4(1+9x^2+6x)=56x+84 eching

$$4(1+9 { x }^{ 2 } +6x)=56x+84$$

x uchun yechish

$$x = -\frac{10}{9} = -1\frac{1}{9} \approx -1,111111111$$

$$x=2$$

Guruhlash orqali faktorlash qadamlari

4 ga 1+9x^{2}+6x ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$4+36x^{2}+24x=56x+84$$

Ikkala tarafdan 56x ni ayirish.

$$4+36x^{2}+24x-56x=84$$

-32x ni olish uchun 24x va -56x ni birlashtirish.

$$4+36x^{2}-32x=84$$

Ikkala tarafdan 84 ni ayirish.

$$4+36x^{2}-32x-84=0$$

-80 olish uchun 4 dan 84 ni ayirish.

$$-80+36x^{2}-32x=0$$

Ikki tarafini 4 ga bo‘ling.

$$-20+9x^{2}-8x=0$$

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

$$9x^{2}-8x-20=0$$

Tenglamani yechish uchun guruhlash orqali chap qoʻl tomonni faktorlang. Avvalo, chap qoʻl tomon 9x^{2}+ax+bx-20 sifatida qayta yozilishi kerak. a va b ni topish uchun yechiladigan tizimni sozlang.

$$a+b=-8 ab=9\left(-20\right)=-180$$

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.

$$a=-18 b=10$$

9x^{2}-8x-20 ni \left(9x^{2}-18x\right)+\left(10x-20\right) sifatida qaytadan yozish.

$$\left(9x^{2}-18x\right)+\left(10x-20\right)$$

Birinchi guruhda 9x ni va ikkinchi guruhda 10 ni faktordan chiqaring.

$$9x\left(x-2\right)+10\left(x-2\right)$$

Distributiv funktsiyasidan foydalangan holda x-2 umumiy terminini chiqaring.

$$\left(x-2\right)\left(9x+10\right)$$

Tenglamani yechish uchun x-2=0 va 9x+10=0 ni yeching.

$$x=2 x=-\frac{10}{9}$$

Kvadrat tenglamalaridan foydalanish qoidalari

4 ga 1+9x^{2}+6x ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$4+36x^{2}+24x=56x+84$$

Ikkala tarafdan 56x ni ayirish.

$$4+36x^{2}+24x-56x=84$$

-32x ni olish uchun 24x va -56x ni birlashtirish.

$$4+36x^{2}-32x=84$$

Ikkala tarafdan 84 ni ayirish.

$$4+36x^{2}-32x-84=0$$

-80 olish uchun 4 dan 84 ni ayirish.

$$-80+36x^{2}-32x=0$$

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.

$$36x^{2}-32x-80=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 36 ni a, -32 ni b va -80 ni c bilan almashtiring.

$$x=\frac{-\left(-32\right)±\sqrt{\left(-32\right)^{2}-4\times 36\left(-80\right)}}{2\times 36}$$

-32 kvadratini chiqarish.

$$x=\frac{-\left(-32\right)±\sqrt{1024-4\times 36\left(-80\right)}}{2\times 36}$$

-4 ni 36 marotabaga ko'paytirish.

$$x=\frac{-\left(-32\right)±\sqrt{1024-144\left(-80\right)}}{2\times 36}$$

-144 ni -80 marotabaga ko'paytirish.

$$x=\frac{-\left(-32\right)±\sqrt{1024+11520}}{2\times 36}$$

1024 ni 11520 ga qo'shish.

$$x=\frac{-\left(-32\right)±\sqrt{12544}}{2\times 36}$$

12544 ning kvadrat ildizini chiqarish.

$$x=\frac{-\left(-32\right)±112}{2\times 36}$$

-32 ning teskarisi 32 ga teng.

$$x=\frac{32±112}{2\times 36}$$

2 ni 36 marotabaga ko'paytirish.

$$x=\frac{32±112}{72}$$

x=\frac{32±112}{72} tenglamasini yeching, bunda ± musbat. 32 ni 112 ga qo'shish.

$$x=\frac{144}{72}$$

144 ni 72 ga bo'lish.

$$x=2$$

x=\frac{32±112}{72} tenglamasini yeching, bunda ± manfiy. 32 dan 112 ni ayirish.

$$x=-\frac{80}{72}$$

\frac{-80}{72} ulushini 8 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

$$x=-\frac{10}{9}$$

Tenglama yechildi.

$$x=2 x=-\frac{10}{9}$$

Kvadratni yakunlash bo'yicha qadamlar

4 ga 1+9x^{2}+6x ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$4+36x^{2}+24x=56x+84$$

Ikkala tarafdan 56x ni ayirish.

$$4+36x^{2}+24x-56x=84$$

-32x ni olish uchun 24x va -56x ni birlashtirish.

$$4+36x^{2}-32x=84$$

Ikkala tarafdan 4 ni ayirish.

$$36x^{2}-32x=84-4$$

80 olish uchun 84 dan 4 ni ayirish.

$$36x^{2}-32x=80$$

Ikki tarafini 36 ga bo‘ling.

$$\frac{36x^{2}-32x}{36}=\frac{80}{36}$$

36 ga bo'lish 36 ga ko'paytirishni bekor qiladi.

$$x^{2}+\left(-\frac{32}{36}\right)x=\frac{80}{36}$$

\frac{-32}{36} ulushini 4 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

$$x^{2}-\frac{8}{9}x=\frac{80}{36}$$

\frac{80}{36} ulushini 4 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

$$x^{2}-\frac{8}{9}x=\frac{20}{9}$$

-\frac{8}{9} ni bo‘lish, x shartining koeffitsienti, 2 ga -\frac{4}{9} olish uchun. Keyin, -\frac{4}{9} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}-\frac{8}{9}x+\left(-\frac{4}{9}\right)^{2}=\frac{20}{9}+\left(-\frac{4}{9}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib -\frac{4}{9} kvadratini chiqarish.

$$x^{2}-\frac{8}{9}x+\frac{16}{81}=\frac{20}{9}+\frac{16}{81}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali \frac{20}{9} ni \frac{16}{81} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$x^{2}-\frac{8}{9}x+\frac{16}{81}=\frac{196}{81}$$

x^{2}-\frac{8}{9}x+\frac{16}{81} omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(x-\frac{4}{9}\right)^{2}=\frac{196}{81}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x-\frac{4}{9}\right)^{2}}=\sqrt{\frac{196}{81}}$$

Qisqartirish.

$$x-\frac{4}{9}=\frac{14}{9} x-\frac{4}{9}=-\frac{14}{9}$$

\frac{4}{9} ni tenglamaning ikkala tarafiga qo'shish.

$$x=2 x=-\frac{10}{9}$$