 2xy + 2y2 + 2y2 2xy + 2x2 == 22xx22 ++ 22yy22 ++ 22xx22
++ 22yy22
(2 + 2)(x2 + y2) = (x + y)2 + (y x)2.
;)) 0x = 0 )) x + y = 0 )) 2 = 3 )) (x + 3)2 = x2 + 6x + 9 )) =
0
.ii)) (x + )2 :
)) x2 + 2 )) x2 2x + 2 )) x2 + x + 2 )) x2 + 2x + 2
iiii)) (2 + 1)2 :)) 22 + 4 + 1 )) 42 + 1 )) 42 + 4 + 1 )) 42 + 2
+ 1
iiiiii)) (y 2)2 :)) y2 2y + 4 )) y2 4 )) y2 4y + 4 )) y2 + 4y +
4
(), (), .)) (x y)2 = x2 2x(y) + (y)2
)) ( + )2 = 2 2 + 2
)) (5 + 4)2 = 252 + 16)) (3x y)2 = 3x2 2 3x y + y2
33
22
11
E
77
47
1.5 AA
- = , :
( 1, 2, 5)
1 = .
2 = .
= = = ( 7).
042-052 3-11-06 00:58 47
-
.
ii)) T (x + 1)3 :
)) x3 + 3 x 1 + 13 )) x3 + 13
)) x3 + 3 x2 1 + 3 x 12 + 13 )) x3 + x2 1 + x 12 + 13
iiii)) T ( 2)3 :
)) 3 3 2 + 23 )) 3 23
)) 3 2 2 + 22 23 )) 3 3 2 2 + 3 22 23
() (),
.
)) (x y)3 = x3 3x2y 3xy2 y3
)) (2x + 3)3 = 2x3 + 3 2x2 3 + 3 2x 32 + 33
)) (3x 1)3 = (3x)3 3(3x)2 1 + 3(3x) 12 + 13
)) (x + 2)3 = x3 + 6x2 + 12x + 8
.
ii)) T (y 3)(y + 3) :
)) y2 3 )) 9 y2 )) y2 9 )) 3 y2
iiii)) T (y + x)(x y) :
)) y2 x2 )) x2 y2 )) (x y)2 )) x2 + y2
iiiiii)) T ( 2)( + 2) :
)) 2 22 )) 2 + 42 )) 42 2 )) 2 42
iivv)) T (5 x)(52 + 5x + x2) :
)) 53 + x3 )) x3 53 )) 53 x3 )) 25 x3
vv)) To (x + 2)(x2 2x + 42) :
)) x3 + 23 )) x3 (2)3 )) x3 23 )) x3 + 83
, .
11.. x2 2xy + y2
22.. x3 y3
33.. x3 3x2y + 3xy2 + y3
44.. y2 x2
55.. x2 + 2xy + y2
66.. x2 y2
77.. x3 + y3
88.. x3 3x2y + 3xy2 y3
.. (x + y)(y x)
.. (x + y)2
.. (y x)2
.. (x y)(x2 + xy + y2)
.. (x + y)(x2 xy + y2)
.. (x y)3
B
77
66
55
44
48
M - 1
042-052 3-11-06 00:58 48
-
N :)) (x + 2)2 )) (y + 5)2 )) (2 + 1)2 )) ( + 2)2
)) (3y + 2)2 )) (x2 + 1)2 )) ( y2 + y)2 )) (2x2 + 3x)2
)) (x + 2)2 )) (x + y )2 )) ( + )2 )) ( + )2
N :)) (x 3)2 )) (y 5)2 )) (3 1)2 )) (2 )2
)) (3y 2)2 )) (x2 2)2 )) (y2 y)2 )) (2x2 5x)2
)) (x 3 )2 )) (x y )2 )) ( )2 )) ( )2
X :)) (3 + 1)2 )) (6 + 5)2 )) (2 3)2 )) (1 7)2
:)) (x ... .....)2 = ..... + ..... + 9 )) ( ..... ... 4)2 = y2
..... ... .....)) (..... .....)2 = 16x2 ... 8x ... ..... )) (.....
... 2)2 = ..... 4x2 ... .....
:)) (x + 1)3 )) (y + 4)3 )) (2 + 1)3 )) (3 + 2)3
)) (x2 + 3)3 )) (y2 + y)3 )) (x 2)3 )) (y 5)3
)) (3 1)3 )) (2x 3y)3 )) (y2 2)3 )) (2 2)3
:)) (x 1)(x + 1) )) (y 2)(y + 2) )) (3 )(3 + ))) (x + 4)(4 x) ))
(x y)(x y) )) (x + y)(x y))) (2x + 7y)(2x 7y) )) (x 2)(x +2) )) (x
+ y )(x y )
(x) = (x 3)2 + (3x + 1)2 10(x 1)(x + 1) .
)) ( )( + )(2 + 2)(4 + 4) = 8 8.)) : 9 11 101 10001.
, , .
)) )) )) )) 1223 + 6
53 + 2
67 3
15 1
99
88
77
66
55
44
33
2
32
22
4
12
11
49
1.5 AA
042-052 3-11-06 00:58 49
-
:)) (x 3)(x2 + 3x + 9) )) (y + 2)(y2 2y + 4))) (2 + 1)(42 2 + 1)
)) (1 )(1 + + 2)
:)) (x 4)2 + (2x + 5)2 )) (x2 1)2 (x2 3)(x2 + 3))) (x + y)2 (x
2y)(x + 2y) + (2x y)2 )) (3x 4)2 + (3x + 4)2 2(3x 4)(3x + 4))) (2 +
1)3 + (2 1)3 )) ( + 2)3 ( + 2)(2 2 + 4))) (2 + )3 (2 )3 )) (4 1)3
(8 + 1)(8 1)
:)) (x 2y)2 (2x y)2 + 3x2 = 3y2
)) ( 3)2 + (3 + 2)(3 2) (3 )2 = 2 + 42
)) (x 1)(x + 1)3 2x(x 1)(x + 1) = x4 1)) (2 + 2)2 (2)2 = (2
2)2
)) ( 4)2 + (2 3)2 = 2 + (2 5)2
)) (2x2 + 2x)2 + (2x + 1)2 = (2x2 + 2x + 1)2
A x = 3 + 5 y = 3 5, :)) xy )) x2 y2 )) x2 + y2 )) x3 + y3
)) ( + )2 ( )2 = 20)) x = (2005 + )2 (2005 )2
A , .
. . .
.
. 4
. ;
)) = .
)) , 10 cm, 2 cm.
2 + 2 ( )2
21177
1166
1155
1401
1401
5
51144
1133
1122
1111
1100
50
M - 1
11
55xx++22
33xx++22
44xx++11
042-052 3-11-06 00:58 50
-
, . ; .
+
11 (+)0= 1111 11 (+)1= 11 + 11
11 22 11 (+)2= 112 + 22 + 112
11 33 33 11 (+)3= 113 + 332 + 332 +113
11 44 66 44 11 (+)4= 114+ 443 + 6622 + 443 +114
11 11 (+)5= .... + .... + .... + .... + .... + ....
11 11(+)6= .... .... ... .... ... .... ... .... ... .... ...
.... ... .... .... ...
+ .
11.. , . Blaise Pascal (1623 - 1662) . 1. ;
22.. ( + )5 ( + )6, .
33.. ( )6, , , +.
.. ( )2 = 2 2 + 2, ( )3 = 3 32 + 32 3
44.. ;
1188
51
1.5 AA
++
042-052 3-11-06 00:58 51
-
A , , , ,
2 = 2 + 2 (1) ; , , , (1), . 5, 4, 3 52 = 42 + 32., , ;
(6 ..)
, , , ( = 3, 5, 7, ...)
.) ;) .
(5 4 ..)
+ 1, 1, , ( = 4, 6, 8, ...)
.) ;)
.O (3 ..) , ( ). 2 + 2, 2 2, 2, , , .) ;)
.
