4n^2-7n=11 eching

$$4 n ^ { 2 } - 7 n = 11$$

n uchun yechish

$$n=-1$$

$$n = \frac{11}{4} = 2\frac{3}{4} = 2,75$$

Guruhlash orqali faktorlash qadamlari

Ikkala tarafdan 11 ni ayirish.

$$4n^{2}-7n-11=0$$

Tenglamani yechish uchun guruhlash orqali chap qoʻl tomonni faktorlang. Avvalo, chap qoʻl tomon 4n^{2}+an+bn-11 sifatida qayta yozilishi kerak. a va b ni topish uchun yechiladigan tizimni sozlang.

$$a+b=-7 ab=4\left(-11\right)=-44$$

ab manfiy boʻlganda, a va b da qarama-qarshi belgilar bor. a+b manfiy boʻlganda, manfiy sonda musbatga nisbatdan kattaroq mutlaq qiymat bor. -44-mahsulotni beruvchi bunday butun juftliklarni roʻyxat qiling.

$$1,-44 2,-22 4,-11$$

Har bir juftlik yigʻindisini hisoblang.

$$1-44=-43 2-22=-20 4-11=-7$$

Yechim – -7 yigʻindisini beruvchi juftlik.

$$a=-11 b=4$$

4n^{2}-7n-11 ni \left(4n^{2}-11n\right)+\left(4n-11\right) sifatida qaytadan yozish.

$$\left(4n^{2}-11n\right)+\left(4n-11\right)$$

4n^{2}-11n ichida n ni ajrating.

$$n\left(4n-11\right)+4n-11$$

Distributiv funktsiyasidan foydalangan holda 4n-11 umumiy terminini chiqaring.

$$\left(4n-11\right)\left(n+1\right)$$

Tenglamani yechish uchun 4n-11=0 va n+1=0 ni yeching.

$$n=\frac{11}{4} n=-1$$

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.

$$4n^{2}-7n=11$$

Tenglamaning ikkala tarafidan 11 ni ayirish.

$$4n^{2}-7n-11=11-11$$

O‘zidan 11 ayirilsa 0 qoladi.

$$4n^{2}-7n-11=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 4 ni a, -7 ni b va -11 ni c bilan almashtiring.

$$n=\frac{-\left(-7\right)±\sqrt{\left(-7\right)^{2}-4\times 4\left(-11\right)}}{2\times 4}$$

-7 kvadratini chiqarish.

$$n=\frac{-\left(-7\right)±\sqrt{49-4\times 4\left(-11\right)}}{2\times 4}$$

-4 ni 4 marotabaga ko'paytirish.

$$n=\frac{-\left(-7\right)±\sqrt{49-16\left(-11\right)}}{2\times 4}$$

-16 ni -11 marotabaga ko'paytirish.

$$n=\frac{-\left(-7\right)±\sqrt{49+176}}{2\times 4}$$

49 ni 176 ga qo'shish.

$$n=\frac{-\left(-7\right)±\sqrt{225}}{2\times 4}$$

225 ning kvadrat ildizini chiqarish.

$$n=\frac{-\left(-7\right)±15}{2\times 4}$$

-7 ning teskarisi 7 ga teng.

$$n=\frac{7±15}{2\times 4}$$

2 ni 4 marotabaga ko'paytirish.

$$n=\frac{7±15}{8}$$

n=\frac{7±15}{8} tenglamasini yeching, bunda ± musbat. 7 ni 15 ga qo'shish.

$$n=\frac{22}{8}$$

\frac{22}{8} ulushini 2 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

$$n=\frac{11}{4}$$

n=\frac{7±15}{8} tenglamasini yeching, bunda ± manfiy. 7 dan 15 ni ayirish.

$$n=-\frac{8}{8}$$

-8 ni 8 ga bo'lish.

$$n=-1$$

Tenglama yechildi.

$$n=\frac{11}{4} n=-1$$

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.

$$4n^{2}-7n=11$$

Ikki tarafini 4 ga bo‘ling.

$$\frac{4n^{2}-7n}{4}=\frac{11}{4}$$

4 ga bo'lish 4 ga ko'paytirishni bekor qiladi.

$$n^{2}-\frac{7}{4}n=\frac{11}{4}$$

-\frac{7}{4} ni bo‘lish, x shartining koeffitsienti, 2 ga -\frac{7}{8} olish uchun. Keyin, -\frac{7}{8} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$n^{2}-\frac{7}{4}n+\left(-\frac{7}{8}\right)^{2}=\frac{11}{4}+\left(-\frac{7}{8}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib -\frac{7}{8} kvadratini chiqarish.

$$n^{2}-\frac{7}{4}n+\frac{49}{64}=\frac{11}{4}+\frac{49}{64}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali \frac{11}{4} ni \frac{49}{64} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$n^{2}-\frac{7}{4}n+\frac{49}{64}=\frac{225}{64}$$

n^{2}-\frac{7}{4}n+\frac{49}{64} omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(n-\frac{7}{8}\right)^{2}=\frac{225}{64}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(n-\frac{7}{8}\right)^{2}}=\sqrt{\frac{225}{64}}$$

Qisqartirish.

$$n-\frac{7}{8}=\frac{15}{8} n-\frac{7}{8}=-\frac{15}{8}$$

\frac{7}{8} ni tenglamaning ikkala tarafiga qo'shish.

$$n=\frac{11}{4} n=-1$$