13n^2-41n-120=0 eching

$$13 n ^ { 2 } - 41 n - 120 = 0$$

n uchun yechish

$$n = -\frac{24}{13} = -1\frac{11}{13} \approx -1,846153846$$

$$n=5$$

Guruhlash orqali faktorlash qadamlari

Tenglamani yechish uchun guruhlash orqali chap qoʻl tomonni faktorlang. Avvalo, chap qoʻl tomon 13n^{2}+an+bn-120 sifatida qayta yozilishi kerak. a va b ni topish uchun yechiladigan tizimni sozlang.

$$a+b=-41 ab=13\left(-120\right)=-1560$$

ab manfiy boʻlganda, a va b da qarama-qarshi belgilar bor. a+b manfiy boʻlganda, manfiy sonda musbatga nisbatdan kattaroq mutlaq qiymat bor. -1560-mahsulotni beruvchi bunday butun juftliklarni roʻyxat qiling.

$$1,-1560 2,-780 3,-520 4,-390 5,-312 6,-260 8,-195 10,-156 12,-130 13,-120 15,-104 20,-78 24,-65 26,-60 30,-52 39,-40$$

Har bir juftlik yigʻindisini hisoblang.

$$1-1560=-1559 2-780=-778 3-520=-517 4-390=-386 5-312=-307 6-260=-254 8-195=-187 10-156=-146 12-130=-118 13-120=-107 15-104=-89 20-78=-58 24-65=-41 26-60=-34 30-52=-22 39-40=-1$$

Yechim – -41 yigʻindisini beruvchi juftlik.

$$a=-65 b=24$$

13n^{2}-41n-120 ni \left(13n^{2}-65n\right)+\left(24n-120\right) sifatida qaytadan yozish.

$$\left(13n^{2}-65n\right)+\left(24n-120\right)$$

Birinchi guruhda 13n ni va ikkinchi guruhda 24 ni faktordan chiqaring.

$$13n\left(n-5\right)+24\left(n-5\right)$$

Distributiv funktsiyasidan foydalangan holda n-5 umumiy terminini chiqaring.

$$\left(n-5\right)\left(13n+24\right)$$

Tenglamani yechish uchun n-5=0 va 13n+24=0 ni yeching.

$$n=5 n=-\frac{24}{13}$$

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.

$$13n^{2}-41n-120=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 13 ni a, -41 ni b va -120 ni c bilan almashtiring.

$$n=\frac{-\left(-41\right)±\sqrt{\left(-41\right)^{2}-4\times 13\left(-120\right)}}{2\times 13}$$

-41 kvadratini chiqarish.

$$n=\frac{-\left(-41\right)±\sqrt{1681-4\times 13\left(-120\right)}}{2\times 13}$$

-4 ni 13 marotabaga ko'paytirish.

$$n=\frac{-\left(-41\right)±\sqrt{1681-52\left(-120\right)}}{2\times 13}$$

-52 ni -120 marotabaga ko'paytirish.

$$n=\frac{-\left(-41\right)±\sqrt{1681+6240}}{2\times 13}$$

1681 ni 6240 ga qo'shish.

$$n=\frac{-\left(-41\right)±\sqrt{7921}}{2\times 13}$$

7921 ning kvadrat ildizini chiqarish.

$$n=\frac{-\left(-41\right)±89}{2\times 13}$$

-41 ning teskarisi 41 ga teng.

$$n=\frac{41±89}{2\times 13}$$

2 ni 13 marotabaga ko'paytirish.

$$n=\frac{41±89}{26}$$

n=\frac{41±89}{26} tenglamasini yeching, bunda ± musbat. 41 ni 89 ga qo'shish.

$$n=\frac{130}{26}$$

130 ni 26 ga bo'lish.

$$n=5$$

n=\frac{41±89}{26} tenglamasini yeching, bunda ± manfiy. 41 dan 89 ni ayirish.

$$n=-\frac{48}{26}$$

\frac{-48}{26} ulushini 2 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

$$n=-\frac{24}{13}$$

Tenglama yechildi.

$$n=5 n=-\frac{24}{13}$$

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.

$$13n^{2}-41n-120=0$$

120 ni tenglamaning ikkala tarafiga qo'shish.

$$13n^{2}-41n-120-\left(-120\right)=-\left(-120\right)$$

O‘zidan -120 ayirilsa 0 qoladi.

$$13n^{2}-41n=-\left(-120\right)$$

0 dan -120 ni ayirish.

$$13n^{2}-41n=120$$

Ikki tarafini 13 ga bo‘ling.

$$\frac{13n^{2}-41n}{13}=\frac{120}{13}$$

13 ga bo'lish 13 ga ko'paytirishni bekor qiladi.

$$n^{2}-\frac{41}{13}n=\frac{120}{13}$$

-\frac{41}{13} ni bo‘lish, x shartining koeffitsienti, 2 ga -\frac{41}{26} olish uchun. Keyin, -\frac{41}{26} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$n^{2}-\frac{41}{13}n+\left(-\frac{41}{26}\right)^{2}=\frac{120}{13}+\left(-\frac{41}{26}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib -\frac{41}{26} kvadratini chiqarish.

$$n^{2}-\frac{41}{13}n+\frac{1681}{676}=\frac{120}{13}+\frac{1681}{676}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali \frac{120}{13} ni \frac{1681}{676} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$n^{2}-\frac{41}{13}n+\frac{1681}{676}=\frac{7921}{676}$$

n^{2}-\frac{41}{13}n+\frac{1681}{676} omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(n-\frac{41}{26}\right)^{2}=\frac{7921}{676}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(n-\frac{41}{26}\right)^{2}}=\sqrt{\frac{7921}{676}}$$

Qisqartirish.

$$n-\frac{41}{26}=\frac{89}{26} n-\frac{41}{26}=-\frac{89}{26}$$

\frac{41}{26} ni tenglamaning ikkala tarafiga qo'shish.

$$n=5 n=-\frac{24}{13}$$