Ko'rsatkichli tenglamalar
![Ko’rsat k ichli
t e nglamalar](/data/documents/55cae6a4-4a8c-4231-a7a7-62a2adc0f1fe/page_1.png)
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uning y echilish sxemasi
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![p (x) = 0 ko’rinishdagi tenglamalar .
1 . Chiziqli tenglama .
2. Kvadrat Tenglama
3. Ixtiyoriy darajadagi ikki o’zgaruvchili tenglamalar.
4. Bikvadrat tenglamalar.nn](/data/documents/55cae6a4-4a8c-4231-a7a7-62a2adc0f1fe/page_4.png)
![1) Chiziqli tenglama ( darajasi n =1) :
2) Kvadrat tenglama ( n =2):. )0 ( ,0
a
b
x b ax a b ax
, ) 0 ( 0 2 a c bx ax
. 4 2 ac b D
Agar D < 0 Agar D = 0 Agar D > 0
a
b
x
2
0
a
ac b b
x
2
4 2
2.1
Ildizi yo’q
. )0 ( ,0
a
b
x b ax a b ax
, ) 0 ( 0 2 a c bx ax
. 4 2 ac b D
a
b
x
2
0
a acbb
x
2 4 2
2.1
](/data/documents/55cae6a4-4a8c-4231-a7a7-62a2adc0f1fe/page_5.png)
![3) Ixtiyoriy darajadagi ikki o’zgaruvchili tenglamalar ( n ≥ 2):
Agar n = 2k + 1 (toq) bo'lsa, u holda c ning istalgan qiymati uchun tenglama yagona
yechimga ega:
Agar n = 2 k juft bo’lsa , u holda :
с < 0. c x
n
, 0 b ax
n
n c x 0
n
c x 2,1
Ildiz yo’q с = 0
х = 0 с > 01)
2)
. c x
n
,0 bax n
n c x 0
n
c x 2,1](/data/documents/55cae6a4-4a8c-4231-a7a7-62a2adc0f1fe/page_6.png)
![4) Bikvadrat tenglama ( a ≠ 0):
.0,
0
22
24
cbzaz zx
cbxax
Bikublik tenglama ( a ≠ 0) :0 ax
3 6
c bx
. 0
,
2
3
c bz az
z x
. 0
,
0 2
2
c bz az
z x
c bx ax
n
n n
, 0 , , , , ,0
2
a R c b a N n c bx ax
n n
Umumiy yechilish sxemasi
. 0
,
0 2
2
2 4
c bz az
z x
c bx ax
0 ax
3 6
c bx
. 0
,
2
3
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. 0
,
0 2
2
c bz az
z x
c bx ax
n
n n
, 0 , , , , , 0
2
a R c b a N n c bx ax
n n
](/data/documents/55cae6a4-4a8c-4231-a7a7-62a2adc0f1fe/page_7.png)
![Darajasi 3 dan katta bo’lgan bo’lgn
tenglamalarni yechish usullari
O’riniga
qo’yish usuli Yoyish usuli](/data/documents/55cae6a4-4a8c-4231-a7a7-62a2adc0f1fe/page_8.png)
![O’rniga qo’yish usuli, ... ) ( ) ( 0 )) ( ( 2 1 z x z x x f
. 0 ) ( z f
0 ) ( x F . 0 )) ( ( x f
, ) ( z x . 0 ) ( z f
Tenglamani yechamiz
... , 2 1 z z z
, ) ( 1z x ,... ) ( 2 z x Boshlang’ich tenglamani
o’rniga almashtiramiz Yechamiz
Qiymatini topamiz
O’rniga qo’yish sxemasi :
Bu yerda tenglama ildizi
... , 2 1 z z ,o’rniga
Qo’yamiz
,...)()(0))((
21 zxzxxf
. 0 ) ( z f
0 ) ( x F . 0 )) ( ( x f
, ) ( z x . 0 ) ( z f
... , 2 1 z z z
, ) ( 1z x ,... ) ( 2 z x
... , 2 1 z z](/data/documents/55cae6a4-4a8c-4231-a7a7-62a2adc0f1fe/page_9.png)
![0. 1 - 3) - (x 2006 9) 6x - (x 2007224
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.
