400/x+400div5*2/x+400div5*3/x-20=11 eching

$$\frac{ 400 }{ x } + \frac{ 400 \div 5 \times 2 }{ x } + \frac{ 400 \div 5 \times 3 }{ x-20 } =11$$

x uchun yechish

$$x=80$$

$$x = \frac{140}{11} = 12\frac{8}{11} \approx 12,727272727$$

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.

$$\left(x-20\right)\times 400+\left(x-20\right)\times \frac{400}{5}\times 2+x\times \frac{400}{5}\times 3=11x\left(x-20\right)$$

x-20 ga 400 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$400x-8000+\left(x-20\right)\times \frac{400}{5}\times 2+x\times \frac{400}{5}\times 3=11x\left(x-20\right)$$

80 ni olish uchun 400 ni 5 ga bo‘ling.

$$400x-8000+\left(x-20\right)\times 80\times 2+x\times \frac{400}{5}\times 3=11x\left(x-20\right)$$

160 hosil qilish uchun 80 va 2 ni ko'paytirish.

$$400x-8000+\left(x-20\right)\times 160+x\times \frac{400}{5}\times 3=11x\left(x-20\right)$$

x-20 ga 160 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$400x-8000+160x-3200+x\times \frac{400}{5}\times 3=11x\left(x-20\right)$$

560x ni olish uchun 400x va 160x ni birlashtirish.

$$560x-8000-3200+x\times \frac{400}{5}\times 3=11x\left(x-20\right)$$

-11200 olish uchun -8000 dan 3200 ni ayirish.

$$560x-11200+x\times \frac{400}{5}\times 3=11x\left(x-20\right)$$

80 ni olish uchun 400 ni 5 ga bo‘ling.

$$560x-11200+x\times 80\times 3=11x\left(x-20\right)$$

240 hosil qilish uchun 80 va 3 ni ko'paytirish.

$$560x-11200+x\times 240=11x\left(x-20\right)$$

800x ni olish uchun 560x va x\times 240 ni birlashtirish.

$$800x-11200=11x\left(x-20\right)$$

11x ga x-20 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$800x-11200=11x^{2}-220x$$

Ikkala tarafdan 11x^{2} ni ayirish.

$$800x-11200-11x^{2}=-220x$$

220x ni ikki tarafga qo’shing.

$$800x-11200-11x^{2}+220x=0$$

1020x ni olish uchun 800x va 220x ni birlashtirish.

$$1020x-11200-11x^{2}=0$$

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.

$$-11x^{2}+1020x-11200=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} -11 ni a, 1020 ni b va -11200 ni c bilan almashtiring.

$$x=\frac{-1020±\sqrt{1020^{2}-4\left(-11\right)\left(-11200\right)}}{2\left(-11\right)}$$

1020 kvadratini chiqarish.

$$x=\frac{-1020±\sqrt{1040400-4\left(-11\right)\left(-11200\right)}}{2\left(-11\right)}$$

-4 ni -11 marotabaga ko'paytirish.

$$x=\frac{-1020±\sqrt{1040400+44\left(-11200\right)}}{2\left(-11\right)}$$

44 ni -11200 marotabaga ko'paytirish.

$$x=\frac{-1020±\sqrt{1040400-492800}}{2\left(-11\right)}$$

1040400 ni -492800 ga qo'shish.

$$x=\frac{-1020±\sqrt{547600}}{2\left(-11\right)}$$

547600 ning kvadrat ildizini chiqarish.

$$x=\frac{-1020±740}{2\left(-11\right)}$$

2 ni -11 marotabaga ko'paytirish.

$$x=\frac{-1020±740}{-22}$$

x=\frac{-1020±740}{-22} tenglamasini yeching, bunda ± musbat. -1020 ni 740 ga qo'shish.

$$x=-\frac{280}{-22}$$

\frac{-280}{-22} ulushini 2 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

$$x=\frac{140}{11}$$

x=\frac{-1020±740}{-22} tenglamasini yeching, bunda ± manfiy. -1020 dan 740 ni ayirish.

$$x=-\frac{1760}{-22}$$

-1760 ni -22 ga bo'lish.

$$x=80$$

Tenglama yechildi.

