12x^2-x-6 eching

$$12 x ^ { 2 } - x - 6$$

Omil

$$\left(4x-3\right)\left(3x+2\right)$$

Yechim qadamlari

Ifodani guruhlash orqali faktorlang. Avvalo, ifoda 12x^{2}+ax+bx-6 sifatida qayta yozilishi kerak. a va b ni topish uchun yechiladigan tizimni sozlang.

$$a+b=-1 ab=12\left(-6\right)=-72$$

ab manfiy boʻlganda, a va b da qarama-qarshi belgilar bor. a+b manfiy boʻlganda, manfiy sonda musbatga nisbatdan kattaroq mutlaq qiymat bor. -72-mahsulotni beruvchi bunday butun juftliklarni roʻyxat qiling.

$$1,-72 2,-36 3,-24 4,-18 6,-12 8,-9$$

Har bir juftlik yigʻindisini hisoblang.

$$1-72=-71 2-36=-34 3-24=-21 4-18=-14 6-12=-6 8-9=-1$$

Yechim – -1 yigʻindisini beruvchi juftlik.

$$a=-9 b=8$$

12x^{2}-x-6 ni \left(12x^{2}-9x\right)+\left(8x-6\right) sifatida qaytadan yozish.

$$\left(12x^{2}-9x\right)+\left(8x-6\right)$$

Birinchi guruhda 3x ni va ikkinchi guruhda 2 ni faktordan chiqaring.

$$3x\left(4x-3\right)+2\left(4x-3\right)$$

Distributiv funktsiyasidan foydalangan holda 4x-3 umumiy terminini chiqaring.

$$\left(4x-3\right)\left(3x+2\right)$$

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.

$$12x^{2}-x-6=0$$

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.

$$x=\frac{-\left(-1\right)±\sqrt{1-4\times 12\left(-6\right)}}{2\times 12}$$

-4 ni 12 marotabaga ko'paytirish.

$$x=\frac{-\left(-1\right)±\sqrt{1-48\left(-6\right)}}{2\times 12}$$

-48 ni -6 marotabaga ko'paytirish.

$$x=\frac{-\left(-1\right)±\sqrt{1+288}}{2\times 12}$$

1 ni 288 ga qo'shish.

$$x=\frac{-\left(-1\right)±\sqrt{289}}{2\times 12}$$

289 ning kvadrat ildizini chiqarish.

$$x=\frac{-\left(-1\right)±17}{2\times 12}$$

-1 ning teskarisi 1 ga teng.

$$x=\frac{1±17}{2\times 12}$$

2 ni 12 marotabaga ko'paytirish.

$$x=\frac{1±17}{24}$$

x=\frac{1±17}{24} tenglamasini yeching, bunda ± musbat. 1 ni 17 ga qo'shish.

$$x=\frac{18}{24}$$

\frac{18}{24} ulushini 6 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

$$x=\frac{3}{4}$$

x=\frac{1±17}{24} tenglamasini yeching, bunda ± manfiy. 1 dan 17 ni ayirish.

$$x=-\frac{16}{24}$$

\frac{-16}{24} ulushini 8 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

$$x=-\frac{2}{3}$$

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

$$12x^{2}-x-6=12\left(x-\frac{3}{4}\right)\left(x-\left(-\frac{2}{3}\right)\right)$$

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

$$12x^{2}-x-6=12\left(x-\frac{3}{4}\right)\left(x+\frac{2}{3}\right)$$

Umumiy maxrajni topib va suratlarni ayirib \frac{3}{4} ni x dan ayirish. So'ngra imkoni boricha kasrni eng kichik shartga qisqartirish.

$$12x^{2}-x-6=12\times \frac{4x-3}{4}\left(x+\frac{2}{3}\right)$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali \frac{2}{3} ni x ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$12x^{2}-x-6=12\times \frac{4x-3}{4}\times \frac{3x+2}{3}$$

Raqamlash sonlarini va maxraj sonlariga ko'paytirish orqali \frac{4x-3}{4} ni \frac{3x+2}{3} ga ko'paytirish. So'ngra kasrni imkoni boricha eng kam a'zoga qisqartiring.

$$12x^{2}-x-6=12\times \frac{\left(4x-3\right)\left(3x+2\right)}{4\times 3}$$

4 ni 3 marotabaga ko'paytirish.

$$12x^{2}-x-6=12\times \frac{\left(4x-3\right)\left(3x+2\right)}{12}$$

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

$$12x^{2}-x-6=\left(4x-3\right)\left(3x+2\right)$$

Baholash

$$\left(4x-3\right)\left(3x+2\right)$$