((,,
......)) (( ,,
......))..
:
2
42
4
2 12
2 + 12
52
M - 1
E
042-052 3-11-06 00:59 52
-
53
1. :
)) 7,32 25 + 7,32 75 )) 347 347
2. R = 32,50 m = 7,50 m. .
, , , , .
, , , .
, R2 2 (R2 2) (R + )(R ) = R2 2, :
R2 2 = (R2 2) = (R + )(R )
(R + )(R ) .
, , .
, .
16
76
11..66
RR==3322,,5500 mm
==77,,5500 mm
053-062 3-11-06 01:05 53
-
)) , .
, 3 + 3 3 3, :
3 + 3 3 = 3( + ).
22 2 + 2, 22 22 + 22 1, 22.
, :
22 2 + 2 = 2( + 1).
, 2.
NN ::)) 1122xx22yy 3300xxyy22 ++ 66xx22yy22 )) (( xx)) ++ 33((xx
)) )) 33((22xx 11)) ++ xx((44xx 22))
)) 12x2y 30xy2 + 6x2y2 = 6xy, : = 6xy(2x 5y + xy)
)) , ( x) 3(x ). ( x) + 3(x ) = x, = ( x) 3( x) = 3( x), : = (
x)( 3)
)) A 3(2x 1) + x(4x 2) = 2, = 3(2x 1) + 2x(2x 1)= 2x 1, : = (2x
1)(3 + 2x)
)) (()) x + y + 2x + 2y, . ,
54
M - 1
=
3( + )3 + 3 3
:
((2222)) :: ((22)) == ==
((22)) :: ((22)) == ==
((22)) :: ((22)) == == 112222
2222
222222
053-062 3-11-06 01:05 54
-
2, x + y. , :
x + y + 2x + 2y = (x + y) + 2(x + y) = (x + y) ( + 2)
T . . , :
x + y + 2x + 2y = ( + 2) + y( + 2) = ( + 2) (x + y)
NN ::)) 33xx33 1122xx22 ++ 55xx 2200 )) 33 33 ++ 99 )) 33xx22 ++
55xxyy ++ 22yy22
)) 3x3 12x2 + 5x 20 = 3x2(x 4) + 5(x 4) = (x 4)(3x2 + 5)
)) 3 3 + 9 = ( 3) 3( 3) = ( 3)( 3)
)) 3x2 + 5xy + 2y2 = 3x2 + 3xy + 2xy + 2y2 == 3x(x + y) + 2y(x +
y) == (x + y)(3x + 2y).
)) ( + )( ) = 2 2, :
, , .. 2 9 = 2 32 = ( + 3) ( 3).
NN :: )) 4422 2255 )) ((33xx 11))22 8811 )) 22 77..
)) 42 25 = (2)2 52 = (2 + 5)(2 5)
)) (3x 1)2 81 = (3x 1)2 92 == (3x 1 + 9)(3x 1 9) == (3x + 8)(3x
10)
)) 2 7 = 2 (7 )2 = ( 7 )( + 7 )
2 2 = ( + )( )
55
1.6
M - ,
.. 5xy = 3xy + 2xy
.
053-062 3-11-06 01:05 55
-
)) ( )(2 + + 2) = 3 3 ( + )(2 + 2) = 3 + 3 :
, , ..
x3 64 = x 3 43 = (x 4) (x 2 + x 4 + 42) = (x 4) (x 2 + 4x +
16)
y 3 + 27 = y 3 + 33 = (y + 3) (y 2 y 3 + 32) = (y + 3) (y 2 3y +
9)
NN :: )) xx33 2277 )) xx33 ++ 6644 )) 8833 33
)) x3 27 = x3 33 = (x 3)(x2 + x 3 + 32) == (x 3)(x2 + 3x +
9)
)) x3 + 64= x3 + 43 = (x + 4)(x2 x 4 + 42) == (x + 4)(x2 4x +
16)
)) 83 3 = (2)3 3 = (2 )(2)2 + 2 + 2 = (2 )(42 + 2 + 2)
)) ( + )2 = 2 + 2 + 2 ( )2 = 2 2 + 2 :
, (( )), ..
x2 + 4x + 4 = x2 + 2 x 2 + 22 = (x + 2)2
y 2 6y + 9 = y2 2 y 3 + 32 = (y 3)2
NN ::)) 4422 ++ 1122 ++ 99 )) 22 1100 ++ 225522 )) 44yy22 ++
44yy 11
)) 42 + 12 + 9 = (2)2 + 2 2 3 + 32 = (2 + 3)2
2 2 + 2 = ( )22 + 2 + 2 = ( + )2
3 + 3 = ( + )(2 + 2)3 3 = ( )(2 + + 2)
56
M - 1
.
O (x + 2)2 (y 3)2
, ((xx ++ 22))22 == ((xx ++ 22))((xx ++ 22)) ((yy 33))22 == ((
yy 33))(( yy 33))
053-062 3-11-06 01:05 56
-
)) 2 10 + 252 = 2 2 5 + (5)2 == ( 5)2
)) 4y2 + 4y 1= (4y2 4y + 1) == (2y)2 2 2y 1 + 12 == (2y 1)2
)) xx22 ++ (( ++ ))xx ++ (x + )(x + ) x2 + ( + )x + ,
(x + )(x + ) = x2 + x + x + = x2+ ( + )x + ., x2 + ( + )x +
, x2 + 8x + 12 12 ( ) 8 ( x).Y 12 (.. 1 12, 2 6, 3 4 ...)., 2 6
8. :
2 + 8 + 12 = 2 + (6 + 2) + 6 2 = ( + 6)( + 2)
NN ::)) xx22 88xx ++ 1122 )) xx22 ++ 55xx 66 )) 33yy22 ++ 1122yy
99
)) x2 8x+ 12, 12 8. , . 2 6. x2 8x + 12 = (x 2)(x 6)
)) x2 + 5x 6, , 6 5. 6 1, x2 + 5x 6 = (x + 6)(x 1).
)) 3y2 + 12y 9, - 3, y2 1,
3y2 + 12y 9 = 3(y2 4y + 3) y2 4y + 3, 3 4. 3 1, 3y2 + 12y 9 =
3(y2 4y + 3) = 3(y 3)(y 1).
x2 + ( + )x + = (x + )(x + )
57
1.6
22 ++ 22 ++ 22 22 22 ++ 22
((22))++(( 66))
xx22 88xx ++ 1122 == ((xx 22))((xx 66))
((22))(( 66))
053-062 3-11-06 01:05 57
-
)) 33xx22 1188xx..)) 33xx22 == 1188xx..
)) H 3x2 18x : 3x2 18x = 3x(x 6).
)) 3x2 = 18x 3x2 18x = 0 3x(x 6) = 0. 3x(x 6) , 3x = 0 x 6 = 0,
x = 0 x = 6.
AA ,, .. ..
, ,, = x 1 + y 1 + xy + 1 1 = x + y + xy + 1., x + y + xy + 1=(x
+ xy) + (y + 1) =
= x(1 + y) + (1 + y) = (1 + y)(x + 1)., 1 + y 1 + x.
::)) 778866 4455 ++ 778866 5555 )) 2200005522 1199995522 ))
556655 449999 ++ 556655 6666 44335522..
