21- §.  YIG‘INDINING   KVADRATI.   AYIRMANING   KVADRATI

 

Ikkita son yig‘indisining kvadrati (a+b)2 ni qaraymiz. Ko‘phadni ko‘phadga ko‘paytirish qoidasidan foydalanib, hosil qilamiz:

(a+b)2=(a+b)(a+b)=a2+ab+ab+b2= a2+2ab+b2,

Ïîäïèñü: (a+b)2=a2+2ab+b2ya’ni

                      (1)

 

 
 
Ikki son yig‘indisining kvadrati birinchi son kvadrati plyus birinchi son bilan ikkinchi son ko‘paytmasining ikkilangani plyus ikkinchi son kvadratiga teng.

(1) formulani 13- rasmda tasvirlangan kvadratning yuzini ko‘zdan kechirib, osongina hosil qilish mumkinligini aytib o‘tamiz.

 

 

 

 

 

 

 

 


Endi ikki son ayirmasining kvadratini qaraymiz:

(a–b)2=(a–b)(a–b)=a2–ab–ab+b2=a2–2ab+b2,

Ïîäïèñü: (a–b)2=a2–2ab+b2ya’ni

                   (2)

 

Ikki son ayirmasining kvadrati birinchi son kvadrati minus birinchi son bilan ikkinchi son ko‘paytmasining ikkilangani plyus ikkinchi son kvadratiga teng.

(1) va (2) tengliklarda a va b istalgan sonlar yoki algebraik ifodalardir.

(1) va (2) formulalarni qo‘llashga doir misollar:

1) (2m+3k)2=(2m)2+2(2m)(3k)+(3k)2=4m2+12mk+9k2;

2) (5a2–3) 2=(5a2) 2–2·5a2·3+32=25a4–30a2+9;

3) (–a–3b)2=((–1)(a+3b))2=(–1)2(a+3b)2=(a+3b)2=

    =a2+2a(3b)+(3b)2=a2+6ab+9b2.

Zaruriy hisoblashlarni og‘zaki bajarib, oraliq natijalarni yozmaslik mumkin. Masalan, birdaniga bunday yozish mumkin:

(5a2–7b2)2=25a4–70a2b2+49b4.

Yig‘indi yoki ayirmaning kvadrati formulasini qisqa ko‘paytirish formulalari deyiladi va ba’zi hollarda hisoblashlarni soddalashtirish uchun qo‘llanadi. Masalan:

1)          992 = (100 – 1)2 = 10000 – 200 + 1 = 9801;

2)          522 = (50 + 2)2 = 2500 + 200 + 4 = 2704.

   (1) formula (1+a)2 ifodaning qiymatlarini taqribiy hisoblashlarda ham qo‘llaniladi. a son musbat yoki manfiy son bo‘lib, uning moduli 1 ga nisbatan kichik bo‘lsa (masalan, a=0,0032 yoki a=–0,0021), u holda a2 son yanada kichik bo‘ladi va shu sababli

(1+a)2=1+2a+a2

tenglikni (1+a)2≈1+2a taqribiy tenglik bilan almashtirish mumkin. Masalan:

1)          (1,002)2=(1+0,002)2≈1+2·0,002=1,004;

2)          (0,997)2=(1–0,003)2≈1–2·0,003=0,994.

Yig‘indining kvadrati va ayirmaning kvadrati formulalari ko‘phadni ko‘paytuvchilarga ajratishda ham qo‘llaniladi, masalan:

1)               x2+10x+25=x2+2·5·x+52=(x+5)2;

2)               a4–8a2b3+16b6=(a2)2–2·a2·4b3+(4b3)2=(a2–4b3)2.

Ïîäïèñü: (a+b)3=a3+3a2b+3ab2+b3M a s a l a .  Formulani isbotlang:

                     (3)

 

(a+b)3=(a+b)(a+b)2=(a+b)(a2+2ab+b2)=

=a3+2a2b+ab2+a2b+2ab2+b3=a3+3a2b+3ab2+b3.

Ïîäïèñü: (a–b)3=a3–3a2b+3ab2–b3Xuddi shunga o‘xshash,

                    (4)

formulani ham isbotlash mumkin.

  (3) va (4) formulalar mos ravishda yig‘indining kubi va ayirmaning kubi deb ataladi.

  (3) va (4) formulalar ham qisqa ko‘paytirish formulalari hisoblanadi.

 

 

     TAYANCH   TUSHUNCHALAR:

Yig‘indining kvadrati, ayirmaning kvadratini, yig‘indining kubi, ayirmaning kubi, qisqa ko‘paytirish formulalari.