77r^2+45r-18 eching

$$77 r ^ { 2 } + 45 r - 18$$

Omil

$$\left(11r-3\right)\left(7r+6\right)$$

Yechim qadamlari

Ifodani guruhlash orqali faktorlang. Avvalo, ifoda 77r^{2}+ar+br-18 sifatida qayta yozilishi kerak. a va b ni topish uchun yechiladigan tizimni sozlang.

$$a+b=45 ab=77\left(-18\right)=-1386$$

ab manfiy boʻlganda, a va b da qarama-qarshi belgilar bor. a+b musbat boʻlganda, musbat sonda manfiyga nisbatdan kattaroq mutlaq qiymat bor. -1386-mahsulotni beruvchi bunday butun juftliklarni roʻyxat qiling.

$$-1,1386 -2,693 -3,462 -6,231 -7,198 -9,154 -11,126 -14,99 -18,77 -21,66 -22,63 -33,42$$

Har bir juftlik yigʻindisini hisoblang.

$$-1+1386=1385 -2+693=691 -3+462=459 -6+231=225 -7+198=191 -9+154=145 -11+126=115 -14+99=85 -18+77=59 -21+66=45 -22+63=41 -33+42=9$$

Yechim – 45 yigʻindisini beruvchi juftlik.

$$a=-21 b=66$$

77r^{2}+45r-18 ni \left(77r^{2}-21r\right)+\left(66r-18\right) sifatida qaytadan yozish.

$$\left(77r^{2}-21r\right)+\left(66r-18\right)$$

Birinchi guruhda 7r ni va ikkinchi guruhda 6 ni faktordan chiqaring.

$$7r\left(11r-3\right)+6\left(11r-3\right)$$

Distributiv funktsiyasidan foydalangan holda 11r-3 umumiy terminini chiqaring.

$$\left(11r-3\right)\left(7r+6\right)$$

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.

$$77r^{2}+45r-18=0$$

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.

$$r=\frac{-45±\sqrt{45^{2}-4\times 77\left(-18\right)}}{2\times 77}$$

45 kvadratini chiqarish.

$$r=\frac{-45±\sqrt{2025-4\times 77\left(-18\right)}}{2\times 77}$$

-4 ni 77 marotabaga ko'paytirish.

$$r=\frac{-45±\sqrt{2025-308\left(-18\right)}}{2\times 77}$$

-308 ni -18 marotabaga ko'paytirish.

$$r=\frac{-45±\sqrt{2025+5544}}{2\times 77}$$

2025 ni 5544 ga qo'shish.

$$r=\frac{-45±\sqrt{7569}}{2\times 77}$$

7569 ning kvadrat ildizini chiqarish.

$$r=\frac{-45±87}{2\times 77}$$

2 ni 77 marotabaga ko'paytirish.

$$r=\frac{-45±87}{154}$$

r=\frac{-45±87}{154} tenglamasini yeching, bunda ± musbat. -45 ni 87 ga qo'shish.

$$r=\frac{42}{154}$$

\frac{42}{154} ulushini 14 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

$$r=\frac{3}{11}$$

r=\frac{-45±87}{154} tenglamasini yeching, bunda ± manfiy. -45 dan 87 ni ayirish.

$$r=-\frac{132}{154}$$

\frac{-132}{154} ulushini 22 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

$$r=-\frac{6}{7}$$

ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right) formulasi yordamida amalni hisoblang. x_{1} uchun \frac{3}{11} ga va x_{2} uchun -\frac{6}{7} ga bo‘ling.

$$77r^{2}+45r-18=77\left(r-\frac{3}{11}\right)\left(r-\left(-\frac{6}{7}\right)\right)$$

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

$$77r^{2}+45r-18=77\left(r-\frac{3}{11}\right)\left(r+\frac{6}{7}\right)$$

Umumiy maxrajni topib va suratlarni ayirib \frac{3}{11} ni r dan ayirish. So'ngra imkoni boricha kasrni eng kichik shartga qisqartirish.

$$77r^{2}+45r-18=77\times \frac{11r-3}{11}\left(r+\frac{6}{7}\right)$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali \frac{6}{7} ni r ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$77r^{2}+45r-18=77\times \frac{11r-3}{11}\times \frac{7r+6}{7}$$

Raqamlash sonlarini va maxraj sonlariga ko'paytirish orqali \frac{11r-3}{11} ni \frac{7r+6}{7} ga ko'paytirish. So'ngra kasrni imkoni boricha eng kam a'zoga qisqartiring.

$$77r^{2}+45r-18=77\times \frac{\left(11r-3\right)\left(7r+6\right)}{11\times 7}$$

11 ni 7 marotabaga ko'paytirish.

$$77r^{2}+45r-18=77\times \frac{\left(11r-3\right)\left(7r+6\right)}{77}$$

77 va 77 ichida eng katta umumiy 77 faktorini bekor qiling.

$$77r^{2}+45r-18=\left(11r-3\right)\left(7r+6\right)$$

Baholash

$$\left(11r-3\right)\left(7r+6\right)$$