)) 786 45 + 786 55 = 786(45 + 55) = 786 100 = 78600)) 20052
19952 = (2005 1995)(2005 + 1995) = 10 4000 = 40000)) 565 499 + 565
66 4352 = 565(499 + 66) 4352 = 5652 4352 =
(565 435)(565 + 435) = 130 1000 = 130000
::)) 33xx22 yy 1122yy33 )) 55xx22 yy ++ 1100xx22 ++ 55xxyy ++
1100xx )) xx44 1166yy44
)) 116633 5544 )) xx22 44xx ++ 44 yy22 )) 33xx33 ++ 1122xx22
1155xx
)) 3x2y 12y 3 = 3y(x2 4y2) = 3y x2 (2y)2 = 3y(x 2y)(x + 2y)))
5x2y + 10x2 + 5xy + 10x = 5x(xy + 2x + y + 2) = 5xy(x + 1) + 2(x +
1) =
= 5x(x + 1)(y + 2)
44
33
22
11
58
M - 1
xx
xx
yy
11
11
yy
11
11
xx
xx
yy
11
11
11
11
yy
053-062 3-11-06 01:05 58
-
)) x4 16y4 = (x2)2 (4y2)2 = (x2 + 4y2)(x2 4y2) = (x2 + 4y2) x2
(2y)2 == (x2 + 4y2)(x 2y)(x + 2y)
)) 163 54 = 2(83 27) = 2(2)3 33 = 2(2 3)(42 + 6 + 9)
)) x2 4x + 4 y2 = (x2 2 x 2 + 22) y2 = (x 2)2 y2 = (x 2 + y)(x 2
y).
)) 3x3 + 12x2 15x = 3x(x2 + 4x 5)
To x2 + 4x 5 , 5 4, 5 1., 3x2 + 12x2 15x = 3x(x2 + 4x 5) = 3x(x
+ 5)(x 1).
;)) 2(x y)(x + y) )) 2 + (x y)(x + y) )) 4( )2 )) 4 + ( )2
)) (x + 2y)x y )) (x + 2y)(x y) )) ( + )( + 3))) ( + )( + 3) +
1.
.)) 8x + 16 = 8(.............) )) 3y y2 = y(.............))) 6x2
+ 12x = .............(x + 2) )) 4x2 + 8x = 4x(.............))) 2x +
2 = 2(.............) )) (x 1)2 (x 1) = (x 1)(.............)
. 3x3 + 3x2 + x + 1 :)) 3x2(x + 1) )) (x + 3)(3x2 1) )) (x +
1)(3x2 + 1) )) x(3x2 + x + 1).
(), (), .)) x2 22 = (x 2)(x + 2) )) x2 9 = (x 9)(x + 9))) 1122
122 = 100 124 )) 4y2 1 = (4y 1)(4y + 1))) 4x2 2 = (2x )(2x + ) )) 2
( 1)2 = ( + 1)( 1)
(x y)(x + y), ; .
.)) 3 23 = ( 2)(................................))) 3 + 33 = ( +
3)(................................))) (2x)3 1 = (2x
1)(................................))) 1 + (5y)3 = (1 +
5y)(...............................)
66
55
44
33
22
11
E
59
1.6
xx
xx
yy
yy
053-062 3-11-06 01:05 59
-
(), (), .)) x3 53 = (x 5)(x2 5x + 25))) 8 + 3 = (2 + )(22 2 +
2))) (3y)3 + 1 = (3y + 1)(3y2 3y + 1))) 1 (2)3 = (1 2)(1 + 2 +
42)
.)) x2 + 6x + 9 = (.............)2 )) 42 4 + 1 =
(.............)2
)) y4 2y2 + 1 = (.............)2 )) 25 + 10x3 + x6 =
(.............)2
2 + 2 + , > 0,
)) + 2 )) 2 + 1 )) + 1 )) ( + 1)
.
.)) x2 + ( + 2)x + 2 = (x + .....) (x + .....)
)) x2 + (2 + 3 )x + 6 = (x + .....) (x + .....)
N :)) 3 + 6 )) 2x 8 )) 82 + 6)) 9x2 6x )) 82 + 42 )) 2x2 2xy +
2x)) 2 + 2 )) 23 42 + 62 )) 2 xy 18 y + 8 y2
N :)) x( ) + y( ) )) (x + y) + (x + y) )) (3x 1)(x 2) (x + 4)(x
2))) 2( 2) 3(2 ) )) 4x(x 1) x + 1 )) 2x2(x 3) 6x(x 3)2
22
11
1111
xx22 ++ xx 22
xx22 xx 22
xx22 ++ 55xx ++ 66
xx22 ++ 55xx 66
xx22 33xx ++ 22
xx22 ++ 33xx ++ 22
((xx ++ )) ((xx ++ )) ++ xx22 ++ (( ++ ))xx ++
1100
99
88
77
60
M - 1
053-062 3-11-06 01:05 60
-
ii)) :)) x2 + x )) 2y2 5y )) ( 3) 2(3 ) )) (3 + 1) 4
iiii)) :)) x2 + x = 0 )) 2y2 = 5y )) ( 3) 2(3 ) = 0 )) (3 + 1) =
4
:)) x2 + xy + x + y )) x3 x2 + x 1 )) x3 5x2 + 4x 20)) 2x3 3x2 +
4x 6 )) 4x2 8x x + 2 )) 9 182 + 10 5)) 12x2 8xy 15x + 10y )) x3 + 2
x2 + x + 2 )) 6x2 + 22x 3x 2
:)) 72 + 10 + 32 )) 5x2 8xy + 3y2 )) 3x2 xy 2y2
)) 2 + 2 .)) , 2 + 2 = + ,
, .
:)) 22 2 + + x x )) 2 4 + 5 10 + 2 4
:)) x2 9 )) 16x2 1 )) 2 92
)) 22 4 )) 362 ( + 5)2 )) 4(x + 1)2 9(x 2)2
)) 16 )) x2 3 )) x2 2y2
:)) 2x2 32 )) 28 7y2 )) 2x3 2x )) 5x2 80 )) 2(x 1)2 8
, :)) = 53, = 28 )) = 0,37, = 0,12)) = 26, = 10
:)) x2 49 = 0 )) 9x3 4x = 0 )) x(x + 1)2 = 4x )) (x + 2)3 = x +
2
:)) x3 27 )) y3 + 8 )) 3 + 64 )) 8x3 1 )) 27y3 + 1
:)) 3x3 24 )) 164 + 2 )) R3 3 )) 4 + 4
:)) x3 ..... = (x 3)(..... + ..... + 9) )) ..... + y3 = (2x +
y)(4x2 ..... + .....))) 3 ..... = ( 2)(..... + ..... + 42) )) 3 +
..... = ( + 5)(..... ..... + 252)
1144
43
43
1133
1122
1111
1100
99
1x2
88
77
66
55
44
33
61
1.6
053-062 3-11-06 01:05 61
-
:)) x2 2x + 1 )) y2 + 4y + 4 )) 2 6 + 9 )) 2 + 10 + 25)) 1 4 +
42 )) 9x4 + 6x2 + 1 )) 4y2 12y + 9 )) 16x2 + 8xy + y2
)) 252 10 + 2 )) ( + )2 2( + ) + 1 )) 2y +9 )) x2 + x +
:)) 3x2 + 24x + 48 )) y2 + 4y 4 )) 22 8 + 82 )) 43 + 122 + 9
:))
.))
.
, .
:)) x2 + 3x + 2 )) y2 4y + 3 )) 2 + 5 + 6 )) 2 + 6 + 5)) x2 7x +
12 )) y2 y 12 )) 2 9 + 18 )) 2 + 3 10
:)) x2 + (2 + 3)x + 23 )) x2 + (2 + 3)x + 6 )) x2 + (3 2)x
32
:)) 22 + 10 + 8 )) 32 12 15 )) x2 7x + 6
- .)) 1453 1821 1453 821 )) 8012 + 199 801 )) 9982 4)) 999 1001
+ 1 )) 9992 + 2 999 + 1 )) 972 + 6 97 + 9
:)) x2y2 4y2 x2 + 4 )) x4 1 + x3 x )) x3(x2 1) + 1 x2
)) (x2 + 9)2 36x2 )) 2 2 + 2 + )) x2 2xy + y2 2
)) 1 2 + 2 2 )) y2 x2 10y + 25 )) 2(x1)(x24)5(x1)(x2)2
)) (y2 4)2 (y + 2)2 )) (2 + 2 2)2422 )) (x2+9)(2+4) (x+6)2
E x, y, . xy x 2y + 2, .
