3n^2-13=3n eching

$$3 n ^ { 2 } - 13 = 3 n$$

n uchun yechish

$$n=\frac{\sqrt{165}}{6}+\frac{1}{2}\approx 2,640872096$$

$$n=-\frac{\sqrt{165}}{6}+\frac{1}{2}\approx -1,640872096$$

Kvadrat tenglamalaridan foydalanish qoidalari

Ikkala tarafdan 3n ni ayirish.

$$3n^{2}-13-3n=0$$

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.

$$3n^{2}-3n-13=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 3 ni a, -3 ni b va -13 ni c bilan almashtiring.

$$n=\frac{-\left(-3\right)±\sqrt{\left(-3\right)^{2}-4\times 3\left(-13\right)}}{2\times 3}$$

-3 kvadratini chiqarish.

$$n=\frac{-\left(-3\right)±\sqrt{9-4\times 3\left(-13\right)}}{2\times 3}$$

-4 ni 3 marotabaga ko'paytirish.

$$n=\frac{-\left(-3\right)±\sqrt{9-12\left(-13\right)}}{2\times 3}$$

-12 ni -13 marotabaga ko'paytirish.

$$n=\frac{-\left(-3\right)±\sqrt{9+156}}{2\times 3}$$

9 ni 156 ga qo'shish.

$$n=\frac{-\left(-3\right)±\sqrt{165}}{2\times 3}$$

-3 ning teskarisi 3 ga teng.

$$n=\frac{3±\sqrt{165}}{2\times 3}$$

2 ni 3 marotabaga ko'paytirish.

$$n=\frac{3±\sqrt{165}}{6}$$

n=\frac{3±\sqrt{165}}{6} tenglamasini yeching, bunda ± musbat. 3 ni \sqrt{165} ga qo'shish.

$$n=\frac{\sqrt{165}+3}{6}$$

3+\sqrt{165} ni 6 ga bo'lish.

$$n=\frac{\sqrt{165}}{6}+\frac{1}{2}$$

n=\frac{3±\sqrt{165}}{6} tenglamasini yeching, bunda ± manfiy. 3 dan \sqrt{165} ni ayirish.

$$n=\frac{3-\sqrt{165}}{6}$$

3-\sqrt{165} ni 6 ga bo'lish.

$$n=-\frac{\sqrt{165}}{6}+\frac{1}{2}$$

Tenglama yechildi.

$$n=\frac{\sqrt{165}}{6}+\frac{1}{2} n=-\frac{\sqrt{165}}{6}+\frac{1}{2}$$

Kvadratni yakunlash bo'yicha qadamlar

Ikkala tarafdan 3n ni ayirish.

$$3n^{2}-13-3n=0$$

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

$$3n^{2}-3n=13$$

Ikki tarafini 3 ga bo‘ling.

$$\frac{3n^{2}-3n}{3}=\frac{13}{3}$$

3 ga bo'lish 3 ga ko'paytirishni bekor qiladi.

$$n^{2}+\left(-\frac{3}{3}\right)n=\frac{13}{3}$$

-3 ni 3 ga bo'lish.

$$n^{2}-n=\frac{13}{3}$$

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

$$n^{2}-n+\left(-\frac{1}{2}\right)^{2}=\frac{13}{3}+\left(-\frac{1}{2}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib -\frac{1}{2} kvadratini chiqarish.

$$n^{2}-n+\frac{1}{4}=\frac{13}{3}+\frac{1}{4}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali \frac{13}{3} ni \frac{1}{4} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$n^{2}-n+\frac{1}{4}=\frac{55}{12}$$

n^{2}-n+\frac{1}{4} omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(n-\frac{1}{2}\right)^{2}=\frac{55}{12}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(n-\frac{1}{2}\right)^{2}}=\sqrt{\frac{55}{12}}$$

Qisqartirish.

$$n-\frac{1}{2}=\frac{\sqrt{165}}{6} n-\frac{1}{2}=-\frac{\sqrt{165}}{6}$$

\frac{1}{2} ni tenglamaning ikkala tarafiga qo'shish.

$$n=\frac{\sqrt{165}}{6}+\frac{1}{2} n=-\frac{\sqrt{165}}{6}+\frac{1}{2}$$