2007 1
3,2 xx
Javob :
.
2007
1
3 , 2 4,3 2,1 x x
0. 1 - 3) - (x 2006 9) 6x - (x 2007 2 2 4
3, - x y 2
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.
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.
2007
1
3 , 2 4,3 2,1 x x](/data/documents/55cae6a4-4a8c-4231-a7a7-62a2adc0f1fe/page_10.png)
![Misol № 3.. 82 ) 2 (
4 4
x x
, 1
2
) 2 (
z x
x x
, 1 2 , 1 z x z x
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zzSimmetriyaga asoslangan chiziqli
o’rniga qo’yish .
1x 2 x
o x
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4 4
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) 2 (
z x
x x
, 1 2 , 1 z x z x
, 82 ) 1 ( ) 1 (
4 4
z z
1x 2 x
o x](/data/documents/55cae6a4-4a8c-4231-a7a7-62a2adc0f1fe/page_11.png)
![ 0 ) 10 )( 4 ( 0 40 6 2 2 2 4 z z z z
. 2 10 422 z z z
, 2 1 x
. 3 1 2 1 1 2 x и x .0406822122 2424
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Javob :
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zzzz
. 2 10 4
22 z z z
, 2 1 x
. 3 1 2 1 1 2 x и x
. 0 40 6 82 2 12 2 2 4 2 4 z z z z
. 3 ,1 2 1 x x](/data/documents/55cae6a4-4a8c-4231-a7a7-62a2adc0f1fe/page_12.png)
![Misol № 4., 3 z x
, 40 ) 2 )( 1 )( 1 )( 2 ( *) 2( z z z z
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![, 0 2 z u . 4 u,392
zz
, 3 3 3 z x
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, 6 2 y x x . 5 6 2 u x x
. 4 2 5 1 1) Bu yerdan
II-usul
Guruhlaymiz
Hosil qilamzi
Mumkin bo'lgan almashtirishlar
Javob :
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и .40)5)(4)(2)(1( xxxxMumkin bo'lgan almashtirishlar
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![Yoyish usuli .
,0)( xpn
, 0 ) ( ) ( x p x p m k
, 0 ) ( deg k x p k , 0 ) ( deg m x p
m
, n m k . , n m k
. 0 ) ( 0 ) ( x p x p
mk
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agar . bo’lsa
Yechish sxemasi :,0)( xp
n
, 0 ) ( ) ( x p x p m k
, 0 ) ( deg k x p k , 0 ) ( deg m x p m
, n m k . , n m k
. 0 ) ( 0 ) ( x p x p m k
0 ) ( ) ( 0 ) ( x p x p x p m k n](/data/documents/55cae6a4-4a8c-4231-a7a7-62a2adc0f1fe/page_15.png)
![Mi sol № 5 . . 2 2 4 2
3 3 3
x x x
, 2 2 4 4 2 2 4 2
3 2 2
x x x x x x x
, 2 2 16 8 8 2 4 4 2 2
3 2 2 2 x x x x x x x x
, 0 4 8 4 28 2 2 2
22 x x x x x
0. 24 6x -3x- 0 2 2x 2
yoki
2. x 4, - x 1.- x 3 2 1
Javob :
2. x 4,- x 1,- x
321
. 2 2 4 2 3 3 3 x x x
, 2 2 4 4 2 2 4 2
3 2 2
x x x x x x x
, 2 2 16 8 8 2 4 4 2 2
3 2 2 2 x x x x x x x x
, 0 4 8 4 28 2 2 2 22 x x x x x
0. 24 6x - 3x- 0 2 2x 2 yoki
2. x 4, - x 1.- x 3 2 1
2. x 4,- x 1,- x 3 2 1 ](/data/documents/55cae6a4-4a8c-4231-a7a7-62a2adc0f1fe/page_16.png)
![Misol № 6 .0. 2 - 13x 8x - x 2 3
-2, -1, 1, 2.