2244
2233
2222
2211
2200
1199
1188
1177
1166
14
y29
1155
62
M - 1
xx
xx
22xx yy
yy
BB
11
xx ++ 22
xxxx
xx
yy
053-062 3-11-06 01:05 62
-
63
(x) (x).
(x) (x).
1. A 325 19 , ; .
2. (x) (x) = x2 x, (x) = 2x2 3x 1 (x) = 7x 4.
, () () 0 :: , () (), :
= 0, == . .
, = 325 = 19, 325 :: 19, = 17 = 2,
325 = 19 17 + 2 2 < 19
, ((xx)) () ((xx)) () (x)0 ((xx)) :: ((xx)), ((xx)) () ((xx))
(), :
((TT ))
(x) (x).
, (x) = 2x2 5x3 + 2x4 4 + 8x (x) = x2 x.
(x) = (x)(x) + (x)
= + <
11..77
325 1919 17
135133
2
063-067 3-11-06 01:08 63
-
, :2x4 5x3 + 2x2 + 8x 4 = (x2 x) (2x2 3x 1) + (7x 4)
() == () () ++ () .
64
M - 1
x.
2x4 x2 ( = 2x2)To 2x2 .
2x4x2
2x4 5x3 + 2x2 + 8x 4 x2 x
2x 4 5x3 + 2x2 + 8x 4 x2 x22
2x2, , x2 x 2x2(x2 x) = 2x4 2x3 . , -
1 = 3x3 + 2x2 + 8x 4.
3x3
1 x2
( = 3x). 3x .
3x3x2
3x, - , x2 x 3x(x2 x) = 3x3 + 3x2 - 1 2 = x2 + 8x 4.
(( )) x2 x (( )), .
2x4 5x3 + 2x2 + 8x 4 x2 x2x4 + 2x3 22
3x3 + 2x2 + 8x 4
2x 4 5x3 + 2x2 + 8x 4 x2 x2x4 + 2x3 22 3
3x3 + 2x2 + 8x 4
2x 4 5x3 + 2x2 + 8x 4 x2 x2x4 + 2x3 22 3
3x3 +2x2 + 8x 43x3 3x2
x2 + 8x 4
2x4 5x3 + 2x2 + 8x 4 x2 x2x4 + 2x3 2231
3x3 + 2x2 + 8x 43x3 3x2
x2 + 8x 42
7 4
063-067 7-11-06 21:32 64
-
(8x4 + 8x3 + 17x 5) : (2x2 + 3x 1), :
, , :
8x4 + 8x3 + 17x 5 = (2x2 + 3x 1) (4x2 2x + 5)() == () ()
T = 2x2 + 3x 1 = 4x2 2x + 5 = 8x4 + 8x3 + 17x 5.
, :: , , == .
)) ((44xx44 ++ 33xx22 11)) :: ((22xx 11))..)) 44xx44 ++ 33xx22
11 ..
))
, :4x4+3x2 1=(2x1)(2x3+x2+2x+1). :
2x3 + x2 + 2x + 1 == x2(2x + 1) + (2x + 1) == (2x + 1)(x2 +
1).E, 4x4 + 3x2 1 :4x4 + 3x2 1 = (2x 1)(2x + 1)(x2 + 1).
11
65
1.7
2 ,, - , - .
8x4 + 8x3 + 17x 5 2x2+3x18x4123 + 42 422+5
4x3 + 4x2 + 17x 5+ 4x3 + 6x2 2
102 + 15x 5102 15x + 5
0
)) 4x4 + 3x2 1 2x 14x4 + 2x3 2x3+x2+2x+1
2x3 + 3x2 12x3 + x2
4x2 1 4x2 + 2x
2x 12x+ 1
0
063-067 3-11-06 01:08 65
-
== 33xx ++ 22 == 33xx33 44xx22 22xx ++ 2233..
, : . : .
:
3x3 4x2 2x + 23 == (3x + 2)(x2 2x + 2), 3x + 2
3x3 4x2 2x + 23
:ii)) 4x + 7 :
)) 1 )) 2 )) 3 )) .iiii)) To x2 4x + 9
:)) 5 )) 3x 2 )) x2 + 3 )) 4x.
iiiiii)) (x) 2x2 + x + 5 x4 + x 2, (x) :)) 4 )) 6 )) 8 )) .
:
(), (), :)) (2x + 1)(x + 3) 2x + 1 x + 3.)) x + 6 x2 + 2.)) 6 2
,
3 .)) x 4 x2 16.)) To (x3 + 1) :: (x + 1) x2 x + 1.
33
3627
38
22
11
E
22
66
M - 1
3x3 4x2 2x + 23 3x + 23x3 2x2
6x2 2x + 23 x2 2x + 2
6x2 + 42x+ 32x + 23
32x 23
0
063-067 3-11-06 01:08 66
-
N .)) (2x3 + x2 3x + 6) :: (x + 2) )) (6x3 x2 10x + 5) :: (3x +
1))) (6x4 x2 + 2x 7) :: (x 1) )) (4x3 + 5x 8) :: (2x 1))) (x5 x4 +
3x2 + 2) :: (x2 x + 2) )) (9x4 x2 + 2x 1) :: (3x2 x + 1))) (8x4 6x2
9) :: (2x2 3) )) (3x5 2x3 4) :: (3x2 1)
N , .)) 6x2 + ..... + ..... ..... + 2 )) ..... + ..... + 2x + 20
x + .....
6x2 ..... 2x + ..... ..... 6x2 2x2 + ..... .....18x + ..... 4x2
+ ..... + 20
18x ..... ..... .....0 10x + .....
..... + ..........
x2 x + 1 2x + 3 3x + 2;
N Q(x) (x), :)) (x) = 6x3 7x2 + 9x 18 Q(x) = 2x 3)) (x) = 2x4 x2
+ 5x 3 Q(x) = x2 + x 1.
)) (x4 2x3 8x2 + 18x 9) : (x2 9))) N x4 2x3 8x2 + 18x 9.
)) x + 1 x4 + 4x3 + 6x2 + 4x + 1.)) N x4 + 4x3 + 6x2 + 4x +
1.
3 + 3 , + . , ;
(x) = (x3 + 2)(x2 5) + 4x2 6x + 7. N )) (x) :: (x3 + 2) )) (x)
:: (x2 5)
(6x3 + ) :: (x 1) , .
2x3 x2 4x + 3 (x 1)2, .
, 45 ,56 16 . 5x + 4y, ;
1111
1100
99
88
77
66
55
44
33
22
11
67
1.7
xx
yy
yy
yy
xx
xx
063-067 3-11-06 01:08 67
-
68
: .
1. 12, 24, 300 ... ... .
2. ... ... 12x3y2, 24x2y3, 300x4y 3(x y)(x + y), 18(x y)2, 9(x
y).
... ... . , 12, 24 300, , :
12 = 22 3 24 = 23 3 300 = 22 3 52
,...(12, 24, 300) = 23 3 52 = 600 (
)....(12, 24, 300) = 22 3 = 12 (
).
, ... ... - . :
(......) , .
(......) , .
. , ..., ... ... ... .
... ... 11..88
068-070 7-11-06 21:35 68
-
, 12x3y2, 24x2y3, 300x4y
... = 600x4y3 ... = 12x2y 3(x y)(x + y), 18(x y)2, 9(x y)
... = 18( y)2(x + y) ... = 3( y)
NN ...... ...... 66xx33yy,, 99xx22yy22,, 33xxyy44..
O 6, 9, 3 ... = 18 ... = 3, ... = 18x3y42 ... = 3xy.
NN ...... ...... :: == 1122xx33 1122xx22,, BB == 1188xx22 3366xx
++ 1188 == 99xx22 99xx..
