27x^2-12x-4 eching

$$27 x ^ { 2 } - 12 x - 4$$

Omil

$$\left(3x-2\right)\left(9x+2\right)$$

Yechim qadamlari

Ifodani guruhlash orqali faktorlang. Avvalo, ifoda 27x^{2}+ax+bx-4 sifatida qayta yozilishi kerak. a va b ni topish uchun yechiladigan tizimni sozlang.

$$a+b=-12 ab=27\left(-4\right)=-108$$

ab manfiy boʻlganda, a va b da qarama-qarshi belgilar bor. a+b manfiy boʻlganda, manfiy sonda musbatga nisbatdan kattaroq mutlaq qiymat bor. -108-mahsulotni beruvchi bunday butun juftliklarni roʻyxat qiling.

$$1,-108 2,-54 3,-36 4,-27 6,-18 9,-12$$

Har bir juftlik yigʻindisini hisoblang.

$$1-108=-107 2-54=-52 3-36=-33 4-27=-23 6-18=-12 9-12=-3$$

Yechim – -12 yigʻindisini beruvchi juftlik.

$$a=-18 b=6$$

27x^{2}-12x-4 ni \left(27x^{2}-18x\right)+\left(6x-4\right) sifatida qaytadan yozish.

$$\left(27x^{2}-18x\right)+\left(6x-4\right)$$

Birinchi guruhda 9x ni va ikkinchi guruhda 2 ni faktordan chiqaring.

$$9x\left(3x-2\right)+2\left(3x-2\right)$$

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

$$\left(3x-2\right)\left(9x+2\right)$$

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.

$$27x^{2}-12x-4=0$$

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.

$$x=\frac{-\left(-12\right)±\sqrt{\left(-12\right)^{2}-4\times 27\left(-4\right)}}{2\times 27}$$

-12 kvadratini chiqarish.

$$x=\frac{-\left(-12\right)±\sqrt{144-4\times 27\left(-4\right)}}{2\times 27}$$

-4 ni 27 marotabaga ko'paytirish.

$$x=\frac{-\left(-12\right)±\sqrt{144-108\left(-4\right)}}{2\times 27}$$

-108 ni -4 marotabaga ko'paytirish.

$$x=\frac{-\left(-12\right)±\sqrt{144+432}}{2\times 27}$$

144 ni 432 ga qo'shish.

$$x=\frac{-\left(-12\right)±\sqrt{576}}{2\times 27}$$

576 ning kvadrat ildizini chiqarish.

$$x=\frac{-\left(-12\right)±24}{2\times 27}$$

-12 ning teskarisi 12 ga teng.

$$x=\frac{12±24}{2\times 27}$$

2 ni 27 marotabaga ko'paytirish.

$$x=\frac{12±24}{54}$$

x=\frac{12±24}{54} tenglamasini yeching, bunda ± musbat. 12 ni 24 ga qo'shish.

$$x=\frac{36}{54}$$

\frac{36}{54} ulushini 18 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

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

x=\frac{12±24}{54} tenglamasini yeching, bunda ± manfiy. 12 dan 24 ni ayirish.

$$x=-\frac{12}{54}$$

\frac{-12}{54} ulushini 6 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

$$x=-\frac{2}{9}$$

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{2}{9} ga bo‘ling.

$$27x^{2}-12x-4=27\left(x-\frac{2}{3}\right)\left(x-\left(-\frac{2}{9}\right)\right)$$

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

$$27x^{2}-12x-4=27\left(x-\frac{2}{3}\right)\left(x+\frac{2}{9}\right)$$

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

$$27x^{2}-12x-4=27\times \frac{3x-2}{3}\left(x+\frac{2}{9}\right)$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali \frac{2}{9} ni x ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$27x^{2}-12x-4=27\times \frac{3x-2}{3}\times \frac{9x+2}{9}$$

Raqamlash sonlarini va maxraj sonlariga ko'paytirish orqali \frac{3x-2}{3} ni \frac{9x+2}{9} ga ko'paytirish. So'ngra kasrni imkoni boricha eng kam a'zoga qisqartiring.

$$27x^{2}-12x-4=27\times \frac{\left(3x-2\right)\left(9x+2\right)}{3\times 9}$$

3 ni 9 marotabaga ko'paytirish.

$$27x^{2}-12x-4=27\times \frac{\left(3x-2\right)\left(9x+2\right)}{27}$$

27 va 27 ichida eng katta umumiy 27 faktorini bekor qiling.

$$27x^{2}-12x-4=\left(3x-2\right)\left(9x+2\right)$$

Baholash

$$\left(3x-2\right)\left(9x+2\right)$$