0
2
2
12 6
13 6
1 6
2
2
2 13 8
22 2
23 23
x
x
x x
x x
x x
x
x x
x x xIldizlar extimoli bilan : x = 2 - ildiz deb olsak .
0, 1) 6x - (x 2) -(x 2
, 0 1 6x - x 0 2 - x
2
yoki
2. x 1
.8 3x
2,3
Javob :
. 8 3 x 2, x 2,3 1
0. 2 - 13x 8x - x 2 3
0, 1) 6x - (x 2) - (x 2
, 0 1 6x - x 0 2 - x 2 yoki
2. x 1 . 8 3 x 2,3
. 8 3 x 2, x 2,3 1 ](/data/documents/55cae6a4-4a8c-4231-a7a7-62a2adc0f1fe/page_17.png)
![Misol № 7 . 0. 1 x - x x - x
2 5 8
1- x 1 x
00 yoki
, 0 x- x 0, x - 1
52 5.(-1)p и 111-11-1(1)p
88
Agar х < 0
0, (x) p 8
х > 1
, x x , x x 2 5 8
0, (x) p 8
Agar 1)
2) ildizga ega emas
3)
ildizga ega emas
0, x )x-(x x)-(1 (x)p 852
8 ),1;0(x
4)
Ildizga ega emas . Tenglamaning ildizi emas
Javob : .
0. 1 x - x x - x
2 5 8
1- x 1 x 0 0 yoki
, 0 x- x 0, x -1 52
5. (-1) p и 1 1 1- 1 1- 1 (1) p 8 8
0, (x) p 8
, x x , x x 2 5 8
0, (x) p 8
0, x ) x - (x x) - (1 (x) p 8 5 2
8
),1;0( x](/data/documents/55cae6a4-4a8c-4231-a7a7-62a2adc0f1fe/page_18.png)
![Misol № 8 .
2 2 2
12 6 4 6 3 x x x x x
, 12
6
4
6
3
x
x
x
x
,
6
x
x y
,12127 ,1243
2
yy yy
, 7 , 0 2 1 y y
, 0
6
x
x , 7
6
x
x
. 1 , 6 2 1 x x
, 0 : 2 x
Javob : Yildizi yo’q .
.1 ,6 2 1 x x
2 2 2
12 6 4 6 3 x x x x x
, 12
6
4
6
3
x
x
x
x
,
6
x
x y
, 12 12 7
, 12 4 3
2
y y
y y
, 7 , 0 2 1 y y
, 0
6
x
x , 7
6
x
x
. 1 , 6 2 1 x x
, 0 : 2 x
.1 ,6 2 1 x x](/data/documents/55cae6a4-4a8c-4231-a7a7-62a2adc0f1fe/page_19.png)
![Kasrli rat sional t englamalarni y echishning
ba'zi usullari .
.0)( ,0)(
0
)( )(
xQ xP
xQ xP
. 0
, 0 ) ( ) (
d cx
d cx m b ax d cx kx
m
d cx
b ax
kx
. 0 ) (
, 0 ) (
0
) (
) (
x Q
x P
x Q
x P
. 0
, 0 ) ( ) (
d cx
d cx m b ax d cx kx
m
d cx
b ax
kx](/data/documents/55cae6a4-4a8c-4231-a7a7-62a2adc0f1fe/page_20.png)
![Misol № 9 . . 18
4
24
2 2
2
x x
x . 0 4
2
x x
. 18
4
24
4 4 2
2
x x
x x
x x y 4
2
, 0 y
. 18
24
4
y
y
. 2 , 12 2 1 y y
12 4
2
x x . 2 4
2
x x
. 6 2 , 2 , 6 4,3 2 1 x x x
Javob : yoki
. 6 2 .2 ,6 4,3 2 1 x x x
. 18
4
24
2 2
2
x x
x . 0 4
2
x x
. 18
4
24
4 4 2
2
x x
x x
x x y 4
2
, 0 y .1824
4
yy
. 2 , 12 2 1 y y
12 4
2
x x . 2 4
2
x x
. 6 2 , 2 , 6 4,3 2 1 x x x
. 6 2 .2 ,6 4,3 2 1 x x x](/data/documents/55cae6a4-4a8c-4231-a7a7-62a2adc0f1fe/page_21.png)
![Misol № 10 . . 6 43
18 6
324
2
4
x
x x
x
,643
186 3632436
2 224
x
xx xxx
, 6 43
18 6
36 18
2
2 2 2
x
x x
x x
. 6 43
18 6
6 18 6 18
2 22
x
x x
x x x x
. 0 18 6
2
x x
, 6 43 18 6
2
x x x
. 5 , 5 2 1 x x
Javob :
. 5 2,1 x Kasrni qisqartiramiz .