$$x=\frac{140}{11} x=80$$

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.

$$\left(x-20\right)\times 400+\left(x-20\right)\times \frac{400}{5}\times 2+x\times \frac{400}{5}\times 3=11x\left(x-20\right)$$

x-20 ga 400 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$400x-8000+\left(x-20\right)\times \frac{400}{5}\times 2+x\times \frac{400}{5}\times 3=11x\left(x-20\right)$$

80 ni olish uchun 400 ni 5 ga bo‘ling.

$$400x-8000+\left(x-20\right)\times 80\times 2+x\times \frac{400}{5}\times 3=11x\left(x-20\right)$$

160 hosil qilish uchun 80 va 2 ni ko'paytirish.

$$400x-8000+\left(x-20\right)\times 160+x\times \frac{400}{5}\times 3=11x\left(x-20\right)$$

x-20 ga 160 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$400x-8000+160x-3200+x\times \frac{400}{5}\times 3=11x\left(x-20\right)$$

560x ni olish uchun 400x va 160x ni birlashtirish.

$$560x-8000-3200+x\times \frac{400}{5}\times 3=11x\left(x-20\right)$$

-11200 olish uchun -8000 dan 3200 ni ayirish.

$$560x-11200+x\times \frac{400}{5}\times 3=11x\left(x-20\right)$$

80 ni olish uchun 400 ni 5 ga bo‘ling.

$$560x-11200+x\times 80\times 3=11x\left(x-20\right)$$

240 hosil qilish uchun 80 va 3 ni ko'paytirish.

$$560x-11200+x\times 240=11x\left(x-20\right)$$

800x ni olish uchun 560x va x\times 240 ni birlashtirish.

$$800x-11200=11x\left(x-20\right)$$

11x ga x-20 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$800x-11200=11x^{2}-220x$$

Ikkala tarafdan 11x^{2} ni ayirish.

$$800x-11200-11x^{2}=-220x$$

220x ni ikki tarafga qo’shing.

$$800x-11200-11x^{2}+220x=0$$

1020x ni olish uchun 800x va 220x ni birlashtirish.

$$1020x-11200-11x^{2}=0$$

11200 ni ikki tarafga qo’shing. Har qanday songa nolni qo‘shsangiz, o‘zi chiqadi.

$$1020x-11x^{2}=11200$$

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

$$-11x^{2}+1020x=11200$$

Ikki tarafini -11 ga bo‘ling.

$$\frac{-11x^{2}+1020x}{-11}=\frac{11200}{-11}$$

-11 ga bo'lish -11 ga ko'paytirishni bekor qiladi.

$$x^{2}+\frac{1020}{-11}x=\frac{11200}{-11}$$

1020 ni -11 ga bo'lish.

$$x^{2}-\frac{1020}{11}x=\frac{11200}{-11}$$

11200 ni -11 ga bo'lish.

$$x^{2}-\frac{1020}{11}x=-\frac{11200}{11}$$

-\frac{1020}{11} ni bo‘lish, x shartining koeffitsienti, 2 ga -\frac{510}{11} olish uchun. Keyin, -\frac{510}{11} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}-\frac{1020}{11}x+\left(-\frac{510}{11}\right)^{2}=-\frac{11200}{11}+\left(-\frac{510}{11}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib -\frac{510}{11} kvadratini chiqarish.

$$x^{2}-\frac{1020}{11}x+\frac{260100}{121}=-\frac{11200}{11}+\frac{260100}{121}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali -\frac{11200}{11} ni \frac{260100}{121} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$x^{2}-\frac{1020}{11}x+\frac{260100}{121}=\frac{136900}{121}$$

x^{2}-\frac{1020}{11}x+\frac{260100}{121} omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(x-\frac{510}{11}\right)^{2}=\frac{136900}{121}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x-\frac{510}{11}\right)^{2}}=\sqrt{\frac{136900}{121}}$$

Qisqartirish.

$$x-\frac{510}{11}=\frac{370}{11} x-\frac{510}{11}=-\frac{370}{11}$$

\frac{510}{11} ni tenglamaning ikkala tarafiga qo'shish.

$$x=80 x=\frac{140}{11}$$