12a^2-11a-36 eching

$$12 a ^ { 2 } - 11 a - 36$$

Omil

$$\left(4a-9\right)\left(3a+4\right)$$

Yechim qadamlari

Ifodani guruhlash orqali faktorlang. Avvalo, ifoda 12a^{2}+pa+qa-36 sifatida qayta yozilishi kerak. p va q ni topish uchun yechiladigan tizimni sozlang.

$$p+q=-11 pq=12\left(-36\right)=-432$$

pq manfiy boʻlganda, p va q da qarama-qarshi belgilar bor. p+q manfiy boʻlganda, manfiy sonda musbatga nisbatdan kattaroq mutlaq qiymat bor. -432-mahsulotni beruvchi bunday butun juftliklarni roʻyxat qiling.

$$1,-432 2,-216 3,-144 4,-108 6,-72 8,-54 9,-48 12,-36 16,-27 18,-24$$

Har bir juftlik yigʻindisini hisoblang.

$$1-432=-431 2-216=-214 3-144=-141 4-108=-104 6-72=-66 8-54=-46 9-48=-39 12-36=-24 16-27=-11 18-24=-6$$

Yechim – -11 yigʻindisini beruvchi juftlik.

$$p=-27 q=16$$

12a^{2}-11a-36 ni \left(12a^{2}-27a\right)+\left(16a-36\right) sifatida qaytadan yozish.

$$\left(12a^{2}-27a\right)+\left(16a-36\right)$$

Birinchi guruhda 3a ni va ikkinchi guruhda 4 ni faktordan chiqaring.

$$3a\left(4a-9\right)+4\left(4a-9\right)$$

Distributiv funktsiyasidan foydalangan holda 4a-9 umumiy terminini chiqaring.

$$\left(4a-9\right)\left(3a+4\right)$$

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.

$$12a^{2}-11a-36=0$$

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.

$$a=\frac{-\left(-11\right)±\sqrt{\left(-11\right)^{2}-4\times 12\left(-36\right)}}{2\times 12}$$

-11 kvadratini chiqarish.

$$a=\frac{-\left(-11\right)±\sqrt{121-4\times 12\left(-36\right)}}{2\times 12}$$

-4 ni 12 marotabaga ko'paytirish.

$$a=\frac{-\left(-11\right)±\sqrt{121-48\left(-36\right)}}{2\times 12}$$

-48 ni -36 marotabaga ko'paytirish.

$$a=\frac{-\left(-11\right)±\sqrt{121+1728}}{2\times 12}$$

121 ni 1728 ga qo'shish.

$$a=\frac{-\left(-11\right)±\sqrt{1849}}{2\times 12}$$

1849 ning kvadrat ildizini chiqarish.

$$a=\frac{-\left(-11\right)±43}{2\times 12}$$

-11 ning teskarisi 11 ga teng.

$$a=\frac{11±43}{2\times 12}$$

2 ni 12 marotabaga ko'paytirish.

$$a=\frac{11±43}{24}$$

a=\frac{11±43}{24} tenglamasini yeching, bunda ± musbat. 11 ni 43 ga qo'shish.

$$a=\frac{54}{24}$$

\frac{54}{24} ulushini 6 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

$$a=\frac{9}{4}$$

a=\frac{11±43}{24} tenglamasini yeching, bunda ± manfiy. 11 dan 43 ni ayirish.

$$a=-\frac{32}{24}$$

\frac{-32}{24} ulushini 8 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

$$a=-\frac{4}{3}$$

ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right) formulasi yordamida amalni hisoblang. x_{1} uchun \frac{9}{4} ga va x_{2} uchun -\frac{4}{3} ga bo‘ling.

$$12a^{2}-11a-36=12\left(a-\frac{9}{4}\right)\left(a-\left(-\frac{4}{3}\right)\right)$$

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

$$12a^{2}-11a-36=12\left(a-\frac{9}{4}\right)\left(a+\frac{4}{3}\right)$$

Umumiy maxrajni topib va suratlarni ayirib \frac{9}{4} ni a dan ayirish. So'ngra imkoni boricha kasrni eng kichik shartga qisqartirish.

$$12a^{2}-11a-36=12\times \frac{4a-9}{4}\left(a+\frac{4}{3}\right)$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali \frac{4}{3} ni a ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$12a^{2}-11a-36=12\times \frac{4a-9}{4}\times \frac{3a+4}{3}$$

Raqamlash sonlarini va maxraj sonlariga ko'paytirish orqali \frac{4a-9}{4} ni \frac{3a+4}{3} ga ko'paytirish. So'ngra kasrni imkoni boricha eng kam a'zoga qisqartiring.

$$12a^{2}-11a-36=12\times \frac{\left(4a-9\right)\left(3a+4\right)}{4\times 3}$$

4 ni 3 marotabaga ko'paytirish.

$$12a^{2}-11a-36=12\times \frac{\left(4a-9\right)\left(3a+4\right)}{12}$$

12 va 12 ichida eng katta umumiy 12 faktorini bekor qiling.

$$12a^{2}-11a-36=\left(4a-9\right)\left(3a+4\right)$$

Baholash

$$\left(4a-9\right)\left(3a+4\right)$$