175a^2+5a-60= eching

$$175 a ^ { 2 } + 5 a - 60 =$$

Omil

$$5\left(7a-4\right)\left(5a+3\right)$$

Yechim qadamlari

5 omili.

$$5\left(35a^{2}+a-12\right)$$

Hisoblang: 35a^{2}+a-12. Ifodani guruhlash orqali faktorlang. Avvalo, ifoda 35a^{2}+pa+qa-12 sifatida qayta yozilishi kerak. p va q ni topish uchun yechiladigan tizimni sozlang.

$$p+q=1 pq=35\left(-12\right)=-420$$

pq manfiy boʻlganda, p va q da qarama-qarshi belgilar bor. p+q musbat boʻlganda, musbat sonda manfiyga nisbatdan kattaroq mutlaq qiymat bor. -420-mahsulotni beruvchi bunday butun juftliklarni roʻyxat qiling.

$$-1,420 -2,210 -3,140 -4,105 -5,84 -6,70 -7,60 -10,42 -12,35 -14,30 -15,28 -20,21$$

Har bir juftlik yigʻindisini hisoblang.

$$-1+420=419 -2+210=208 -3+140=137 -4+105=101 -5+84=79 -6+70=64 -7+60=53 -10+42=32 -12+35=23 -14+30=16 -15+28=13 -20+21=1$$

Yechim – 1 yigʻindisini beruvchi juftlik.

$$p=-20 q=21$$

35a^{2}+a-12 ni \left(35a^{2}-20a\right)+\left(21a-12\right) sifatida qaytadan yozish.

$$\left(35a^{2}-20a\right)+\left(21a-12\right)$$

Birinchi guruhda 5a ni va ikkinchi guruhda 3 ni faktordan chiqaring.

$$5a\left(7a-4\right)+3\left(7a-4\right)$$

Distributiv funktsiyasidan foydalangan holda 7a-4 umumiy terminini chiqaring.

$$\left(7a-4\right)\left(5a+3\right)$$

Toʻliq ajratilgan ifodani qaytadan yozing.

$$5\left(7a-4\right)\left(5a+3\right)$$

Kvadrat tenglamalaridan foydalanish qoidalari

Kvadrat koʻp tenglama bu orqali hisoblanadi: ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right), bu yerda x_{1} va x_{2} ax^{2}+bx+c=0 kvadrat tenglamaning yechimlari.

$$175a^{2}+5a-60=0$$

ax^{2}+bx+c=0 shaklidagi barcha tenglamalarni kvadrat formulasi bilan yechish mumkin: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. Kvadrat formula ikki yechmni taqdim qiladi, biri ± qo'shish bo'lganda, va ikkinchisi ayiruv bo'lganda.

$$a=\frac{-5±\sqrt{5^{2}-4\times 175\left(-60\right)}}{2\times 175}$$

5 kvadratini chiqarish.

$$a=\frac{-5±\sqrt{25-4\times 175\left(-60\right)}}{2\times 175}$$

-4 ni 175 marotabaga ko'paytirish.

$$a=\frac{-5±\sqrt{25-700\left(-60\right)}}{2\times 175}$$

-700 ni -60 marotabaga ko'paytirish.

$$a=\frac{-5±\sqrt{25+42000}}{2\times 175}$$

25 ni 42000 ga qo'shish.

$$a=\frac{-5±\sqrt{42025}}{2\times 175}$$

42025 ning kvadrat ildizini chiqarish.

$$a=\frac{-5±205}{2\times 175}$$

2 ni 175 marotabaga ko'paytirish.

$$a=\frac{-5±205}{350}$$

a=\frac{-5±205}{350} tenglamasini yeching, bunda ± musbat. -5 ni 205 ga qo'shish.

$$a=\frac{200}{350}$$

\frac{200}{350} ulushini 50 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

$$a=\frac{4}{7}$$

a=\frac{-5±205}{350} tenglamasini yeching, bunda ± manfiy. -5 dan 205 ni ayirish.

$$a=-\frac{210}{350}$$

\frac{-210}{350} ulushini 70 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

$$a=-\frac{3}{5}$$

ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right) formulasi yordamida amalni hisoblang. x_{1} uchun \frac{4}{7} ga va x_{2} uchun -\frac{3}{5} ga bo‘ling.

$$175a^{2}+5a-60=175\left(a-\frac{4}{7}\right)\left(a-\left(-\frac{3}{5}\right)\right)$$

p-\left(-q\right) shaklining barcha amallarigani p+q ga soddalashtiring.

$$175a^{2}+5a-60=175\left(a-\frac{4}{7}\right)\left(a+\frac{3}{5}\right)$$

Umumiy maxrajni topib va suratlarni ayirib \frac{4}{7} ni a dan ayirish. So'ngra imkoni boricha kasrni eng kichik shartga qisqartirish.

$$175a^{2}+5a-60=175\times \frac{7a-4}{7}\left(a+\frac{3}{5}\right)$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali \frac{3}{5} ni a ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$175a^{2}+5a-60=175\times \frac{7a-4}{7}\times \frac{5a+3}{5}$$

Raqamlash sonlarini va maxraj sonlariga ko'paytirish orqali \frac{7a-4}{7} ni \frac{5a+3}{5} ga ko'paytirish. So'ngra kasrni imkoni boricha eng kam a'zoga qisqartiring.

$$175a^{2}+5a-60=175\times \frac{\left(7a-4\right)\left(5a+3\right)}{7\times 5}$$

7 ni 5 marotabaga ko'paytirish.

$$175a^{2}+5a-60=175\times \frac{\left(7a-4\right)\left(5a+3\right)}{35}$$

175 va 35 ichida eng katta umumiy 35 faktorini bekor qiling.

$$175a^{2}+5a-60=5\left(7a-4\right)\left(5a+3\right)$$

Baholash

$$5\left(35a^{2}+a-12\right)$$