. 6 43
18 6
324
2
4
x
x x
x
,643
186 3632436
2
2 2 4
x
xx xxx
, 6 43
18 6
36 18
2
2 2 2
x
x x
x x
. 6 43
18 6
6 18 6 18
2
2 2
x
x x
x x x x
. 0 18 6
2
x x
, 6 43 18 6
2
x x x
. 5 , 5 2 1 x x
. 5 2,1 x](/data/documents/55cae6a4-4a8c-4231-a7a7-62a2adc0f1fe/page_22.png)
![Xulosa
Tenglamalar nazariyasi barcha zamonlar va xalqlar
matematiklarini qiziqtirgan. Ularga ilmiy risolalar bag‘ishlangan, hatto
tarixning buyuk shaxslari ham she’rlar yozgan.
Biz taqdimotda tenglamalar nazariyasiga oid bilgan bilimlarimizni
tizimlashtirishga, tenglamalarni yechishning ayrim usullarining
go‘zalligi va nafisligini ko‘rsatishga harakat qildik.](/data/documents/55cae6a4-4a8c-4231-a7a7-62a2adc0f1fe/page_23.png)
Ko’rsat k ichli t e nglamalar
, 0 ) ( ) ( x Q x P ]. [ , x R Q P . 0 ) ( , 0 ) ( 0 ) ( ) ( x Q x P x Q x PAlgebraik t englama v a uning y echilish sxemasi , 0 ) ( ) ( x Q x P ]. [ , x R Q P . 0 ) ( , 0 ) ( 0 ) ( ) ( x Q x P x Q x P
Misol № 1 .*)1( . 1 1 1 2 2 x x x . 0 , 0 )1 ( ;0 , 0 0 0 )1 ( 1 0 1 1 1 *)1( 2 2 2 2 2 2 2 2 x x x x x x x x x x x x x x x . 1 x Javob : .1 ;0 ,1,0 x x xx *)1( . 1 1 1 2 2 x x x . 0 , 0 )1 ( ;0 , 0 0 0 )1 ( 1 0 1 1 1 *)1( 2 2 2 2 2 2 2 2 x x x x x x x x x x x x x x x . 1 x .1 ;0 ,1,0 x x xx
p (x) = 0 ko’rinishdagi tenglamalar . 1 . Chiziqli tenglama . 2. Kvadrat Tenglama 3. Ixtiyoriy darajadagi ikki o’zgaruvchili tenglamalar. 4. Bikvadrat tenglamalar.nn
1) Chiziqli tenglama ( darajasi n =1) : 2) Kvadrat tenglama ( n =2):. )0 ( ,0 a b x b ax a b ax , ) 0 ( 0 2 a c bx ax . 4 2 ac b D Agar D < 0 Agar D = 0 Agar D > 0 a b x 2 0 a ac b b x 2 4 2 2.1 Ildizi yo’q . )0 ( ,0 a b x b ax a b ax , ) 0 ( 0 2 a c bx ax . 4 2 ac b D a b x 2 0 a acbb x 2 4 2 2.1