= 12x2 12 = 12(x2 1) = 12(x 1)(x + 1) = 18x2 36x + 18 = 18(x2 2x
+ 1) = 18(x 1)2
= 9x2 9x = 9x(x 1)
12, 18, 9 ... = 36 ... = 3. , , ... = 36x(x 1)2(x + 1) ... = 3(x
1).
- , ... .
11.. 6x2(x + 2)2
22.. x3(x + 2)3
33.. 6x2(x + 2)
44.. x4(x + 2)3
.. x4(x + 2)2, x(x + 2)3
.. x3(x + 2), x(x + 2)3
.. 6x2(x + 2), 2x(x + 2)2
B
11
E
22
11
69
1.8 ...... ......
.
... ... .
... ... .
068-070 3-11-06 01:12 69
-
, ... , .
- , ... .
... , .
... ... :)) 12x3y22, 18x2y3, 24x2y34
)) 15xy3, 10x22, 5y2
)) 2x2(x + y)2, 3xy3(x + y)2, 8x2y(x y)(x + y)
... ... :)) 6(x2 y2), 4(x y)2, 12(x y)3
) 2 3 + 2, 2 4, 3 4) 3 2, (2 )(2 1), 3 22 +
22
11
33xx33((xx 22))3322 xx33((xx 22))66 xx((xx 22))22
66((xx 22))33xx44((xx 22))2233xx 22BB
44
11.. 6x2(x + 1)2
22.. 3x(x + 1)2
33.. 3x2(x + 1)
44.. x2(x + 1)2
.. 6x3(x + 1)2, 3x(x + 1)3
.. 2x2(x + 1)3, 3x4(x + 1)2
.. 3x2(x + 1), 6x3(x + 1)2
B
33
88xx 55xx22((xx 11))
66xx 2299((xx 11))2222xx((xx 11))44xx 33BB
22
70
M - 1
068-070 3-11-06 01:12 70
-
71
.
.
1. x = 0;
M x = 1;
2. ;, ,
3. ;, ,
M (.. , , ) , . , .
, , x 1.
, , .
, , , .
, , ,
3x. A
, = = = 2y
H , ,
= = 2y3 2y
36y3x
2y1
6y : 3x3x : 3x
6y3x
6xy3x
6x + y3x
x3 + 4x 1
2x2 + 4
xyx + y
x3 + 4x 1
6xy3x
6xy3 + x
6x + y3x
6 2 73 2
6 2 73 + 2
6 2 + 73 2
x3 + 4x 1
11..99
071-074 3-11-06 01:18 71
-
A , :
..
, :
= = =
;;
)) )) ))
)) H , x
, x 0.
)) O , x + 2 0, x 2.
)) , x, y ,
x y 0, x y.
NN ::
)) )) ))
)) ,
=
))
= = =
)) = = x yx2 + xy + y2
(x y)2
(x y)(x2 + xy + y2)x2 2xy + y2
x3 y3
x + 12x
3(x 1)(x + 1)6x(x 1)
3(x2 1)6x(x 1)
3x2 36x2 6x
3x22
2y212x3y2
8xy3
12x3y2
8xy3
xx22 22xxyy ++ yy22
xx33 yy 3333xx22 3366xx22 66xx
1122xx33yy 22
88xxyy 33
22
x2 + y2
x y
x2 + 6x + 2
x2 + 7x + 2x
xx22 ++ yy22
xx yyxx22 ++ 66xx ++ 22
xx22 ++ 77xx ++ 22xx
11
5x + 2
5(x 2)(x 2)(x + 2)
5(x 2)x2 22
5x 10x2 4
5x 10x2 4
72
M - 1
071-074 7-11-06 21:36 72
-
, .
(), (), .
)) = x + 1 )) = x + 1
)) = )) =
)) = x + y )) = x + y
:
)) = )) = 1 )) = x
)) = x + 1 )) = )) =
x,
, =
x 4. ;
1x 4
xx(x 4)
xx (x 4)
44
3x + 2
3(x + 2)
............1
+ ............2(+)2
x(x + 1)
............
x(x + 1)
............
( + )(............)( )(............)
7x 2
7xx(............)
33
(x y)2/x y
x2 y2
x y
x + 24
x + 2(x + 1)4(x + 1)
x + 24
(x + 2)(x + 1)4(x + 1)
x (x + 1)x
x2/ + 1x
22
11.. x 1
22.. x 0 x 1
33.. x 1
44.. x 1 x 1
55..
66.. x 0
.. 1x
.. x 1x + 1
.. xx2 1
.. 2(x 1)x 1
.. 3x2 + 1
B
11
E
73
1.9
071-074 8-11-06 15:41 73
-
:
)) )) )) ))
:
)) )) )) ))
)) )) )) ))
:
)) )) )) ))
)) )) )) ))
:
)) )) ))
:
)) ))
)) ))
5 m/sec. 4 m/sec2. t sec
t + m/sec52
66
( + 1)( 2)2 4( + 1)3 + 2
(2 + 1)2 ( + 2)2
4 1
y(y 3) + y2 94y2 9
x(x 1) + 4(x 1)x2 + 2x 3
55
3 22 + 3
y2 5y + 4y2 6y + 8
x2 + 3x + 2x2 + 4x + 4
44
2 + + 2
3 36x2 + 3x4x2 2
y2 1y2 + 2y + 1
x2 16x2 4x
52 20( 2)2
x2 + x2 + x
3y 9y2 3y
6x2x2 + 4x
33
( )( )( )( )
2(2 )2
y 11 y
x + 44 + x
523
1022x2
8x23y2
12y4x6x
22
6x + 1x(x 3)
2( + 1)2
y + 32y 5
1x 4
11
74
M - 1
071-074 3-11-06 01:18 74
-
75
.
.
1. : 4 , , ::
2. :
2x , , ::
- , .
. , 3x = =
= = =
, .
. ,
:: = = = = x 12xx(x 1)(x + 1)
2x2(x + 1)x(x2 1)
(x + 1) 2x2x2 1
2x2x
x + 12x2
x2 1x
x + 1
: = =
2x3
2x(x 1)(x + 1)3(x + 1)(x 1)
(x2 1) 2x(3x + 3)(x 1)
2xx 1
x2 13x + 3
15x32y
3x 5x22y
5x22y
=
=
3x25y + 5
9xy + 1
229xy
3x22
3xy5
47
25
34
25
35
A
11..1100
075-084 3-11-06 01:26 75
-
, , ::
.
, = =
NN :: )) ((55xx22 ++ 55xx)) ))
)) (5x2 + 5x) = = =
)) = = = =
NN ::
)) :: ))
)) :: = = = =
= =
)) = = = x2
( x)( + x)22/ ( + x)
(2 x2)2 (2 + 2x)
2 x2
22 + 2x
x2 + xx2 + x + 2
(x + )xx2 + x + 2
(x )(x + )x2/
x(x )(x2 + x + 2)(x2 2)x2
x(x3 3)x2
x3 3x2 2
xx3 3
x2x2 2
x
22 xx22
2222 ++ 22xx
xx33 33
xx22xx22 22
xx
22
x2 x3
(x 1)x3
(x 1)2/ x(x + 2)3(x + 2)(x 1)
(x22x+1)(x2+2x)(3x + 6)(x 1)
x2+2xx 1
x2 2x + 13x + 6
15x2
25x(x + 1)3x
2(x + 1)(5x2 + 5x)3x
2x + 23x
2x + 2
xx22 ++ 22xxxx 11
xx22 22xx ++ 1133xx ++ 66
33xx22xx ++ 22
11
x2
22x24x
22x4 x2
=
76
M - 1
=
075-084 3-11-06 01:26 76
-
(), (), .
