4800/x+10=4200/x-20 eching

$$\frac{ 4800 }{ x } +10= \frac{ 4200 }{ x-20 }$$

x uchun yechish

$$x=-120$$

$$x=80$$

Guruhlash orqali faktorlash qadamlari

x qiymati 0,20 qiymatlaridan birortasiga teng bo‘lmaydi, chunki nolga bo‘lish mumkin emas. Tenglamaning ikkala tarafini x\left(x-20\right) ga, x,x-20 ning eng kichik karralisiga ko‘paytiring.

$$\left(x-20\right)\times 4800+x\left(x-20\right)\times 10=x\times 4200$$

x-20 ga 4800 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$4800x-96000+x\left(x-20\right)\times 10=x\times 4200$$

x ga x-20 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$4800x-96000+\left(x^{2}-20x\right)\times 10=x\times 4200$$

x^{2}-20x ga 10 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$4800x-96000+10x^{2}-200x=x\times 4200$$

4600x ni olish uchun 4800x va -200x ni birlashtirish.

$$4600x-96000+10x^{2}=x\times 4200$$

Ikkala tarafdan x\times 4200 ni ayirish.

$$4600x-96000+10x^{2}-x\times 4200=0$$

400x ni olish uchun 4600x va -x\times 4200 ni birlashtirish.

$$400x-96000+10x^{2}=0$$

Ikki tarafini 10 ga bo‘ling.

$$40x-9600+x^{2}=0$$

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

$$x^{2}+40x-9600=0$$

Tenglamani yechish uchun guruhlash orqali chap qoʻl tomonni faktorlang. Avvalo, chap qoʻl tomon x^{2}+ax+bx-9600 sifatida qayta yozilishi kerak. a va b ni topish uchun yechiladigan tizimni sozlang.

$$a+b=40 ab=1\left(-9600\right)=-9600$$

ab manfiy boʻlganda, a va b da qarama-qarshi belgilar bor. a+b musbat boʻlganda, musbat sonda manfiyga nisbatdan kattaroq mutlaq qiymat bor. -9600-mahsulotni beruvchi bunday butun juftliklarni roʻyxat qiling.

$$-1,9600 -2,4800 -3,3200 -4,2400 -5,1920 -6,1600 -8,1200 -10,960 -12,800 -15,640 -16,600 -20,480 -24,400 -25,384 -30,320 -32,300 -40,240 -48,200 -50,192 -60,160 -64,150 -75,128 -80,120 -96,100$$

Har bir juftlik yigʻindisini hisoblang.

$$-1+9600=9599 -2+4800=4798 -3+3200=3197 -4+2400=2396 -5+1920=1915 -6+1600=1594 -8+1200=1192 -10+960=950 -12+800=788 -15+640=625 -16+600=584 -20+480=460 -24+400=376 -25+384=359 -30+320=290 -32+300=268 -40+240=200 -48+200=152 -50+192=142 -60+160=100 -64+150=86 -75+128=53 -80+120=40 -96+100=4$$

Yechim – 40 yigʻindisini beruvchi juftlik.

$$a=-80 b=120$$

x^{2}+40x-9600 ni \left(x^{2}-80x\right)+\left(120x-9600\right) sifatida qaytadan yozish.

$$\left(x^{2}-80x\right)+\left(120x-9600\right)$$

Birinchi guruhda x ni va ikkinchi guruhda 120 ni faktordan chiqaring.

$$x\left(x-80\right)+120\left(x-80\right)$$

Distributiv funktsiyasidan foydalangan holda x-80 umumiy terminini chiqaring.

$$\left(x-80\right)\left(x+120\right)$$

Tenglamani yechish uchun x-80=0 va x+120=0 ni yeching.

$$x=80 x=-120$$

Kvadrat tenglamalaridan foydalanish qoidalari

x qiymati 0,20 qiymatlaridan birortasiga teng bo‘lmaydi, chunki nolga bo‘lish mumkin emas. Tenglamaning ikkala tarafini x\left(x-20\right) ga, x,x-20 ning eng kichik karralisiga ko‘paytiring.

$$\left(x-20\right)\times 4800+x\left(x-20\right)\times 10=x\times 4200$$

x-20 ga 4800 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$4800x-96000+x\left(x-20\right)\times 10=x\times 4200$$

x ga x-20 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$4800x-96000+\left(x^{2}-20x\right)\times 10=x\times 4200$$

x^{2}-20x ga 10 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$4800x-96000+10x^{2}-200x=x\times 4200$$

4600x ni olish uchun 4800x va -200x ni birlashtirish.

