24x^2-119x-48 eching

$$24 x ^ { 2 } - 119 x - 48$$

Omil

$$\left(3x-16\right)\left(8x+3\right)$$

Yechim qadamlari

Ifodani guruhlash orqali faktorlang. Avvalo, ifoda 24x^{2}+ax+bx-48 sifatida qayta yozilishi kerak. a va b ni topish uchun yechiladigan tizimni sozlang.

$$a+b=-119 ab=24\left(-48\right)=-1152$$

ab manfiy boʻlganda, a va b da qarama-qarshi belgilar bor. a+b manfiy boʻlganda, manfiy sonda musbatga nisbatdan kattaroq mutlaq qiymat bor. -1152-mahsulotni beruvchi bunday butun juftliklarni roʻyxat qiling.

$$1,-1152 2,-576 3,-384 4,-288 6,-192 8,-144 9,-128 12,-96 16,-72 18,-64 24,-48 32,-36$$

Har bir juftlik yigʻindisini hisoblang.

$$1-1152=-1151 2-576=-574 3-384=-381 4-288=-284 6-192=-186 8-144=-136 9-128=-119 12-96=-84 16-72=-56 18-64=-46 24-48=-24 32-36=-4$$

Yechim – -119 yigʻindisini beruvchi juftlik.

$$a=-128 b=9$$

24x^{2}-119x-48 ni \left(24x^{2}-128x\right)+\left(9x-48\right) sifatida qaytadan yozish.

$$\left(24x^{2}-128x\right)+\left(9x-48\right)$$

Birinchi guruhda 8x ni va ikkinchi guruhda 3 ni faktordan chiqaring.

$$8x\left(3x-16\right)+3\left(3x-16\right)$$

Distributiv funktsiyasidan foydalangan holda 3x-16 umumiy terminini chiqaring.

$$\left(3x-16\right)\left(8x+3\right)$$

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.

$$24x^{2}-119x-48=0$$

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.

$$x=\frac{-\left(-119\right)±\sqrt{\left(-119\right)^{2}-4\times 24\left(-48\right)}}{2\times 24}$$

-119 kvadratini chiqarish.

$$x=\frac{-\left(-119\right)±\sqrt{14161-4\times 24\left(-48\right)}}{2\times 24}$$

-4 ni 24 marotabaga ko'paytirish.

$$x=\frac{-\left(-119\right)±\sqrt{14161-96\left(-48\right)}}{2\times 24}$$

-96 ni -48 marotabaga ko'paytirish.

$$x=\frac{-\left(-119\right)±\sqrt{14161+4608}}{2\times 24}$$

14161 ni 4608 ga qo'shish.

$$x=\frac{-\left(-119\right)±\sqrt{18769}}{2\times 24}$$

18769 ning kvadrat ildizini chiqarish.

$$x=\frac{-\left(-119\right)±137}{2\times 24}$$

-119 ning teskarisi 119 ga teng.

$$x=\frac{119±137}{2\times 24}$$

2 ni 24 marotabaga ko'paytirish.

$$x=\frac{119±137}{48}$$

x=\frac{119±137}{48} tenglamasini yeching, bunda ± musbat. 119 ni 137 ga qo'shish.

$$x=\frac{256}{48}$$

\frac{256}{48} ulushini 16 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

$$x=\frac{16}{3}$$

x=\frac{119±137}{48} tenglamasini yeching, bunda ± manfiy. 119 dan 137 ni ayirish.

$$x=-\frac{18}{48}$$

\frac{-18}{48} ulushini 6 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

$$x=-\frac{3}{8}$$

ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right) formulasi yordamida amalni hisoblang. x_{1} uchun \frac{16}{3} ga va x_{2} uchun -\frac{3}{8} ga bo‘ling.

$$24x^{2}-119x-48=24\left(x-\frac{16}{3}\right)\left(x-\left(-\frac{3}{8}\right)\right)$$

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

$$24x^{2}-119x-48=24\left(x-\frac{16}{3}\right)\left(x+\frac{3}{8}\right)$$

Umumiy maxrajni topib va suratlarni ayirib \frac{16}{3} ni x dan ayirish. So'ngra imkoni boricha kasrni eng kichik shartga qisqartirish.

$$24x^{2}-119x-48=24\times \frac{3x-16}{3}\left(x+\frac{3}{8}\right)$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali \frac{3}{8} ni x ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$24x^{2}-119x-48=24\times \frac{3x-16}{3}\times \frac{8x+3}{8}$$

Raqamlash sonlarini va maxraj sonlariga ko'paytirish orqali \frac{3x-16}{3} ni \frac{8x+3}{8} ga ko'paytirish. So'ngra kasrni imkoni boricha eng kam a'zoga qisqartiring.

$$24x^{2}-119x-48=24\times \frac{\left(3x-16\right)\left(8x+3\right)}{3\times 8}$$

3 ni 8 marotabaga ko'paytirish.

$$24x^{2}-119x-48=24\times \frac{\left(3x-16\right)\left(8x+3\right)}{24}$$

24 va 24 ichida eng katta umumiy 24 faktorini bekor qiling.

$$24x^{2}-119x-48=\left(3x-16\right)\left(8x+3\right)$$

Baholash

$$\left(3x-16\right)\left(8x+3\right)$$