125
442.
Motorli qayiqning oqim bo‘yicha tezligi 18 km/soat,
oqimga qarshi tezligi esa 14 km/soat. Daryo oqimining
tezligini va qayiqning turg‘un suvdagi tezligini toping.
O‘zingizni tekshirib ko‘ring!
1.
Ifodani standart ko‘phad ko‘rinishida tasvirlang:
-
+
-
+
+
2
(
3)
(
3)(
3) 6 .
a
a
a
a
2.
Ko‘paytuvchilarga ajrating:
1)
-
2 ;
xy
y
2)
-
2
16
81;
a
3)
-
2
3
3
6 ;
x
x
4)
-
+
2
10
25;
x
x
5)
- +
-
3(
1)
(
1);
x
y x
6)
-
+
2
2
2
4
2 .
a
ab
b
3.
Ko‘phadni ko‘paytuvchilarga
ajrating va uning
=
= -
1
3
1,
a
b
bo‘lgandagi son qiymatini toping:
-
+
-
2
3
3
9 .
a
ab
a
b
I V b o b g a d o i r m a s h q l a r
Ko‘paytuvchilarga ajrating
(443
—
447)
:
443.
1)
(
) (
)
+
+
+
2
6
;
a b
a b
3)
(
) (
)
-
+
-
2
;
a b
b a
2)
(
) (
)
-
+
-
2
4
3
;
x y
x y
4)
(
) (
)
-
-
-
2
.
a b
b a
444.
1)
(
)(
) (
)
+
-
+
+
2
3
;
x y x y
x y
3)
(
) (
)(
)
-
-
+
-
2
5
;
a b
a b b a
2)
(
)
(
)
+
-
+
3
2
;
x y
x x y
4)
(
) (
)
-
-
-
2
2
.
a a b
b a
445.
1)
(
)
(
)
(
)
(
)
+
+
+
-
+
2
2
12
6
12
6
;
y z
x
x
y z
x
x
2)
(
)
(
)
(
)
(
)
-
-
+
-
+
2
2
12
6
12
6
;
y z
x
x
y z
x
x
3)
(
)
(
)
(
)
-
+
-
-
-
2
2
2
6
3
7 6
3
4 6
3 ;
x
x
x
y
x
4)
(
)
(
) (
)
-
-
-
-
-
2 8
4
3 8
4
8
4
.
x x
y
y x
y
x
y
126
446.
1)
-
+
-
2
18
27
14
21 ;
a
ab
ac
bc
2)
+
+
+
2
10
10
5
5 ;
x
xy
x
y
3)
+
-
-
2
35
24
20
42 ;
ax
xy
ay
x
4)
+
-
-
2
2
2
3
48
32
15
10 .
xz
xy
yz
y
447.
1)
-
-
+
2
2
3
2
16
5
10
32
;
ab
b c
c
ac
2)
2
2
3
2
6
15
14
35
;
mnk
m k
n k
mn
+
-
-
3)
-
+
-
+
2
28
35
10
8 ;
ac
c
cx
ax
4)
-
-
+
+
2
24
15
40
9 .
bx
c
bc
cx
448.
Ifodani soddalashtiring:
1)
(
)
(
)
-
-
-
+
2
2
2
1
2 2
3
17;
x
x
2)
(
)
(
)
+
-
-
-
2
2
2
3
2
2
1
7 ;
x
x
x
3)
(
) (
) (
)
-
-
+
-
+
2
2
24
7
2
5
3 5
1 ;
y
y
y
y
4)
(
) (
) (
)
+
-
+
-
-
2
2
3
1 2
3
2
3
10 .
y
y
y
y
449.
Ikkita ketma-ket natural
son kvadratlari ayirmasining
moduli toq son bo‘lishini isbotlang.
450.
Kasrni qisqartiring:
1)
-
-
2
2
2
2
53
27
79
51
;
3)
- ×
×
+
-
2
2
2
2
49
2 49 29 29
49
19
;
2)
-
-
2
2
2
2
38
17
47
19
;
4)
-
+ ×
×
+
2
2
2
2
47
3
.
27
2 27 13 13
451.
x
va
y
ning istalgan qiymatlarida tenglik to‘g‘ri bo‘lishini
isbotlang:
(
)
(
)
(
)(
)
+
-
=
-
+
2
2
2
x y x
y
x y x y
.
1) Oiladagi 6 ta qizning har birining akasi bor. Shu
oilada nechta farzand bor?
2) Muhammadjonning akalari qancha bo‘lsa, opalari ham
shuncha. Katta opasining ukalari soni singillari sonidan
2 marta ko‘p. Shu oilada nechta o‘g‘il, nechta qiz bor?
¹ 8
128
&
T a r i x i y m a ’ l u m o t l a r
Al-Koshiyning „Arifmetika kaliti“ asarida ikkihadni ixti-
yoriy natural darajaga ko‘tarish qoidalari berilgan.
Turli algebraik
formulalarni isbotlashda, tenglamalarni ye-
chishda geometrik mulohazalardan foydalanish qadimgi Xitoy,
Yunoniston,
Hindiston, O‘rta Osiyo matematiklari asarlarida
uchraydi.
U lar
2
2
2
(
)
2
,
a b
a
ab b
+
=
+
+
2
2
2
(
)
2
,
a b
a
ab b
-
=
-
+
a
2
-
b
2
=(
a
-
b
)
½
½
(
a
+
b
) (yoki
2
2
2
(
) (
)
2 (
)
a
b
a b
b a b
-
=
-
+
-
) kabi ayniyatlarni geo-
metrik usulda isbotlaganlar. Masalan,
2
2
(
)(
)
a
b
a b a b
-
=
-
+
for-
mulani isbotlashga shunday yondashilgan:
tomoni
a
ga teng
kvadratdan tomoni
b
ga teng kvadratni qirqib olinsa,
qolgan
shaklning yuzi:
(
)
(
) (
)(
)
a a b
b a b
a b a b
- +
-
=
-
+
ga, yoki baribir,
2
(
)
2 (
)
a b
b a b
-
+
-
ga teng bo‘lishi 21- rasmdan ravshan ko‘rinib
turibdi.
Demak,
2
2
(
)(
)
a
b
a b a b
-
=
-
+
formula to‘g‘ri.
To‘g‘ri burchakli uchburchakning tomonlarini butun (yoki
ratsional) sonlarda ifodalash uchun Xitoy matematiklari
miloddan avvalgi
birinchi ming yillardayoq
-
+
æ
ö
æ
ö
+
=
ç
÷
ç
÷
è
ø
è
ø
2
2
2
2
2
2
2
2
2
(
)
p
q
p
q
pq
tenglikdan foydalanganlar.
a
-
b
a
-
b
a
-
b
a
b
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