2x^2-60x-3600=0 eching

$$2 x ^ { 2 } - 60 x - 3600 = 0$$

x uchun yechish

$$x=-30$$

$$x=60$$

Guruhlash orqali faktorlash qadamlari

Ikki tarafini 2 ga bo‘ling.

$$x^{2}-30x-1800=0$$

Tenglamani yechish uchun guruhlash orqali chap qoʻl tomonni faktorlang. Avvalo, chap qoʻl tomon x^{2}+ax+bx-1800 sifatida qayta yozilishi kerak. a va b ni topish uchun yechiladigan tizimni sozlang.

$$a+b=-30 ab=1\left(-1800\right)=-1800$$

ab manfiy boʻlganda, a va b da qarama-qarshi belgilar bor. a+b manfiy boʻlganda, manfiy sonda musbatga nisbatdan kattaroq mutlaq qiymat bor. -1800-mahsulotni beruvchi bunday butun juftliklarni roʻyxat qiling.

$$1,-1800 2,-900 3,-600 4,-450 5,-360 6,-300 8,-225 9,-200 10,-180 12,-150 15,-120 18,-100 20,-90 24,-75 25,-72 30,-60 36,-50 40,-45$$

Har bir juftlik yigʻindisini hisoblang.

$$1-1800=-1799 2-900=-898 3-600=-597 4-450=-446 5-360=-355 6-300=-294 8-225=-217 9-200=-191 10-180=-170 12-150=-138 15-120=-105 18-100=-82 20-90=-70 24-75=-51 25-72=-47 30-60=-30 36-50=-14 40-45=-5$$

Yechim – -30 yigʻindisini beruvchi juftlik.

$$a=-60 b=30$$

x^{2}-30x-1800 ni \left(x^{2}-60x\right)+\left(30x-1800\right) sifatida qaytadan yozish.

$$\left(x^{2}-60x\right)+\left(30x-1800\right)$$

Birinchi guruhda x ni va ikkinchi guruhda 30 ni faktordan chiqaring.

$$x\left(x-60\right)+30\left(x-60\right)$$

Distributiv funktsiyasidan foydalangan holda x-60 umumiy terminini chiqaring.

$$\left(x-60\right)\left(x+30\right)$$

Tenglamani yechish uchun x-60=0 va x+30=0 ni yeching.

$$x=60 x=-30$$

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.

$$2x^{2}-60x-3600=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 2 ni a, -60 ni b va -3600 ni c bilan almashtiring.

$$x=\frac{-\left(-60\right)±\sqrt{\left(-60\right)^{2}-4\times 2\left(-3600\right)}}{2\times 2}$$

-60 kvadratini chiqarish.

$$x=\frac{-\left(-60\right)±\sqrt{3600-4\times 2\left(-3600\right)}}{2\times 2}$$

-4 ni 2 marotabaga ko'paytirish.

$$x=\frac{-\left(-60\right)±\sqrt{3600-8\left(-3600\right)}}{2\times 2}$$

-8 ni -3600 marotabaga ko'paytirish.

$$x=\frac{-\left(-60\right)±\sqrt{3600+28800}}{2\times 2}$$

3600 ni 28800 ga qo'shish.

$$x=\frac{-\left(-60\right)±\sqrt{32400}}{2\times 2}$$

32400 ning kvadrat ildizini chiqarish.

$$x=\frac{-\left(-60\right)±180}{2\times 2}$$

-60 ning teskarisi 60 ga teng.

$$x=\frac{60±180}{2\times 2}$$

2 ni 2 marotabaga ko'paytirish.

$$x=\frac{60±180}{4}$$

x=\frac{60±180}{4} tenglamasini yeching, bunda ± musbat. 60 ni 180 ga qo'shish.

$$x=\frac{240}{4}$$

240 ni 4 ga bo'lish.

$$x=60$$

x=\frac{60±180}{4} tenglamasini yeching, bunda ± manfiy. 60 dan 180 ni ayirish.

$$x=-\frac{120}{4}$$

-120 ni 4 ga bo'lish.

$$x=-30$$

Tenglama yechildi.

$$x=60 x=-30$$

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.

$$2x^{2}-60x-3600=0$$

3600 ni tenglamaning ikkala tarafiga qo'shish.

$$2x^{2}-60x-3600-\left(-3600\right)=-\left(-3600\right)$$

O‘zidan -3600 ayirilsa 0 qoladi.

$$2x^{2}-60x=-\left(-3600\right)$$

0 dan -3600 ni ayirish.

$$2x^{2}-60x=3600$$

Ikki tarafini 2 ga bo‘ling.

$$\frac{2x^{2}-60x}{2}=\frac{3600}{2}$$

2 ga bo'lish 2 ga ko'paytirishni bekor qiladi.

$$x^{2}+\left(-\frac{60}{2}\right)x=\frac{3600}{2}$$

-60 ni 2 ga bo'lish.

$$x^{2}-30x=\frac{3600}{2}$$

3600 ni 2 ga bo'lish.

$$x^{2}-30x=1800$$

-30 ni bo‘lish, x shartining koeffitsienti, 2 ga -15 olish uchun. Keyin, -15 ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}-30x+\left(-15\right)^{2}=1800+\left(-15\right)^{2}$$

-15 kvadratini chiqarish.

$$x^{2}-30x+225=1800+225$$

1800 ni 225 ga qo'shish.

$$x^{2}-30x+225=2025$$

x^{2}-30x+225 omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(x-15\right)^{2}=2025$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x-15\right)^{2}}=\sqrt{2025}$$

Qisqartirish.

$$x-15=45 x-15=-45$$

15 ni tenglamaning ikkala tarafiga qo'shish.

$$x=60 x=-30$$