)) x = )) x =
)) 3x :: = )) 3x :: =
)) = )) =
)) = 0 )) :: = 1
:
)) 3x = )) = )) :: =
)) = 1 )) :: = 1 )) :: =
:
)) )) )) 12x2
)) )) (52) )) ( ) ( ) :
)) 8x :: )) :: ( ) )) ( ) :: 32 )) ( ) :: ( ) :
)) )) ))
)) )) ))
:
)) :: )) :: )) ( ) :: ( + 2)
)) :: )) :: )) ::
:
)) ( ) :: )) :: ( ) )) ( :: ) x + 2x + 32x + 6x 1x + 2x 1
x + 2x + 3
2x + 6x 1
x + 2x 1
8x 8x + 2
4x + 4x + 2
x 2x + 1
55
x 2x2 2x+4
x2 4x3+8
x2 + xyx y
x + yx2 xy
( + 1)2
+12
+ 2
1 2y1 + y
2y 1y + 1
x + 415
x + 45
44
y2 + 3y2y2 + 3y
4y2 94y2 12y+9
x2 + 5x+6x2 + 3x
x2 + xx2 4
+ 32+2
2 42+ 6
x32x2 2
x x23
2 + y5 y
y 5y + 2
4xx + 3
2x + 6x2
33
x2
42x3
22
33y
1y2
6x
22
42
32
310
642
2332
19x
13x
9x4y
xy
1x2
11
xx + 2
x + 2..........
xy
..........
..........x + 2x 1
..........
..........x + 2x 1
y
..........
4xy
1y2
1..........
xy
6x2y
..........y
22
+ 2
+ 2
2 + 1
2 + 1
x 2x2
x 2x
x
5y
5x 1
x 1y
3x22
2x
32
2x
xy
1y
xxy
1y
11
E
77
1.10
075-084 3-11-06 01:26 77
-
1. : + , + .
2. :
+ , + .
,
. ,
+ = =
= = = = 4
A , ... , . , + :
3x 2 3x = 3x(x 1) 3x 3 = 3(x 1)
... = 6x(x 1)
+ = + =
= = = = 16x
x16x(x1)
4+5x 5 4x6x(x1)
4+5(x1) 4x6x(x1)
23(x1)
56x
23x(x1)
23x 3
56x
23x23x
23x 3
56x
23x2 3x
4(x 2)x 2
4x 8x 2
3x + 2x 1 7 xx 2
3x + (2x 1) (7 + x)x 2
7 + xx 2
2x 1x 2
3xx 2
=
+ + =
13
16
32
7 + xx 2
2x 1x 2
3xx 2
13
16
32
119
199
79
BB
78
M - 1
.
... .
M .
E .
2 x1 2x
075-084 3-11-06 01:26 78
-
NN :: )) ))
)) , :
x2 1 = (x 1)(x + 1) 2x 2 = 2(x 1)T ... 2(x 1)(x + 1).
2 x +1
= = =
= = = =
)) , :x2 2 = (x )(x + ), x2 x = x(x ), x2 + x = x(x + ) ... x(x
)(x + ).
x x + x
= =
= = = =
5500..000000 .. ,, mm22 mm22 mm22
.. (( mm))..
A 50000 , 100000 . x(x 1) m2, x(x + 1) m2 (x2 1) m2.
m2 ,
. :
+ = = =
, m2 m2 m2 .
100000x2 1
100000xx(x 1)(x+1)
50000(x +1) + 50000(x 1)x(x 1)(x +1)
50000x(x +1)
50000x(x 1)
100000x2 1
50000x(x +1)
50000x(x 1)
22
1x(x + )
x x(x )(x + )
4x 2x 2 x + x(x )(x + )
4x 2(x + ) (x )x(x )(x + )
1x(x + )
2x(x )
4(x )(x + )
1x2 + x
2x2 x
4x2 2
32(x +1)
3(x 1)2(x 1)(x +1)
3x 32(x 1)(x +1)
18x + 12 15x 152(x 1)(x +1)
2(9x + 6) 15(x +1)2(x 1)(x +1)
152(x 1)
9x + 6(x 1)(x +1)
152x 2
9x + 6x2 1
11xx22 ++ xx
22xx22 xx
44xx22 22
115522xx 22
99xx ++ 66xx22 11
11
79
1.10
BB
xxxx
xx++11
xx11
1111
xx
xx
AA
x+1 x1
075-084 7-11-06 21:38 79
-
(), (), .
)) + = 1 )) + =
)) = 1 )) + = 0
)) 1 + = )) =
. ;
)) + = = = 1
)) = = = 1
N :
)) .......... = 0 )) +.......... = 1 )) ..........+ =
)) .......... = )) +.......... = 2 )) .......... = 3
:
)) + )) )) ))
:
)) )) ))
)) + )) + ))
:
)) )) )) ))
1 1
1 + 1+
1 1 3
1y 2 + y1 y y
1x x1 1+ x
33
3 + 1
+ 72 + 4 + 3
32 x
9xx2 x
x36 x2
12x + 12
42 4
3 + 62 4
4y + 2
y 6y2 + 2y
3x 3
2x2x 6
22
22 + 1
12
1y
1y2
2x
3x + 1
1y
1x
11
3x + 8x
2x 1x
1x + 2
5x + 2
2xx + 1
xx + 1
xx + 6
xx + 6
33
x + 1x + 1
3x + 2 2x 1x + 1
2x 1x + 1
3x + 2x + 1
22
2x
+ 2x
x
1 + x
x
+
+
4
+ 4
2x + y
1y
1x
1x + 1
xx + 1
11
E
80
M - 1
075-084 3-11-06 01:26 80
-
:
)) + )) +
)) + )) +
:
)) ( )(1 + ) )) + ::
)) ( 1 )( + ) )) ( + 1) :: ( + )
)) + xy = (x + y)2.
)) + 56 44.
)) = = , 2 + 2 = 1.
)) 1, , .
= 3 (1 + )2 + 4( + )1 + ( 2004)20040, = = 3.(( ......
22000022))..
, :)) ( + 3)2+1 + ( 3)2+1 = 0 )) (x y )2 (y + x)2 = 0
= , :
= B =
(x) = 2x2 + 2x + 800.)) (1 x) = P(x).)) (100) (99).
44
2x3 2xy2 + 3y3
x2y + y34x2 6xy + y2
x2 + y2
12
xy33
22
32
12
11
11
999910001
20010001
x2 1x2 + 1
2xx2 + 1
77
563 443
12
x3 y3
x y66
2
2
+
22 + 2
x2 3(x 1)2
x 3(x 1)2
x + 3x2 1
14x 3
x2x 1
x + 32x + 1
55
2xy2
x2 y2y2
x + yx2
x y3
y 32
y 2y2 6
y2 5y + 6
2x + 16yx2 4y2
2x 2y
3x + 2y
8x2 2x
4x 2
x 2x
44
81
1.10
075-084 3-11-06 01:26 81
-
)) 3 + 3 + 3 3 = ( + + )(2 + 2 + 2 ).(( uulleerr))..
)) + + = 0, 3 + 3 + 3 = 3.)) (x y)3 + (y )3 + ( x)3.
A + = = , :
)) 2 + 2 = )) (3 + 1)2 + (3 + 1)2 + 9( + ) = 40
A x, y x, y .)) x4 2y2 = x2(y2 2) )) x3 + y3 = x2y + xy2
)) x2 + 4x + 3, x2 + 2x 3.
)) = + +
= x(x + 3) = (x + 1)(x + 2).)) = + 2 + 1 = ( + 1)2.)) x(x + 1)(x
+ 2)(x + 3) + 1.
)) 16x4 + 8x2 + . .))
4x 4x2 1.
)) A , 2 + .)) ,
6, 1.)) -
8.
)) (x6 1) :: (x 1) 76 1 6.
)) (x5 + 1) :: (x + 1) 215 + 1 9.
)) = .
