15x^2-12=8x eching

$$15 { x }^{ 2 } -12=8x$$

x uchun yechish

$$x=-\frac{2}{3}\approx -0,666666667$$

$$x = \frac{6}{5} = 1\frac{1}{5} = 1,2$$

Guruhlash orqali faktorlash qadamlari

Ikkala tarafdan 8x ni ayirish.

$$15x^{2}-12-8x=0$$

Polinomni standart shaklga keltirish uchun uni qayta tartiblang. Shartlarni eng yuqoridan eng pastki qiymat ko'rsatgichiga joylashtirish.

$$15x^{2}-8x-12=0$$

Tenglamani yechish uchun guruhlash orqali chap qoʻl tomonni faktorlang. Avvalo, chap qoʻl tomon 15x^{2}+ax+bx-12 sifatida qayta yozilishi kerak. a va b ni topish uchun yechiladigan tizimni sozlang.

$$a+b=-8 ab=15\left(-12\right)=-180$$

ab manfiy boʻlganda, a va b da qarama-qarshi belgilar bor. a+b manfiy boʻlganda, manfiy sonda musbatga nisbatdan kattaroq mutlaq qiymat bor. -180-mahsulotni beruvchi bunday butun juftliklarni roʻyxat qiling.

$$1,-180 2,-90 3,-60 4,-45 5,-36 6,-30 9,-20 10,-18 12,-15$$

Har bir juftlik yigʻindisini hisoblang.

$$1-180=-179 2-90=-88 3-60=-57 4-45=-41 5-36=-31 6-30=-24 9-20=-11 10-18=-8 12-15=-3$$

Yechim – -8 yigʻindisini beruvchi juftlik.

$$a=-18 b=10$$

15x^{2}-8x-12 ni \left(15x^{2}-18x\right)+\left(10x-12\right) sifatida qaytadan yozish.

$$\left(15x^{2}-18x\right)+\left(10x-12\right)$$

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

$$3x\left(5x-6\right)+2\left(5x-6\right)$$

Distributiv funktsiyasidan foydalangan holda 5x-6 umumiy terminini chiqaring.

$$\left(5x-6\right)\left(3x+2\right)$$

Tenglamani yechish uchun 5x-6=0 va 3x+2=0 ni yeching.

$$x=\frac{6}{5} x=-\frac{2}{3}$$

Kvadrat tenglamalaridan foydalanish qoidalari

Ikkala tarafdan 8x ni ayirish.

$$15x^{2}-12-8x=0$$

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.

$$15x^{2}-8x-12=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 15 ni a, -8 ni b va -12 ni c bilan almashtiring.

$$x=\frac{-\left(-8\right)±\sqrt{\left(-8\right)^{2}-4\times 15\left(-12\right)}}{2\times 15}$$

-8 kvadratini chiqarish.

$$x=\frac{-\left(-8\right)±\sqrt{64-4\times 15\left(-12\right)}}{2\times 15}$$

-4 ni 15 marotabaga ko'paytirish.

$$x=\frac{-\left(-8\right)±\sqrt{64-60\left(-12\right)}}{2\times 15}$$

-60 ni -12 marotabaga ko'paytirish.

$$x=\frac{-\left(-8\right)±\sqrt{64+720}}{2\times 15}$$

64 ni 720 ga qo'shish.

$$x=\frac{-\left(-8\right)±\sqrt{784}}{2\times 15}$$

784 ning kvadrat ildizini chiqarish.

$$x=\frac{-\left(-8\right)±28}{2\times 15}$$

-8 ning teskarisi 8 ga teng.

$$x=\frac{8±28}{2\times 15}$$

2 ni 15 marotabaga ko'paytirish.

$$x=\frac{8±28}{30}$$

x=\frac{8±28}{30} tenglamasini yeching, bunda ± musbat. 8 ni 28 ga qo'shish.

$$x=\frac{36}{30}$$

\frac{36}{30} ulushini 6 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

$$x=\frac{6}{5}$$

x=\frac{8±28}{30} tenglamasini yeching, bunda ± manfiy. 8 dan 28 ni ayirish.

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

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

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

Tenglama yechildi.

$$x=\frac{6}{5} x=-\frac{2}{3}$$

Kvadratni yakunlash bo'yicha qadamlar

Ikkala tarafdan 8x ni ayirish.

$$15x^{2}-12-8x=0$$

12 ni ikki tarafga qo’shing. Har qanday songa nolni qo‘shsangiz, o‘zi chiqadi.

$$15x^{2}-8x=12$$

Ikki tarafini 15 ga bo‘ling.

$$\frac{15x^{2}-8x}{15}=\frac{12}{15}$$

15 ga bo'lish 15 ga ko'paytirishni bekor qiladi.

$$x^{2}-\frac{8}{15}x=\frac{12}{15}$$

\frac{12}{15} ulushini 3 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

$$x^{2}-\frac{8}{15}x=\frac{4}{5}$$

-\frac{8}{15} ni bo‘lish, x shartining koeffitsienti, 2 ga -\frac{4}{15} olish uchun. Keyin, -\frac{4}{15} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}-\frac{8}{15}x+\left(-\frac{4}{15}\right)^{2}=\frac{4}{5}+\left(-\frac{4}{15}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib -\frac{4}{15} kvadratini chiqarish.

$$x^{2}-\frac{8}{15}x+\frac{16}{225}=\frac{4}{5}+\frac{16}{225}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali \frac{4}{5} ni \frac{16}{225} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$x^{2}-\frac{8}{15}x+\frac{16}{225}=\frac{196}{225}$$

x^{2}-\frac{8}{15}x+\frac{16}{225} omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(x-\frac{4}{15}\right)^{2}=\frac{196}{225}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x-\frac{4}{15}\right)^{2}}=\sqrt{\frac{196}{225}}$$

Qisqartirish.

$$x-\frac{4}{15}=\frac{14}{15} x-\frac{4}{15}=-\frac{14}{15}$$

\frac{4}{15} ni tenglamaning ikkala tarafiga qo'shish.

$$x=\frac{6}{5} x=-\frac{2}{3}$$