8n^2-4(1-2n)(2+8n)=0 eching

$$8 n ^ { 2 } - 4 ( 1 - 2 n ) ( 2 + 8 n ) = 0$$

n uchun yechish

$$n=\frac{\sqrt{10}+1}{9}\approx 0,462475296$$

$$n=\frac{1-\sqrt{10}}{9}\approx -0,240253073$$

Kvadrat tenglamalaridan foydalanish qoidalari

-4 hosil qilish uchun -1 va 4 ni ko'paytirish.

$$8n^{2}-4\left(1-2n\right)\left(2+8n\right)=0$$

-4 ga 1-2n ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$8n^{2}+\left(-4+8n\right)\left(2+8n\right)=0$$

-4+8n ga 2+8n ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

$$8n^{2}-8-16n+64n^{2}=0$$

72n^{2} ni olish uchun 8n^{2} va 64n^{2} ni birlashtirish.

$$72n^{2}-8-16n=0$$

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.

$$72n^{2}-16n-8=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 72 ni a, -16 ni b va -8 ni c bilan almashtiring.

$$n=\frac{-\left(-16\right)±\sqrt{\left(-16\right)^{2}-4\times 72\left(-8\right)}}{2\times 72}$$

-16 kvadratini chiqarish.

$$n=\frac{-\left(-16\right)±\sqrt{256-4\times 72\left(-8\right)}}{2\times 72}$$

-4 ni 72 marotabaga ko'paytirish.

$$n=\frac{-\left(-16\right)±\sqrt{256-288\left(-8\right)}}{2\times 72}$$

-288 ni -8 marotabaga ko'paytirish.

$$n=\frac{-\left(-16\right)±\sqrt{256+2304}}{2\times 72}$$

256 ni 2304 ga qo'shish.

$$n=\frac{-\left(-16\right)±\sqrt{2560}}{2\times 72}$$

2560 ning kvadrat ildizini chiqarish.

$$n=\frac{-\left(-16\right)±16\sqrt{10}}{2\times 72}$$

-16 ning teskarisi 16 ga teng.

$$n=\frac{16±16\sqrt{10}}{2\times 72}$$

2 ni 72 marotabaga ko'paytirish.

$$n=\frac{16±16\sqrt{10}}{144}$$

n=\frac{16±16\sqrt{10}}{144} tenglamasini yeching, bunda ± musbat. 16 ni 16\sqrt{10} ga qo'shish.

$$n=\frac{16\sqrt{10}+16}{144}$$

16+16\sqrt{10} ni 144 ga bo'lish.

$$n=\frac{\sqrt{10}+1}{9}$$

n=\frac{16±16\sqrt{10}}{144} tenglamasini yeching, bunda ± manfiy. 16 dan 16\sqrt{10} ni ayirish.

$$n=\frac{16-16\sqrt{10}}{144}$$

16-16\sqrt{10} ni 144 ga bo'lish.

$$n=\frac{1-\sqrt{10}}{9}$$

Tenglama yechildi.

$$n=\frac{\sqrt{10}+1}{9} n=\frac{1-\sqrt{10}}{9}$$

Kvadratni yakunlash bo'yicha qadamlar

-4 hosil qilish uchun -1 va 4 ni ko'paytirish.

$$8n^{2}-4\left(1-2n\right)\left(2+8n\right)=0$$

-4 ga 1-2n ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$8n^{2}+\left(-4+8n\right)\left(2+8n\right)=0$$

-4+8n ga 2+8n ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

$$8n^{2}-8-16n+64n^{2}=0$$

72n^{2} ni olish uchun 8n^{2} va 64n^{2} ni birlashtirish.

$$72n^{2}-8-16n=0$$

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

$$72n^{2}-16n=8$$

Ikki tarafini 72 ga bo‘ling.

$$\frac{72n^{2}-16n}{72}=\frac{8}{72}$$

72 ga bo'lish 72 ga ko'paytirishni bekor qiladi.

$$n^{2}+\left(-\frac{16}{72}\right)n=\frac{8}{72}$$

\frac{-16}{72} ulushini 8 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

$$n^{2}-\frac{2}{9}n=\frac{8}{72}$$

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

$$n^{2}-\frac{2}{9}n=\frac{1}{9}$$

-\frac{2}{9} ni bo‘lish, x shartining koeffitsienti, 2 ga -\frac{1}{9} olish uchun. Keyin, -\frac{1}{9} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$n^{2}-\frac{2}{9}n+\left(-\frac{1}{9}\right)^{2}=\frac{1}{9}+\left(-\frac{1}{9}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib -\frac{1}{9} kvadratini chiqarish.

$$n^{2}-\frac{2}{9}n+\frac{1}{81}=\frac{1}{9}+\frac{1}{81}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali \frac{1}{9} ni \frac{1}{81} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$n^{2}-\frac{2}{9}n+\frac{1}{81}=\frac{10}{81}$$

n^{2}-\frac{2}{9}n+\frac{1}{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.

$$\left(n-\frac{1}{9}\right)^{2}=\frac{10}{81}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(n-\frac{1}{9}\right)^{2}}=\sqrt{\frac{10}{81}}$$

Qisqartirish.

$$n-\frac{1}{9}=\frac{\sqrt{10}}{9} n-\frac{1}{9}=-\frac{\sqrt{10}}{9}$$

\frac{1}{9} ni tenglamaning ikkala tarafiga qo'shish.

$$n=\frac{\sqrt{10}+1}{9} n=\frac{1-\sqrt{10}}{9}$$