x^2+6/3-7/2=x+15/2 eching

$$\frac{ { x }^{ 2 } +6 }{ 3 } - \frac{ 7 }{ 2 } = \frac{ x+15 }{ 2 }$$

x uchun yechish

$$x = -\frac{9}{2} = -4\frac{1}{2} = -4,5$$

$$x=6$$

Guruhlash orqali faktorlash qadamlari

Tenglamaning ikkala tarafini 6 ga, 3,2 ning eng kichik karralisiga ko‘paytiring.

$$2\left(x^{2}+6\right)-21=3\left(x+15\right)$$

2 ga x^{2}+6 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$2x^{2}+12-21=3\left(x+15\right)$$

-9 olish uchun 12 dan 21 ni ayirish.

$$2x^{2}-9=3\left(x+15\right)$$

3 ga x+15 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$2x^{2}-9=3x+45$$

Ikkala tarafdan 3x ni ayirish.

$$2x^{2}-9-3x=45$$

Ikkala tarafdan 45 ni ayirish.

$$2x^{2}-9-3x-45=0$$

-54 olish uchun -9 dan 45 ni ayirish.

$$2x^{2}-54-3x=0$$

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

$$2x^{2}-3x-54=0$$

Tenglamani yechish uchun guruhlash orqali chap qoʻl tomonni faktorlang. Avvalo, chap qoʻl tomon 2x^{2}+ax+bx-54 sifatida qayta yozilishi kerak. a va b ni topish uchun yechiladigan tizimni sozlang.

$$a+b=-3 ab=2\left(-54\right)=-108$$

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 – -3 yigʻindisini beruvchi juftlik.

$$a=-12 b=9$$

2x^{2}-3x-54 ni \left(2x^{2}-12x\right)+\left(9x-54\right) sifatida qaytadan yozish.

$$\left(2x^{2}-12x\right)+\left(9x-54\right)$$

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

$$2x\left(x-6\right)+9\left(x-6\right)$$

Distributiv funktsiyasidan foydalangan holda x-6 umumiy terminini chiqaring.

$$\left(x-6\right)\left(2x+9\right)$$

Tenglamani yechish uchun x-6=0 va 2x+9=0 ni yeching.

$$x=6 x=-\frac{9}{2}$$

Kvadrat tenglamalaridan foydalanish qoidalari

Tenglamaning ikkala tarafini 6 ga, 3,2 ning eng kichik karralisiga ko‘paytiring.

$$2\left(x^{2}+6\right)-21=3\left(x+15\right)$$

2 ga x^{2}+6 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$2x^{2}+12-21=3\left(x+15\right)$$

-9 olish uchun 12 dan 21 ni ayirish.

$$2x^{2}-9=3\left(x+15\right)$$

3 ga x+15 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$2x^{2}-9=3x+45$$

Ikkala tarafdan 3x ni ayirish.

$$2x^{2}-9-3x=45$$

Ikkala tarafdan 45 ni ayirish.

$$2x^{2}-9-3x-45=0$$

-54 olish uchun -9 dan 45 ni ayirish.

$$2x^{2}-54-3x=0$$

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.

$$2x^{2}-3x-54=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 2 ni a, -3 ni b va -54 ni c bilan almashtiring.

$$x=\frac{-\left(-3\right)±\sqrt{\left(-3\right)^{2}-4\times 2\left(-54\right)}}{2\times 2}$$

-3 kvadratini chiqarish.

$$x=\frac{-\left(-3\right)±\sqrt{9-4\times 2\left(-54\right)}}{2\times 2}$$

-4 ni 2 marotabaga ko'paytirish.

$$x=\frac{-\left(-3\right)±\sqrt{9-8\left(-54\right)}}{2\times 2}$$

-8 ni -54 marotabaga ko'paytirish.

$$x=\frac{-\left(-3\right)±\sqrt{9+432}}{2\times 2}$$

9 ni 432 ga qo'shish.

$$x=\frac{-\left(-3\right)±\sqrt{441}}{2\times 2}$$

441 ning kvadrat ildizini chiqarish.

$$x=\frac{-\left(-3\right)±21}{2\times 2}$$

-3 ning teskarisi 3 ga teng.

$$x=\frac{3±21}{2\times 2}$$

2 ni 2 marotabaga ko'paytirish.

$$x=\frac{3±21}{4}$$

x=\frac{3±21}{4} tenglamasini yeching, bunda ± musbat. 3 ni 21 ga qo'shish.

$$x=\frac{24}{4}$$

24 ni 4 ga bo'lish.

$$x=6$$

x=\frac{3±21}{4} tenglamasini yeching, bunda ± manfiy. 3 dan 21 ni ayirish.

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

\frac{-18}{4} ulushini 2 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

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

Tenglama yechildi.

$$x=6 x=-\frac{9}{2}$$

Kvadratni yakunlash bo'yicha qadamlar

Tenglamaning ikkala tarafini 6 ga, 3,2 ning eng kichik karralisiga ko‘paytiring.

$$2\left(x^{2}+6\right)-21=3\left(x+15\right)$$

2 ga x^{2}+6 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$2x^{2}+12-21=3\left(x+15\right)$$

-9 olish uchun 12 dan 21 ni ayirish.

$$2x^{2}-9=3\left(x+15\right)$$

3 ga x+15 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$2x^{2}-9=3x+45$$

Ikkala tarafdan 3x ni ayirish.

$$2x^{2}-9-3x=45$$

9 ni ikki tarafga qo’shing.

$$2x^{2}-3x=45+9$$

54 olish uchun 45 va 9'ni qo'shing.

$$2x^{2}-3x=54$$

Ikki tarafini 2 ga bo‘ling.

$$\frac{2x^{2}-3x}{2}=\frac{54}{2}$$

2 ga bo'lish 2 ga ko'paytirishni bekor qiladi.

$$x^{2}-\frac{3}{2}x=\frac{54}{2}$$

54 ni 2 ga bo'lish.

$$x^{2}-\frac{3}{2}x=27$$

-\frac{3}{2} ni bo‘lish, x shartining koeffitsienti, 2 ga -\frac{3}{4} olish uchun. Keyin, -\frac{3}{4} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

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

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib -\frac{3}{4} kvadratini chiqarish.

$$x^{2}-\frac{3}{2}x+\frac{9}{16}=27+\frac{9}{16}$$

27 ni \frac{9}{16} ga qo'shish.

$$x^{2}-\frac{3}{2}x+\frac{9}{16}=\frac{441}{16}$$

x^{2}-\frac{3}{2}x+\frac{9}{16} omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(x-\frac{3}{4}\right)^{2}=\frac{441}{16}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x-\frac{3}{4}\right)^{2}}=\sqrt{\frac{441}{16}}$$

Qisqartirish.

$$x-\frac{3}{4}=\frac{21}{4} x-\frac{3}{4}=-\frac{21}{4}$$

\frac{3}{4} ni tenglamaning ikkala tarafiga qo'shish.

$$x=6 x=-\frac{9}{2}$$