)) x 2, 3, 4,
..., 2008 + + + ... + =20072008
12007 2008
13 4
12 3
11 2
1x
1x 1
1(x 1)x1133
1122
1111
1100
99
1x2 + 2x 3
1x2 1
1x2 + 4x + 3
88
77
439
73
1366
55
82
M - 1
075-084 3-11-06 01:26 82
-
.. 2x2 3y + 4
, .
.
.. 3x2y (3 , x 2y ). , .. 3x2y , 72y, x2y
, . , - .
.. 2x2y + 32y 2y = 42yAA . .. 6x2 + 2 42 + 3 = 22 + 5
. - , - . .. (3x 2y) (2y3) = 6x3y4
- .
.. (12x4y) : (32y) = 12x4y = = 4x2
, . .. 3x2y 5y + 2 ( 3x2y,, 5y,, 2
).
-- - .
))
.))
.
(x) (x) (x) 0 (x) : (x), (x) (x) : (x) = ()() + () ( ) (x) (x).
A ((xx)) == 00, (x) (x) (x).
12x4y
32y
1
32y
.
EE 11oo KK
83
1.10
075-084 7-11-06 21:40 83
-
- .. :
. :
, ..
..
. , . .
2x2 5 + 4
.
x2 + ( + )x + = (x + )(x + ) x2 + ( + )x +
2 + 2 + 2 = ( + )2
2 2 + 2 = ( )2
3 + 3 = ( + )(2 + 2)3 3 = ( ) (2 + + 2)
2 2 = ( + )( )
x + y + x + y == (x + y) + (x + y) = ( + )(x + y)
x + x = x( + )
.
( + )(2 + 2) = 3 + 3
( ) (2 + + 2) = 3 3
( + )( ) = 2 2
( )3 = 3 32 + 32 3
( + )3 = 3 + 32 + 32+ 3
( )2 = 2 2 + 2
( + )2 = 2 + 2 + 2
.
84
M - 1
075-084 3-11-06 01:26 84
-
E
2.1 x + = 0
2.2 2
2.3 2
2.4
2.5
2
2o
(085-088) 3-11-06 01:45 85
-
86
.
.
, . 1 km 1 km . , :
)) 15 km;)) ;)) ;
, 3x = 12, 4y + 11 = 0,... 1. 11 (( )).
3x = 12, - , x = 4. O 4, 3x = 12, .
, 0x = 3 0x = 0, .
0x = 3 x, 0x 3. , , .
, 0x = 0 x .
:
, 0x = 0 x .
:
00, x + = 0 x = == 00, x + = 0 0x =
00, (), == 00, ( ).
x + = 022..11
BB
(085-088) 3-11-06 01:45 86
-
NN == xx ++ 11
= x + 1
6 6 = 6 x + 6 1
3(x 1) (2x + 1) = 6x + 63x 3 2x 1 = 6x + 6
3x 2x 6x = 6 + 3 + 15x = 10
=
x = 2 x = 2
NN ::)) 33((xx ++ 22)) 33 == 33xx ++ 55 )) 22((xx 11)) xx == xx
22
)) 3(x + 2) 3 = 3x + 5 )) 2(x 1) x = x 2
3x + 6 3 = 3x + 5 2x 2 x = x 23x 3x = 5 6 + 3 2x x x = 2 2
0x = 2 0x = 0H x, .. x, ..
.
11..
22..
33..
.. 3x = 7
.. 0x = 0
.. 0x = 5
.. 5x = 0
11
E
22
105
5x5
2x + 16
x 12
2x + 16
x 12
22xx ++ 1166
xx 1122
11
87
2.1 xx ++ == 00
... .
.
.
.
.
.
(085-088) 3-11-06 01:45 87
-
(), (), .
)) x= 2 x = 6.
)) 4x = 0 .)) 0x = 0 .)) 0x = 6 x = 6.)) 5(x + 1) = 5x + 5 .
:)) 3(x + 2) 2(x 1) = 8 + x )) 4y 2(y 3) = 2y + 1)) 5( + 2) 4 =
6 5 )) (2x + 1)2 + 5 = 4(x2 10)
:
)) = x )) = 1
)) = )) 0,2(3x 4) 5(x 0,4) = 0,4(1 10x)
To 5 10. ;
: , ., , 65 C;
: . 10. 2 -
. 5,
. ;
16 km/h. , 12 km/h . 44 km, ;
66
55
44
33
56
+ 12
2( 1)3
3y10
y2
y + 55
13
x + 36
x 12
22
11
13
22
88
M - 2
(085-088) 3-11-06 01:45 88
-
89
.
.
.
- 9 m 1 m. .)) .
, , .
)) . , ., , , , 34 m2.
)) .
22..22
99 mm
11mm
(089-098) 7-11-06 16:16 89
-
2.
22 (( )).
2 x
, , . . O :
x2 9 = 0 : = 1 = 0 = 9x2 3x = 0 : = 1 = 3 = 0
x2 + 15x 16 = 0 : = 1 = 15 = 16
:
EE xx22 ++ xx == 00 00 x2 = 3x :
EE xx22 ++ == 00 00
x2 9 = 0, :
= 0 = 0 = 0
x2 + x ++ == 00 00
90
M - 2
x2 = 9,
2 3 = 0,
2 + 15 16 = 0
x2 = 3xx2 3x = 0
x(x 3) = 0
x = 0 x 3 = 0x = 0 x = 3
, x = 0 x = 3
x2 9 = 0x2 32 = 0
(x 3) (x + 3) = 0
x 3 = 0 x + 3 = 0x = 3 x = 3
, x = 3 x = 3
. . x(x 3)
x = 0 x 3 = 0.
11 :: -
. . (x 3)(x + 3)
x 3 = 0 x + 3 = 0
(089-098) 3-11-06 01:51 90
-
91
2.2 EE
x2 + 16 = 0, , x2 = 16. H (), 16.
x2 = 0 x = 0. H , x2=0 x x = 0, x = 0 x = 0 ( ).
EE xx22 ++ xx ++ == 00 00 9x2 6x + 1 = 0 :
x2 + 15x 16 = 0 :
x2 + 15x 16 = 0 .
, x2 = ()
x2 9 = 0x2 = 9
x = 9 x = 9x = 3 x = 3
22 :: , x2 =
, x = x =
2 2 + 2 = ( )2
(3x 1)2 = 0 3x 1 = 0
9x2 6x + 1 = 0(3x)2 2 3x 1 + 12 = 0
(3x 1)2 = 03x 1 = 0 x =
, x = 13
13
4, x2.
M 2 + 2 2 2.
- 2.
2 + 2 + 2 = ( + )2
2 2 + 2 = ( )2
x2 + 15x 16 = 04x2 + 60x 64 = 0
(2x)2 + 2 2x 15 = 64(2x)2 + 2 2x 15+ 152 = 64 + 152
(2x + 15)2 = 289
2x + 15 =289 2x + 15 = 2892x + 15 = 17 2x + 15 = 17
2x = 2 2x = 32x = 1 x = 16
, x = 1 x = 16
(089-098) 3-11-06 01:51 91
-
NN :: )) 22xx22 == 77xx )) 33xx22 7755 == 00 )) 22xx22 ++ 88 ==
00
)) 2x2 = 7x )) 3x2 75 = 0 )) 2x2 + 8 = 0
2x2 7x = 0 3x2 = 75 2x2 = 8x(2x 7) = 0 x2 = 25 x2 = 4
x = 0 2x 7 = 0 x = 25 x = 25
x = 0 x = x = 5 x = 5 ( )
NN xx22((22xx 11)) 66xx((22xx 11)) ++ 99((22xx 11)) == 00
x2(2x 1) 6x(2x 1) + 9(2x 1) = 0
(2x 1)(x2 6x + 9) = 0(2x 1)(x 3)2 = 0
2x 1 = 0 x 3 = 0
x = x = 3 ( )
(), (), .)) 0 x2 4x + 3 = 0.)) O 3 x2 4x + 3 = 0.)) O (x 2)(x +
1) = 0 x = 2 x = 1.)) x2 = 16 x = 4.)) H x2 = 9 .
