15x^2-4x-4= eching

$$15 x ^ { 2 } - 4 x - 4 =$$

Omil

$$\left(3x-2\right)\left(5x+2\right)$$

Yechim qadamlari

Ifodani guruhlash orqali faktorlang. Avvalo, ifoda 15x^{2}+ax+bx-4 sifatida qayta yozilishi kerak. a va b ni topish uchun yechiladigan tizimni sozlang.

$$a+b=-4 ab=15\left(-4\right)=-60$$

ab manfiy boʻlganda, a va b da qarama-qarshi belgilar bor. a+b manfiy boʻlganda, manfiy sonda musbatga nisbatdan kattaroq mutlaq qiymat bor. -60-mahsulotni beruvchi bunday butun juftliklarni roʻyxat qiling.

$$1,-60 2,-30 3,-20 4,-15 5,-12 6,-10$$

Har bir juftlik yigʻindisini hisoblang.

$$1-60=-59 2-30=-28 3-20=-17 4-15=-11 5-12=-7 6-10=-4$$

Yechim – -4 yigʻindisini beruvchi juftlik.

$$a=-10 b=6$$

15x^{2}-4x-4 ni \left(15x^{2}-10x\right)+\left(6x-4\right) sifatida qaytadan yozish.

$$\left(15x^{2}-10x\right)+\left(6x-4\right)$$

Birinchi guruhda 5x ni va ikkinchi guruhda 2 ni faktordan chiqaring.

$$5x\left(3x-2\right)+2\left(3x-2\right)$$

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

$$\left(3x-2\right)\left(5x+2\right)$$

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.

$$15x^{2}-4x-4=0$$

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.

$$x=\frac{-\left(-4\right)±\sqrt{\left(-4\right)^{2}-4\times 15\left(-4\right)}}{2\times 15}$$

-4 kvadratini chiqarish.

$$x=\frac{-\left(-4\right)±\sqrt{16-4\times 15\left(-4\right)}}{2\times 15}$$

-4 ni 15 marotabaga ko'paytirish.

$$x=\frac{-\left(-4\right)±\sqrt{16-60\left(-4\right)}}{2\times 15}$$

-60 ni -4 marotabaga ko'paytirish.

$$x=\frac{-\left(-4\right)±\sqrt{16+240}}{2\times 15}$$

16 ni 240 ga qo'shish.

$$x=\frac{-\left(-4\right)±\sqrt{256}}{2\times 15}$$

256 ning kvadrat ildizini chiqarish.

$$x=\frac{-\left(-4\right)±16}{2\times 15}$$

-4 ning teskarisi 4 ga teng.

$$x=\frac{4±16}{2\times 15}$$

2 ni 15 marotabaga ko'paytirish.

$$x=\frac{4±16}{30}$$

x=\frac{4±16}{30} tenglamasini yeching, bunda ± musbat. 4 ni 16 ga qo'shish.

$$x=\frac{20}{30}$$

\frac{20}{30} ulushini 10 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

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

x=\frac{4±16}{30} tenglamasini yeching, bunda ± manfiy. 4 dan 16 ni ayirish.

$$x=-\frac{12}{30}$$

\frac{-12}{30} ulushini 6 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

$$x=-\frac{2}{5}$$

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}{5} ga bo‘ling.

$$15x^{2}-4x-4=15\left(x-\frac{2}{3}\right)\left(x-\left(-\frac{2}{5}\right)\right)$$

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

$$15x^{2}-4x-4=15\left(x-\frac{2}{3}\right)\left(x+\frac{2}{5}\right)$$

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

$$15x^{2}-4x-4=15\times \frac{3x-2}{3}\left(x+\frac{2}{5}\right)$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali \frac{2}{5} ni x ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$15x^{2}-4x-4=15\times \frac{3x-2}{3}\times \frac{5x+2}{5}$$

Raqamlash sonlarini va maxraj sonlariga ko'paytirish orqali \frac{3x-2}{3} ni \frac{5x+2}{5} ga ko'paytirish. So'ngra kasrni imkoni boricha eng kam a'zoga qisqartiring.

$$15x^{2}-4x-4=15\times \frac{\left(3x-2\right)\left(5x+2\right)}{3\times 5}$$

3 ni 5 marotabaga ko'paytirish.

$$15x^{2}-4x-4=15\times \frac{\left(3x-2\right)\left(5x+2\right)}{15}$$

15 va 15 ichida eng katta umumiy 15 faktorini bekor qiling.

$$15x^{2}-4x-4=\left(3x-2\right)\left(5x+2\right)$$

Baholash

$$\left(3x-2\right)\left(5x+2\right)$$