8n^2+33n+31=0 eching

$$8 n ^ { 2 } + 33 n + 31 = 0$$

n uchun yechish

$$n=\frac{\sqrt{97}-33}{16}\approx -1,446946387$$

$$n=\frac{-\sqrt{97}-33}{16}\approx -2,678053613$$

Kvadrat tenglamalaridan foydalanish qoidalari

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.

$$8n^{2}+33n+31=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 8 ni a, 33 ni b va 31 ni c bilan almashtiring.

$$n=\frac{-33±\sqrt{33^{2}-4\times 8\times 31}}{2\times 8}$$

33 kvadratini chiqarish.

$$n=\frac{-33±\sqrt{1089-4\times 8\times 31}}{2\times 8}$$

-4 ni 8 marotabaga ko'paytirish.

$$n=\frac{-33±\sqrt{1089-32\times 31}}{2\times 8}$$

-32 ni 31 marotabaga ko'paytirish.

$$n=\frac{-33±\sqrt{1089-992}}{2\times 8}$$

1089 ni -992 ga qo'shish.

$$n=\frac{-33±\sqrt{97}}{2\times 8}$$

2 ni 8 marotabaga ko'paytirish.

$$n=\frac{-33±\sqrt{97}}{16}$$

n=\frac{-33±\sqrt{97}}{16} tenglamasini yeching, bunda ± musbat. -33 ni \sqrt{97} ga qo'shish.

$$n=\frac{\sqrt{97}-33}{16}$$

n=\frac{-33±\sqrt{97}}{16} tenglamasini yeching, bunda ± manfiy. -33 dan \sqrt{97} ni ayirish.

$$n=\frac{-\sqrt{97}-33}{16}$$

Tenglama yechildi.

$$n=\frac{\sqrt{97}-33}{16} n=\frac{-\sqrt{97}-33}{16}$$

Kvadratni yakunlash bo'yicha qadamlar

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

$$8n^{2}+33n+31=0$$

Tenglamaning ikkala tarafidan 31 ni ayirish.

$$8n^{2}+33n+31-31=-31$$

O‘zidan 31 ayirilsa 0 qoladi.

$$8n^{2}+33n=-31$$

Ikki tarafini 8 ga bo‘ling.

$$\frac{8n^{2}+33n}{8}=-\frac{31}{8}$$

8 ga bo'lish 8 ga ko'paytirishni bekor qiladi.

$$n^{2}+\frac{33}{8}n=-\frac{31}{8}$$

\frac{33}{8} ni bo‘lish, x shartining koeffitsienti, 2 ga \frac{33}{16} olish uchun. Keyin, \frac{33}{16} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$n^{2}+\frac{33}{8}n+\left(\frac{33}{16}\right)^{2}=-\frac{31}{8}+\left(\frac{33}{16}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib \frac{33}{16} kvadratini chiqarish.

$$n^{2}+\frac{33}{8}n+\frac{1089}{256}=-\frac{31}{8}+\frac{1089}{256}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali -\frac{31}{8} ni \frac{1089}{256} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$n^{2}+\frac{33}{8}n+\frac{1089}{256}=\frac{97}{256}$$

n^{2}+\frac{33}{8}n+\frac{1089}{256} omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(n+\frac{33}{16}\right)^{2}=\frac{97}{256}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(n+\frac{33}{16}\right)^{2}}=\sqrt{\frac{97}{256}}$$

Qisqartirish.

$$n+\frac{33}{16}=\frac{\sqrt{97}}{16} n+\frac{33}{16}=-\frac{\sqrt{97}}{16}$$

Tenglamaning ikkala tarafidan \frac{33}{16} ni ayirish.

$$n=\frac{\sqrt{97}-33}{16} n=\frac{-\sqrt{97}-33}{16}$$