)) (x 2)2 = 0 x = 2.
(), (), .)) H 5x 6 = x2 2 .)) x2 + 3x + 8 = x(x + 2) 2 .)) (
2)x2 + 5x + 3 = 0
i) 1o , = 2ii) 2 , 2.
x2 = 6x x x = 6. - x = 0. x = 0;
33
22
11
E
12
22
72
11
92
M - 2
B 2x 1.
O .
(089-098) 3-11-06 01:51 92
-
93
2.2 EE
:)) (x 4)(x + 1) = 0 )) y(y + 5) = 0 )) (3 )(2 + 1) = 0
)) 7x(x 7) = 0 )) 3y( 2) = 0 )) ( )(2 1) = 0 :)) x2 = 7x )) y2 =
9y )) 22 72 = 0
)) 2t2 18 = 0 )) 0,22 + 3,2 = 0 )) 0,5z = 0
:)) (2x 1)2 1 = 0 )) 3(x + 2)2 = 12 )) (x + 1)2 = 2x
)) = 27 )) (3x 1)2 4x2 = 0 )) (x + 3)2 3 = 0
:)) (3x + 1)2 = 5(3x + 1) )) 0,5(1 y)2 = 18 )) (22 + 1)(2 16) =
0
:)) x(x 4) = 4 )) y2 + y 12 = 0 )) 2 2 15 = 0)) 2t2 7t + 6 = 0
)) 32 + 1 = 4 )) 5z2 3z 8 = 0
:)) 25x2 + 10x + 1 = 0 )) y2(y 2) + 4y(y 2) + 4y 8 = 0)) 2 +
2006 2007 = 0
:)) x2 ( + )x + = 0 )) x2 ( 3 1)x 3 = 0
OO::11.. M x2 = 12x
x2 + 225 = 30x22.. x(x + 4) + 8(x + 4) = 033.. x2 10x + 9 =
044..
x2 = 25 H x2 = 32x
::11.. x2 20x + 100 = 022.. To x(x 15) = x 1533.. To (x 5)2 (x
5) = 044.. M x2 144 = 055.. x2(x 12) + 2007(x 12) = 0
88
77
66
55
44
(x 9)2
3
33
z26
22
12
y3
11
11 22 33 44 5511
22
33
44
(089-098) 3-11-06 01:51 93
-
x2 + 15x 16 = 0. ,x2 + x + = 0 0. :
2 4 , (2x + )2 = :
>> 00, : == 00, :2x + = (2x + )2 = 02x = 2x + = 0
x = 2x =
x =
, ,
x = x = x =
> 00, x =
== 00, x =
-
NN ::)) 22xx22 ++ 55xx ++ 33 == 00 )) 66xx22 55xx ++ 22 == 00 ))
1166xx22 ++ 88xx 11 == 00
)) 2x2 + 5x + 3 = 0 = 2, = 5, = 3,
= 2 4 = 52 4 2 3 = 25 24 = 1 > 0.
, x = = = ,
x = = 1 x = =
)) 6x2 5x + 2 = 0 = 6, = 5, = 2, = 2 4 = (5)2 4 6 2 = 25 48 = 23
< 0. ().
)) 16x2 + 8x 1 = 0 = 16, = 8, = 1, - = 2 4 = 82 4 (16) (1) = 64
64 = 0.
, x = = =
NN ::)) 99xx22 ((55xx 11))22 == 22xx )) ==
)) 9x2 (5x 1)2 = 2x )) =
9x2 (25x2 10x + 1) = 2x9x2 25x2 + 10x 1 2x = 0 6 6 = 6 16x2 + 8x
1 = 0
x = ( )2x(x + 3) (x 6) = 3
( 1)2x2 + 6x x + 6 3 = 02x2 + 5x + 3 = 0
x = 1 x = ( 1)
)) 22xx22 88xx ++ 66 == 00..)) 22xx22 88xx ++ 66..
)) 2x2 8x + 6 = 0 = 2, = 8, = 6,
= 2 4 = (8)2 4 2 6 = 64 48 = 16 > 0.
, x = x = = ,
x = 3 x = 1.
)) 2x2 8x + 6 = 2(x2 4x + 3) = 2(x2 3x x + 3) = 2 x(x 3) (x 3)
== 2(x 3)(x 1)
8 44
(8) 162 2
2
33
32
14
12
x 66
x(x + 3)3
12
x 66
x(x + 3)3
1122
xx 6666
xx((xx ++ 33))33
22
14
82 (16)
2
32
5 14
5 + 14
5 14
5 12 2
2
11
95
2.2 EE
(089-098) 7-11-06 16:17 95
-
96
M - 2
: 2x2 8x + 6 = 0 33 11. 2x2 8x + 6 :
22x2 8x + 6 = 22(x 33)(x 11)
11,, 22 xx22 ++ xx ++ == 00 00,, xx22 ++ xx ++
xx22 ++ xx ++ == ((xx 11))((xx 22))
, 2x2 + 5x + 3 = 0 1 ( 1).
2x2 + 5x + 3
2x2 + 5x + 3 = 2x (1) x ( ) = 2(x + 1)(x + )O 16x2 + 8x 1 = 0 ,
x = ( 1).
16x2 + 8x 1
16x2 + 8x 1 = 16(x )(x ) = 16(x )2
A x2 + x + = 0 0, - () ().
N (), (), .)) 2 , .)) 2 ,
.
)) 2x2 + 4x 6 = 0 1 3, 2x2 + 4x 6 2x2 + 4x 6 = (x 1)(x + 3).
22
11.. .
22.. .
33.. .
44.. .
.. > 0
.. = 0
.. 0
.. < 0
11
E
14
14
14
14
32
32
32
(089-098) 3-11-06 01:51 96
-
N )) 2x2 = 7x )) 3x2 2x + 8 = 0 )) 2x2 + 50 = 0 )) 5x2 + x 4 =
0
x2 + x + = 0 .
:)) x2 x 2 = 0 )) 4y2 + 3y 1 = 0 )) 22 + + 6 = 0)) 2z2 3z + 1 =
0 )) 25t2 + 10t 1 = 0 )) 4x2 12x + 9 = 0)) 3x2 + 18x + 27 = 0 )) x2
4x = 5 )) x2 3x + 7 = 0
: )) x2 7x = 0 )) x2 16 = 0ii)) iiii))
:)) 3x2 2(x 1) = 2x + 1 )) (y + 2)2 + (y 1)2 = 5(2y + 3))) (2
3)2 ( 2)2 = 22 11 )) (8 ) (3 + 1)( + 2) = 1
:
)) = x 2 )) = 2
)) 0,5t2 0,4(t + 2) = 0,7(t 2) )) (3 7) = 3
:)) x2 + 4x 12 )) 3y2 8y + 5 )) 22 + 5 3)) x2 16x + 64 )) 9y2 +
12y + 4 )) 2 + 10 25
, 0, )) x2 x + 1 = 0 )) x2 + ( + )x + = 0
( + )x2 2x + ( ) = 0, , , . , .
88
77
66
2
y 26
6y + 14
y2
3x + 3
5x2 1
3
55
44
33
22
(x 1)2 = 2(x2 x)
3x2 + 4 = 2(x + 2)
x (x 1) = 2
xx 22 ++ xx ++ == 00EE
11
33
97
2.2 EE
(089-098) 3-11-06 01:51 97
-
( 1650 ..) (*): , 870. . , , ( ). x, x2 x = 870.O B :
1 .
, .
870 870 .
870 , 29 .
29 ( )
30.
.
N
2 .
;
.
, . ;
(*)( . : , , , 1997.
34
12
12
12
14
14
14
14
12
12
12
To 1 x.( - ).
- .
, - (.. x2 x) , (.. x2 + x)
98
(089-098) 7-11-06 16:18 98
