(x-4)(3x+6)+(x-4)(12x+48)=0 eching

$$( x - 4 ) ( 3 x + 6 ) + ( x - 4 ) ( 12 x + 48 ) = 0$$

x uchun yechish

$$x = -\frac{18}{5} = -3\frac{3}{5} = -3,6$$

$$x=4$$

Guruhlash orqali faktorlash qadamlari

x-4 ga 3x+6 ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

$$3x^{2}-6x-24+\left(x-4\right)\left(12x+48\right)=0$$

x-4 ga 12x+48 ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

$$3x^{2}-6x-24+12x^{2}-192=0$$

15x^{2} ni olish uchun 3x^{2} va 12x^{2} ni birlashtirish.

$$15x^{2}-6x-24-192=0$$

-216 olish uchun -24 dan 192 ni ayirish.

$$15x^{2}-6x-216=0$$

Ikki tarafini 3 ga bo‘ling.

$$5x^{2}-2x-72=0$$

Tenglamani yechish uchun guruhlash orqali chap qoʻl tomonni faktorlang. Avvalo, chap qoʻl tomon 5x^{2}+ax+bx-72 sifatida qayta yozilishi kerak. a va b ni topish uchun yechiladigan tizimni sozlang.

$$a+b=-2 ab=5\left(-72\right)=-360$$

ab manfiy boʻlganda, a va b da qarama-qarshi belgilar bor. a+b manfiy boʻlganda, manfiy sonda musbatga nisbatdan kattaroq mutlaq qiymat bor. -360-mahsulotni beruvchi bunday butun juftliklarni roʻyxat qiling.

$$1,-360 2,-180 3,-120 4,-90 5,-72 6,-60 8,-45 9,-40 10,-36 12,-30 15,-24 18,-20$$

Har bir juftlik yigʻindisini hisoblang.

$$1-360=-359 2-180=-178 3-120=-117 4-90=-86 5-72=-67 6-60=-54 8-45=-37 9-40=-31 10-36=-26 12-30=-18 15-24=-9 18-20=-2$$

Yechim – -2 yigʻindisini beruvchi juftlik.

$$a=-20 b=18$$

5x^{2}-2x-72 ni \left(5x^{2}-20x\right)+\left(18x-72\right) sifatida qaytadan yozish.

$$\left(5x^{2}-20x\right)+\left(18x-72\right)$$

Birinchi guruhda 5x ni va ikkinchi guruhda 18 ni faktordan chiqaring.

$$5x\left(x-4\right)+18\left(x-4\right)$$

Distributiv funktsiyasidan foydalangan holda x-4 umumiy terminini chiqaring.

$$\left(x-4\right)\left(5x+18\right)$$

Tenglamani yechish uchun x-4=0 va 5x+18=0 ni yeching.

$$x=4 x=-\frac{18}{5}$$

Kvadrat tenglamalaridan foydalanish qoidalari

x-4 ga 3x+6 ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

$$3x^{2}-6x-24+\left(x-4\right)\left(12x+48\right)=0$$

x-4 ga 12x+48 ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

$$3x^{2}-6x-24+12x^{2}-192=0$$

15x^{2} ni olish uchun 3x^{2} va 12x^{2} ni birlashtirish.

$$15x^{2}-6x-24-192=0$$

-216 olish uchun -24 dan 192 ni ayirish.

$$15x^{2}-6x-216=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 15 ni a, -6 ni b va -216 ni c bilan almashtiring.

$$x=\frac{-\left(-6\right)±\sqrt{\left(-6\right)^{2}-4\times 15\left(-216\right)}}{2\times 15}$$

-6 kvadratini chiqarish.

$$x=\frac{-\left(-6\right)±\sqrt{36-4\times 15\left(-216\right)}}{2\times 15}$$

-4 ni 15 marotabaga ko'paytirish.

$$x=\frac{-\left(-6\right)±\sqrt{36-60\left(-216\right)}}{2\times 15}$$

-60 ni -216 marotabaga ko'paytirish.

$$x=\frac{-\left(-6\right)±\sqrt{36+12960}}{2\times 15}$$

36 ni 12960 ga qo'shish.

$$x=\frac{-\left(-6\right)±\sqrt{12996}}{2\times 15}$$

12996 ning kvadrat ildizini chiqarish.

$$x=\frac{-\left(-6\right)±114}{2\times 15}$$

-6 ning teskarisi 6 ga teng.

$$x=\frac{6±114}{2\times 15}$$

2 ni 15 marotabaga ko'paytirish.

$$x=\frac{6±114}{30}$$

x=\frac{6±114}{30} tenglamasini yeching, bunda ± musbat. 6 ni 114 ga qo'shish.

$$x=\frac{120}{30}$$

120 ni 30 ga bo'lish.

$$x=4$$

x=\frac{6±114}{30} tenglamasini yeching, bunda ± manfiy. 6 dan 114 ni ayirish.

$$x=-\frac{108}{30}$$

\frac{-108}{30} ulushini 6 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

$$x=-\frac{18}{5}$$

Tenglama yechildi.

$$x=4 x=-\frac{18}{5}$$

Kvadratni yakunlash bo'yicha qadamlar

x-4 ga 3x+6 ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

$$3x^{2}-6x-24+\left(x-4\right)\left(12x+48\right)=0$$

x-4 ga 12x+48 ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

$$3x^{2}-6x-24+12x^{2}-192=0$$

15x^{2} ni olish uchun 3x^{2} va 12x^{2} ni birlashtirish.

$$15x^{2}-6x-24-192=0$$

-216 olish uchun -24 dan 192 ni ayirish.

$$15x^{2}-6x-216=0$$

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

$$15x^{2}-6x=216$$

Ikki tarafini 15 ga bo‘ling.

$$\frac{15x^{2}-6x}{15}=\frac{216}{15}$$

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

$$x^{2}+\left(-\frac{6}{15}\right)x=\frac{216}{15}$$

\frac{-6}{15} ulushini 3 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

$$x^{2}-\frac{2}{5}x=\frac{216}{15}$$

\frac{216}{15} ulushini 3 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

$$x^{2}-\frac{2}{5}x=\frac{72}{5}$$

-\frac{2}{5} ni bo‘lish, x shartining koeffitsienti, 2 ga -\frac{1}{5} olish uchun. Keyin, -\frac{1}{5} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}-\frac{2}{5}x+\left(-\frac{1}{5}\right)^{2}=\frac{72}{5}+\left(-\frac{1}{5}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib -\frac{1}{5} kvadratini chiqarish.

$$x^{2}-\frac{2}{5}x+\frac{1}{25}=\frac{72}{5}+\frac{1}{25}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali \frac{72}{5} ni \frac{1}{25} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$x^{2}-\frac{2}{5}x+\frac{1}{25}=\frac{361}{25}$$

x^{2}-\frac{2}{5}x+\frac{1}{25} omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(x-\frac{1}{5}\right)^{2}=\frac{361}{25}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x-\frac{1}{5}\right)^{2}}=\sqrt{\frac{361}{25}}$$

Qisqartirish.

$$x-\frac{1}{5}=\frac{19}{5} x-\frac{1}{5}=-\frac{19}{5}$$

\frac{1}{5} ni tenglamaning ikkala tarafiga qo'shish.

$$x=4 x=-\frac{18}{5}$$