$$4600x-96000+10x^{2}=x\times 4200$$

Ikkala tarafdan x\times 4200 ni ayirish.

$$4600x-96000+10x^{2}-x\times 4200=0$$

400x ni olish uchun 4600x va -x\times 4200 ni birlashtirish.

$$400x-96000+10x^{2}=0$$

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.

$$10x^{2}+400x-96000=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 10 ni a, 400 ni b va -96000 ni c bilan almashtiring.

$$x=\frac{-400±\sqrt{400^{2}-4\times 10\left(-96000\right)}}{2\times 10}$$

400 kvadratini chiqarish.

$$x=\frac{-400±\sqrt{160000-4\times 10\left(-96000\right)}}{2\times 10}$$

-4 ni 10 marotabaga ko'paytirish.

$$x=\frac{-400±\sqrt{160000-40\left(-96000\right)}}{2\times 10}$$

-40 ni -96000 marotabaga ko'paytirish.

$$x=\frac{-400±\sqrt{160000+3840000}}{2\times 10}$$

160000 ni 3840000 ga qo'shish.

$$x=\frac{-400±\sqrt{4000000}}{2\times 10}$$

4000000 ning kvadrat ildizini chiqarish.

$$x=\frac{-400±2000}{2\times 10}$$

2 ni 10 marotabaga ko'paytirish.

$$x=\frac{-400±2000}{20}$$

x=\frac{-400±2000}{20} tenglamasini yeching, bunda ± musbat. -400 ni 2000 ga qo'shish.

$$x=\frac{1600}{20}$$

1600 ni 20 ga bo'lish.

$$x=80$$

x=\frac{-400±2000}{20} tenglamasini yeching, bunda ± manfiy. -400 dan 2000 ni ayirish.

$$x=-\frac{2400}{20}$$

-2400 ni 20 ga bo'lish.

$$x=-120$$

Tenglama yechildi.

$$x=80 x=-120$$

Kvadratni yakunlash bo'yicha qadamlar

x qiymati 0,20 qiymatlaridan birortasiga teng bo‘lmaydi, chunki nolga bo‘lish mumkin emas. Tenglamaning ikkala tarafini x\left(x-20\right) ga, x,x-20 ning eng kichik karralisiga ko‘paytiring.

$$\left(x-20\right)\times 4800+x\left(x-20\right)\times 10=x\times 4200$$

x-20 ga 4800 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$4800x-96000+x\left(x-20\right)\times 10=x\times 4200$$

x ga x-20 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$4800x-96000+\left(x^{2}-20x\right)\times 10=x\times 4200$$

x^{2}-20x ga 10 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$4800x-96000+10x^{2}-200x=x\times 4200$$

4600x ni olish uchun 4800x va -200x ni birlashtirish.

$$4600x-96000+10x^{2}=x\times 4200$$

Ikkala tarafdan x\times 4200 ni ayirish.

$$4600x-96000+10x^{2}-x\times 4200=0$$

400x ni olish uchun 4600x va -x\times 4200 ni birlashtirish.

$$400x-96000+10x^{2}=0$$

96000 ni ikki tarafga qo’shing. Har qanday songa nolni qo‘shsangiz, o‘zi chiqadi.

$$400x+10x^{2}=96000$$

Bu kabi kvadrat tenglamalarni kvadratni yakunlab yechish mumkin. Kvadratni yechish uchun tenglama avval ushbu shaklda bo'lishi shart: x^{2}+bx=c.

$$10x^{2}+400x=96000$$

Ikki tarafini 10 ga bo‘ling.

$$\frac{10x^{2}+400x}{10}=\frac{96000}{10}$$

10 ga bo'lish 10 ga ko'paytirishni bekor qiladi.

$$x^{2}+\frac{400}{10}x=\frac{96000}{10}$$

400 ni 10 ga bo'lish.

$$x^{2}+40x=\frac{96000}{10}$$

96000 ni 10 ga bo'lish.

$$x^{2}+40x=9600$$

40 ni bo‘lish, x shartining koeffitsienti, 2 ga 20 olish uchun. Keyin, 20 ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}+40x+20^{2}=9600+20^{2}$$

20 kvadratini chiqarish.

$$x^{2}+40x+400=9600+400$$

9600 ni 400 ga qo'shish.

$$x^{2}+40x+400=10000$$

x^{2}+40x+400 omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(x+20\right)^{2}=10000$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x+20\right)^{2}}=\sqrt{10000}$$

Qisqartirish.

$$x+20=100 x+20=-100$$

Tenglamaning ikkala tarafidan 20 ni ayirish.

$$x=80 x=-120$$