-9x=2(3x^2+4)+10x eching

$$- 9 x = 2 ( 3 x ^ { 2 } + 4 ) + 10 x$$

x uchun yechish

$$x=-\frac{1}{2}=-0,5$$

$$x = -\frac{8}{3} = -2\frac{2}{3} \approx -2,666666667$$

Guruhlash orqali faktorlash qadamlari

2 ga 3x^{2}+4 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$-9x=6x^{2}+8+10x$$

Ikkala tarafdan 6x^{2} ni ayirish.

$$-9x-6x^{2}=8+10x$$

Ikkala tarafdan 8 ni ayirish.

$$-9x-6x^{2}-8=10x$$

Ikkala tarafdan 10x ni ayirish.

$$-9x-6x^{2}-8-10x=0$$

-19x ni olish uchun -9x va -10x ni birlashtirish.

$$-19x-6x^{2}-8=0$$

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

$$-6x^{2}-19x-8=0$$

Tenglamani yechish uchun guruhlash orqali chap qoʻl tomonni faktorlang. Avvalo, chap qoʻl tomon -6x^{2}+ax+bx-8 sifatida qayta yozilishi kerak. a va b ni topish uchun yechiladigan tizimni sozlang.

$$a+b=-19 ab=-6\left(-8\right)=48$$

ab musbat boʻlganda, a va b da bir xil belgi bor. a+b manfiy boʻlganda, a va b ikkisi ham manfiy. 48-mahsulotni beruvchi bunday butun juftliklarni roʻyxat qiling.

$$-1,-48 -2,-24 -3,-16 -4,-12 -6,-8$$

Har bir juftlik yigʻindisini hisoblang.

$$-1-48=-49 -2-24=-26 -3-16=-19 -4-12=-16 -6-8=-14$$

Yechim – -19 yigʻindisini beruvchi juftlik.

$$a=-3 b=-16$$

-6x^{2}-19x-8 ni \left(-6x^{2}-3x\right)+\left(-16x-8\right) sifatida qaytadan yozish.

$$\left(-6x^{2}-3x\right)+\left(-16x-8\right)$$

Birinchi guruhda -3x ni va ikkinchi guruhda -8 ni faktordan chiqaring.

$$-3x\left(2x+1\right)-8\left(2x+1\right)$$

Distributiv funktsiyasidan foydalangan holda 2x+1 umumiy terminini chiqaring.

$$\left(2x+1\right)\left(-3x-8\right)$$

Tenglamani yechish uchun 2x+1=0 va -3x-8=0 ni yeching.

$$x=-\frac{1}{2} x=-\frac{8}{3}$$

Kvadrat tenglamalaridan foydalanish qoidalari

2 ga 3x^{2}+4 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$-9x=6x^{2}+8+10x$$

Ikkala tarafdan 6x^{2} ni ayirish.

$$-9x-6x^{2}=8+10x$$

Ikkala tarafdan 8 ni ayirish.

$$-9x-6x^{2}-8=10x$$

Ikkala tarafdan 10x ni ayirish.

$$-9x-6x^{2}-8-10x=0$$

-19x ni olish uchun -9x va -10x ni birlashtirish.

$$-19x-6x^{2}-8=0$$

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.

$$-6x^{2}-19x-8=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} -6 ni a, -19 ni b va -8 ni c bilan almashtiring.

$$x=\frac{-\left(-19\right)±\sqrt{\left(-19\right)^{2}-4\left(-6\right)\left(-8\right)}}{2\left(-6\right)}$$

-19 kvadratini chiqarish.

$$x=\frac{-\left(-19\right)±\sqrt{361-4\left(-6\right)\left(-8\right)}}{2\left(-6\right)}$$

-4 ni -6 marotabaga ko'paytirish.

$$x=\frac{-\left(-19\right)±\sqrt{361+24\left(-8\right)}}{2\left(-6\right)}$$

24 ni -8 marotabaga ko'paytirish.

$$x=\frac{-\left(-19\right)±\sqrt{361-192}}{2\left(-6\right)}$$

361 ni -192 ga qo'shish.

$$x=\frac{-\left(-19\right)±\sqrt{169}}{2\left(-6\right)}$$

169 ning kvadrat ildizini chiqarish.

$$x=\frac{-\left(-19\right)±13}{2\left(-6\right)}$$

-19 ning teskarisi 19 ga teng.

$$x=\frac{19±13}{2\left(-6\right)}$$

2 ni -6 marotabaga ko'paytirish.

$$x=\frac{19±13}{-12}$$

x=\frac{19±13}{-12} tenglamasini yeching, bunda ± musbat. 19 ni 13 ga qo'shish.

$$x=\frac{32}{-12}$$

\frac{32}{-12} ulushini 4 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

$$x=-\frac{8}{3}$$

x=\frac{19±13}{-12} tenglamasini yeching, bunda ± manfiy. 19 dan 13 ni ayirish.

$$x=\frac{6}{-12}$$

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

$$x=-\frac{1}{2}$$

Tenglama yechildi.

$$x=-\frac{8}{3} x=-\frac{1}{2}$$

Kvadratni yakunlash bo'yicha qadamlar

2 ga 3x^{2}+4 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$-9x=6x^{2}+8+10x$$

Ikkala tarafdan 6x^{2} ni ayirish.

$$-9x-6x^{2}=8+10x$$

Ikkala tarafdan 10x ni ayirish.

$$-9x-6x^{2}-10x=8$$

-19x ni olish uchun -9x va -10x ni birlashtirish.

$$-19x-6x^{2}=8$$

Bu kabi kvadrat tenglamalarni kvadratni yakunlab yechish mumkin. Kvadratni yechish uchun tenglama avval ushbu shaklda bo'lishi shart: x^{2}+bx=c.

$$-6x^{2}-19x=8$$

Ikki tarafini -6 ga bo‘ling.

$$\frac{-6x^{2}-19x}{-6}=\frac{8}{-6}$$

-6 ga bo'lish -6 ga ko'paytirishni bekor qiladi.

$$x^{2}+\left(-\frac{19}{-6}\right)x=\frac{8}{-6}$$

-19 ni -6 ga bo'lish.

$$x^{2}+\frac{19}{6}x=\frac{8}{-6}$$

\frac{8}{-6} ulushini 2 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

$$x^{2}+\frac{19}{6}x=-\frac{4}{3}$$

\frac{19}{6} ni bo‘lish, x shartining koeffitsienti, 2 ga \frac{19}{12} olish uchun. Keyin, \frac{19}{12} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}+\frac{19}{6}x+\left(\frac{19}{12}\right)^{2}=-\frac{4}{3}+\left(\frac{19}{12}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib \frac{19}{12} kvadratini chiqarish.

$$x^{2}+\frac{19}{6}x+\frac{361}{144}=-\frac{4}{3}+\frac{361}{144}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali -\frac{4}{3} ni \frac{361}{144} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$x^{2}+\frac{19}{6}x+\frac{361}{144}=\frac{169}{144}$$

x^{2}+\frac{19}{6}x+\frac{361}{144} omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(x+\frac{19}{12}\right)^{2}=\frac{169}{144}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x+\frac{19}{12}\right)^{2}}=\sqrt{\frac{169}{144}}$$

Qisqartirish.

$$x+\frac{19}{12}=\frac{13}{12} x+\frac{19}{12}=-\frac{13}{12}$$

Tenglamaning ikkala tarafidan \frac{19}{12} ni ayirish.

$$x=-\frac{1}{2} x=-\frac{8}{3}$$