7n^2=-8+15n eching

$$7 n ^ { 2 } = - 8 + 15 n$$

n uchun yechish

$$n=1$$

$$n = \frac{8}{7} = 1\frac{1}{7} \approx 1,142857143$$

Guruhlash orqali faktorlash qadamlari

Ikkala tarafdan -8 ni ayirish.

$$7n^{2}-\left(-8\right)=15n$$

-8 ning teskarisi 8 ga teng.

$$7n^{2}+8=15n$$

Ikkala tarafdan 15n ni ayirish.

$$7n^{2}+8-15n=0$$

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

$$7n^{2}-15n+8=0$$

Tenglamani yechish uchun guruhlash orqali chap qoʻl tomonni faktorlang. Avvalo, chap qoʻl tomon 7n^{2}+an+bn+8 sifatida qayta yozilishi kerak. a va b ni topish uchun yechiladigan tizimni sozlang.

$$a+b=-15 ab=7\times 8=56$$

ab musbat boʻlganda, a va b da bir xil belgi bor. a+b manfiy boʻlganda, a va b ikkisi ham manfiy. 56-mahsulotni beruvchi bunday butun juftliklarni roʻyxat qiling.

$$-1,-56 -2,-28 -4,-14 -7,-8$$

Har bir juftlik yigʻindisini hisoblang.

$$-1-56=-57 -2-28=-30 -4-14=-18 -7-8=-15$$

Yechim – -15 yigʻindisini beruvchi juftlik.

$$a=-8 b=-7$$

7n^{2}-15n+8 ni \left(7n^{2}-8n\right)+\left(-7n+8\right) sifatida qaytadan yozish.

$$\left(7n^{2}-8n\right)+\left(-7n+8\right)$$

Birinchi guruhda n ni va ikkinchi guruhda -1 ni faktordan chiqaring.

$$n\left(7n-8\right)-\left(7n-8\right)$$

Distributiv funktsiyasidan foydalangan holda 7n-8 umumiy terminini chiqaring.

$$\left(7n-8\right)\left(n-1\right)$$

Tenglamani yechish uchun 7n-8=0 va n-1=0 ni yeching.

$$n=\frac{8}{7} n=1$$

Kvadrat tenglamalaridan foydalanish qoidalari

Ikkala tarafdan -8 ni ayirish.

$$7n^{2}-\left(-8\right)=15n$$

-8 ning teskarisi 8 ga teng.

$$7n^{2}+8=15n$$

Ikkala tarafdan 15n ni ayirish.

$$7n^{2}+8-15n=0$$

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.

$$7n^{2}-15n+8=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 7 ni a, -15 ni b va 8 ni c bilan almashtiring.

$$n=\frac{-\left(-15\right)±\sqrt{\left(-15\right)^{2}-4\times 7\times 8}}{2\times 7}$$

-15 kvadratini chiqarish.

$$n=\frac{-\left(-15\right)±\sqrt{225-4\times 7\times 8}}{2\times 7}$$

-4 ni 7 marotabaga ko'paytirish.

$$n=\frac{-\left(-15\right)±\sqrt{225-28\times 8}}{2\times 7}$$

-28 ni 8 marotabaga ko'paytirish.

$$n=\frac{-\left(-15\right)±\sqrt{225-224}}{2\times 7}$$

225 ni -224 ga qo'shish.

$$n=\frac{-\left(-15\right)±\sqrt{1}}{2\times 7}$$

1 ning kvadrat ildizini chiqarish.

$$n=\frac{-\left(-15\right)±1}{2\times 7}$$

-15 ning teskarisi 15 ga teng.

$$n=\frac{15±1}{2\times 7}$$

2 ni 7 marotabaga ko'paytirish.

$$n=\frac{15±1}{14}$$

n=\frac{15±1}{14} tenglamasini yeching, bunda ± musbat. 15 ni 1 ga qo'shish.

$$n=\frac{16}{14}$$

\frac{16}{14} ulushini 2 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

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

n=\frac{15±1}{14} tenglamasini yeching, bunda ± manfiy. 15 dan 1 ni ayirish.

$$n=\frac{14}{14}$$

14 ni 14 ga bo'lish.

$$n=1$$

Tenglama yechildi.

$$n=\frac{8}{7} n=1$$

Kvadratni yakunlash bo'yicha qadamlar

Ikkala tarafdan 15n ni ayirish.

$$7n^{2}-15n=-8$$

Ikki tarafini 7 ga bo‘ling.

$$\frac{7n^{2}-15n}{7}=-\frac{8}{7}$$

7 ga bo'lish 7 ga ko'paytirishni bekor qiladi.

$$n^{2}-\frac{15}{7}n=-\frac{8}{7}$$

-\frac{15}{7} ni bo‘lish, x shartining koeffitsienti, 2 ga -\frac{15}{14} olish uchun. Keyin, -\frac{15}{14} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$n^{2}-\frac{15}{7}n+\left(-\frac{15}{14}\right)^{2}=-\frac{8}{7}+\left(-\frac{15}{14}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib -\frac{15}{14} kvadratini chiqarish.

$$n^{2}-\frac{15}{7}n+\frac{225}{196}=-\frac{8}{7}+\frac{225}{196}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali -\frac{8}{7} ni \frac{225}{196} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$n^{2}-\frac{15}{7}n+\frac{225}{196}=\frac{1}{196}$$

n^{2}-\frac{15}{7}n+\frac{225}{196} omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(n-\frac{15}{14}\right)^{2}=\frac{1}{196}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(n-\frac{15}{14}\right)^{2}}=\sqrt{\frac{1}{196}}$$

Qisqartirish.

$$n-\frac{15}{14}=\frac{1}{14} n-\frac{15}{14}=-\frac{1}{14}$$

\frac{15}{14} ni tenglamaning ikkala tarafiga qo'shish.

$$n=\frac{8}{7} n=1$$