3x^2-31x-60=0 eching

$$3 { x }^{ 2 } -31x-60=0$$

x uchun yechish

$$x = -\frac{5}{3} = -1\frac{2}{3} \approx -1,666666667$$

$$x=12$$

Guruhlash orqali faktorlash qadamlari

Tenglamani yechish uchun guruhlash orqali chap qoʻl tomonni faktorlang. Avvalo, chap qoʻl tomon 3x^{2}+ax+bx-60 sifatida qayta yozilishi kerak. a va b ni topish uchun yechiladigan tizimni sozlang.

$$a+b=-31 ab=3\left(-60\right)=-180$$

ab manfiy boʻlganda, a va b da qarama-qarshi belgilar bor. a+b manfiy boʻlganda, manfiy sonda musbatga nisbatdan kattaroq mutlaq qiymat bor. -180-mahsulotni beruvchi bunday butun juftliklarni roʻyxat qiling.

$$1,-180 2,-90 3,-60 4,-45 5,-36 6,-30 9,-20 10,-18 12,-15$$

Har bir juftlik yigʻindisini hisoblang.

$$1-180=-179 2-90=-88 3-60=-57 4-45=-41 5-36=-31 6-30=-24 9-20=-11 10-18=-8 12-15=-3$$

Yechim – -31 yigʻindisini beruvchi juftlik.

$$a=-36 b=5$$

3x^{2}-31x-60 ni \left(3x^{2}-36x\right)+\left(5x-60\right) sifatida qaytadan yozish.

$$\left(3x^{2}-36x\right)+\left(5x-60\right)$$

Birinchi guruhda 3x ni va ikkinchi guruhda 5 ni faktordan chiqaring.

$$3x\left(x-12\right)+5\left(x-12\right)$$

Distributiv funktsiyasidan foydalangan holda x-12 umumiy terminini chiqaring.

$$\left(x-12\right)\left(3x+5\right)$$

Tenglamani yechish uchun x-12=0 va 3x+5=0 ni yeching.

$$x=12 x=-\frac{5}{3}$$

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.

$$3x^{2}-31x-60=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 3 ni a, -31 ni b va -60 ni c bilan almashtiring.

$$x=\frac{-\left(-31\right)±\sqrt{\left(-31\right)^{2}-4\times 3\left(-60\right)}}{2\times 3}$$

-31 kvadratini chiqarish.

$$x=\frac{-\left(-31\right)±\sqrt{961-4\times 3\left(-60\right)}}{2\times 3}$$

-4 ni 3 marotabaga ko'paytirish.

$$x=\frac{-\left(-31\right)±\sqrt{961-12\left(-60\right)}}{2\times 3}$$

-12 ni -60 marotabaga ko'paytirish.

$$x=\frac{-\left(-31\right)±\sqrt{961+720}}{2\times 3}$$

961 ni 720 ga qo'shish.

$$x=\frac{-\left(-31\right)±\sqrt{1681}}{2\times 3}$$

1681 ning kvadrat ildizini chiqarish.

$$x=\frac{-\left(-31\right)±41}{2\times 3}$$

-31 ning teskarisi 31 ga teng.

$$x=\frac{31±41}{2\times 3}$$

2 ni 3 marotabaga ko'paytirish.

$$x=\frac{31±41}{6}$$

x=\frac{31±41}{6} tenglamasini yeching, bunda ± musbat. 31 ni 41 ga qo'shish.

$$x=\frac{72}{6}$$

72 ni 6 ga bo'lish.

$$x=12$$

x=\frac{31±41}{6} tenglamasini yeching, bunda ± manfiy. 31 dan 41 ni ayirish.

$$x=-\frac{10}{6}$$

\frac{-10}{6} ulushini 2 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

$$x=-\frac{5}{3}$$

Tenglama yechildi.

$$x=12 x=-\frac{5}{3}$$

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.

$$3x^{2}-31x-60=0$$

60 ni tenglamaning ikkala tarafiga qo'shish.

$$3x^{2}-31x-60-\left(-60\right)=-\left(-60\right)$$

O‘zidan -60 ayirilsa 0 qoladi.

$$3x^{2}-31x=-\left(-60\right)$$

0 dan -60 ni ayirish.

$$3x^{2}-31x=60$$

Ikki tarafini 3 ga bo‘ling.

$$\frac{3x^{2}-31x}{3}=\frac{60}{3}$$

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

$$x^{2}-\frac{31}{3}x=\frac{60}{3}$$

60 ni 3 ga bo'lish.

$$x^{2}-\frac{31}{3}x=20$$

-\frac{31}{3} ni bo‘lish, x shartining koeffitsienti, 2 ga -\frac{31}{6} olish uchun. Keyin, -\frac{31}{6} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}-\frac{31}{3}x+\left(-\frac{31}{6}\right)^{2}=20+\left(-\frac{31}{6}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib -\frac{31}{6} kvadratini chiqarish.

$$x^{2}-\frac{31}{3}x+\frac{961}{36}=20+\frac{961}{36}$$

20 ni \frac{961}{36} ga qo'shish.

$$x^{2}-\frac{31}{3}x+\frac{961}{36}=\frac{1681}{36}$$

x^{2}-\frac{31}{3}x+\frac{961}{36} omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(x-\frac{31}{6}\right)^{2}=\frac{1681}{36}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x-\frac{31}{6}\right)^{2}}=\sqrt{\frac{1681}{36}}$$

Qisqartirish.

$$x-\frac{31}{6}=\frac{41}{6} x-\frac{31}{6}=-\frac{41}{6}$$

\frac{31}{6} ni tenglamaning ikkala tarafiga qo'shish.

$$x=12 x=-\frac{5}{3}$$