12x^2-5x-2 eching

$$12 x ^ { 2 } - 5 x - 2$$

Omil

$$\left(3x-2\right)\left(4x+1\right)$$

Yechim qadamlari

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

$$a+b=-5 ab=12\left(-2\right)=-24$$

ab manfiy boʻlganda, a va b da qarama-qarshi belgilar bor. a+b manfiy boʻlganda, manfiy sonda musbatga nisbatdan kattaroq mutlaq qiymat bor. -24-mahsulotni beruvchi bunday butun juftliklarni roʻyxat qiling.

$$1,-24 2,-12 3,-8 4,-6$$

Har bir juftlik yigʻindisini hisoblang.

$$1-24=-23 2-12=-10 3-8=-5 4-6=-2$$

Yechim – -5 yigʻindisini beruvchi juftlik.

$$a=-8 b=3$$

12x^{2}-5x-2 ni \left(12x^{2}-8x\right)+\left(3x-2\right) sifatida qaytadan yozish.

$$\left(12x^{2}-8x\right)+\left(3x-2\right)$$

12x^{2}-8x ichida 4x ni ajrating.

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

Distributiv funktsiyasidan foydalangan holda 3x-2 umumiy terminini chiqaring.

$$\left(3x-2\right)\left(4x+1\right)$$

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.

$$12x^{2}-5x-2=0$$

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.

$$x=\frac{-\left(-5\right)±\sqrt{\left(-5\right)^{2}-4\times 12\left(-2\right)}}{2\times 12}$$

-5 kvadratini chiqarish.

$$x=\frac{-\left(-5\right)±\sqrt{25-4\times 12\left(-2\right)}}{2\times 12}$$

-4 ni 12 marotabaga ko'paytirish.

$$x=\frac{-\left(-5\right)±\sqrt{25-48\left(-2\right)}}{2\times 12}$$

-48 ni -2 marotabaga ko'paytirish.

$$x=\frac{-\left(-5\right)±\sqrt{25+96}}{2\times 12}$$

25 ni 96 ga qo'shish.

$$x=\frac{-\left(-5\right)±\sqrt{121}}{2\times 12}$$

121 ning kvadrat ildizini chiqarish.

$$x=\frac{-\left(-5\right)±11}{2\times 12}$$

-5 ning teskarisi 5 ga teng.

$$x=\frac{5±11}{2\times 12}$$

2 ni 12 marotabaga ko'paytirish.

$$x=\frac{5±11}{24}$$

x=\frac{5±11}{24} tenglamasini yeching, bunda ± musbat. 5 ni 11 ga qo'shish.

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

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

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

x=\frac{5±11}{24} tenglamasini yeching, bunda ± manfiy. 5 dan 11 ni ayirish.

$$x=-\frac{6}{24}$$

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

$$x=-\frac{1}{4}$$

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

$$12x^{2}-5x-2=12\left(x-\frac{2}{3}\right)\left(x-\left(-\frac{1}{4}\right)\right)$$

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

$$12x^{2}-5x-2=12\left(x-\frac{2}{3}\right)\left(x+\frac{1}{4}\right)$$

Umumiy maxrajni topib va suratlarni ayirib \frac{2}{3} ni x dan ayirish. So'ngra imkoni boricha kasrni eng kichik shartga qisqartirish.

$$12x^{2}-5x-2=12\times \frac{3x-2}{3}\left(x+\frac{1}{4}\right)$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali \frac{1}{4} ni x ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$12x^{2}-5x-2=12\times \frac{3x-2}{3}\times \frac{4x+1}{4}$$

Raqamlash sonlarini va maxraj sonlariga ko'paytirish orqali \frac{3x-2}{3} ni \frac{4x+1}{4} ga ko'paytirish. So'ngra kasrni imkoni boricha eng kam a'zoga qisqartiring.

$$12x^{2}-5x-2=12\times \frac{\left(3x-2\right)\left(4x+1\right)}{3\times 4}$$

3 ni 4 marotabaga ko'paytirish.

$$12x^{2}-5x-2=12\times \frac{\left(3x-2\right)\left(4x+1\right)}{12}$$

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

$$12x^{2}-5x-2=\left(3x-2\right)\left(4x+1\right)$$

Baholash

$$\left(3x-2\right)\left(4x+1\right)$$