120x^2+8000x-92000=0 eching

$$120 { x }^{ 2 } +8000x-92000=0$$

x uchun yechish

$$x = -\frac{230}{3} = -76\frac{2}{3} \approx -76,666666667$$

$$x=10$$

Guruhlash orqali faktorlash qadamlari

Ikki tarafini 40 ga bo‘ling.

$$3x^{2}+200x-2300=0$$

Tenglamani yechish uchun guruhlash orqali chap qoʻl tomonni faktorlang. Avvalo, chap qoʻl tomon 3x^{2}+ax+bx-2300 sifatida qayta yozilishi kerak. a va b ni topish uchun yechiladigan tizimni sozlang.

$$a+b=200 ab=3\left(-2300\right)=-6900$$

ab manfiy boʻlganda, a va b da qarama-qarshi belgilar bor. a+b musbat boʻlganda, musbat sonda manfiyga nisbatdan kattaroq mutlaq qiymat bor. -6900-mahsulotni beruvchi bunday butun juftliklarni roʻyxat qiling.

$$-1,6900 -2,3450 -3,2300 -4,1725 -5,1380 -6,1150 -10,690 -12,575 -15,460 -20,345 -23,300 -25,276 -30,230 -46,150 -50,138 -60,115 -69,100 -75,92$$

Har bir juftlik yigʻindisini hisoblang.

$$-1+6900=6899 -2+3450=3448 -3+2300=2297 -4+1725=1721 -5+1380=1375 -6+1150=1144 -10+690=680 -12+575=563 -15+460=445 -20+345=325 -23+300=277 -25+276=251 -30+230=200 -46+150=104 -50+138=88 -60+115=55 -69+100=31 -75+92=17$$

Yechim – 200 yigʻindisini beruvchi juftlik.

$$a=-30 b=230$$

3x^{2}+200x-2300 ni \left(3x^{2}-30x\right)+\left(230x-2300\right) sifatida qaytadan yozish.

$$\left(3x^{2}-30x\right)+\left(230x-2300\right)$$

Birinchi guruhda 3x ni va ikkinchi guruhda 230 ni faktordan chiqaring.

$$3x\left(x-10\right)+230\left(x-10\right)$$

Distributiv funktsiyasidan foydalangan holda x-10 umumiy terminini chiqaring.

$$\left(x-10\right)\left(3x+230\right)$$

Tenglamani yechish uchun x-10=0 va 3x+230=0 ni yeching.

$$x=10 x=-\frac{230}{3}$$

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.

$$120x^{2}+8000x-92000=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 120 ni a, 8000 ni b va -92000 ni c bilan almashtiring.

$$x=\frac{-8000±\sqrt{8000^{2}-4\times 120\left(-92000\right)}}{2\times 120}$$

8000 kvadratini chiqarish.

$$x=\frac{-8000±\sqrt{64000000-4\times 120\left(-92000\right)}}{2\times 120}$$

-4 ni 120 marotabaga ko'paytirish.

$$x=\frac{-8000±\sqrt{64000000-480\left(-92000\right)}}{2\times 120}$$

-480 ni -92000 marotabaga ko'paytirish.

$$x=\frac{-8000±\sqrt{64000000+44160000}}{2\times 120}$$

64000000 ni 44160000 ga qo'shish.

$$x=\frac{-8000±\sqrt{108160000}}{2\times 120}$$

108160000 ning kvadrat ildizini chiqarish.

$$x=\frac{-8000±10400}{2\times 120}$$

2 ni 120 marotabaga ko'paytirish.

$$x=\frac{-8000±10400}{240}$$

x=\frac{-8000±10400}{240} tenglamasini yeching, bunda ± musbat. -8000 ni 10400 ga qo'shish.

$$x=\frac{2400}{240}$$

2400 ni 240 ga bo'lish.

$$x=10$$

x=\frac{-8000±10400}{240} tenglamasini yeching, bunda ± manfiy. -8000 dan 10400 ni ayirish.

$$x=-\frac{18400}{240}$$

\frac{-18400}{240} ulushini 80 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

$$x=-\frac{230}{3}$$

Tenglama yechildi.

$$x=10 x=-\frac{230}{3}$$

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.

$$120x^{2}+8000x-92000=0$$

92000 ni tenglamaning ikkala tarafiga qo'shish.

$$120x^{2}+8000x-92000-\left(-92000\right)=-\left(-92000\right)$$

O‘zidan -92000 ayirilsa 0 qoladi.

$$120x^{2}+8000x=-\left(-92000\right)$$

0 dan -92000 ni ayirish.

$$120x^{2}+8000x=92000$$

Ikki tarafini 120 ga bo‘ling.

$$\frac{120x^{2}+8000x}{120}=\frac{92000}{120}$$

120 ga bo'lish 120 ga ko'paytirishni bekor qiladi.

$$x^{2}+\frac{8000}{120}x=\frac{92000}{120}$$

\frac{8000}{120} ulushini 40 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

$$x^{2}+\frac{200}{3}x=\frac{92000}{120}$$

\frac{92000}{120} ulushini 40 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

$$x^{2}+\frac{200}{3}x=\frac{2300}{3}$$

\frac{200}{3} ni bo‘lish, x shartining koeffitsienti, 2 ga \frac{100}{3} olish uchun. Keyin, \frac{100}{3} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}+\frac{200}{3}x+\left(\frac{100}{3}\right)^{2}=\frac{2300}{3}+\left(\frac{100}{3}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib \frac{100}{3} kvadratini chiqarish.

$$x^{2}+\frac{200}{3}x+\frac{10000}{9}=\frac{2300}{3}+\frac{10000}{9}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali \frac{2300}{3} ni \frac{10000}{9} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$x^{2}+\frac{200}{3}x+\frac{10000}{9}=\frac{16900}{9}$$

x^{2}+\frac{200}{3}x+\frac{10000}{9} omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(x+\frac{100}{3}\right)^{2}=\frac{16900}{9}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x+\frac{100}{3}\right)^{2}}=\sqrt{\frac{16900}{9}}$$

Qisqartirish.

$$x+\frac{100}{3}=\frac{130}{3} x+\frac{100}{3}=-\frac{130}{3}$$

Tenglamaning ikkala tarafidan \frac{100}{3} ni ayirish.

$$x=10 x=-\frac{230}{3}$$