400/x+20+400/52/x+20+frac{400/53}{x}=11 eching

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

x uchun yechish

$$x = -\frac{80}{11} = -7\frac{3}{11} \approx -7,272727273$$

$$x=60$$

Guruhlash orqali faktorlash qadamlari

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

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

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

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

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

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

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

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

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

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

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

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

x+20 ga 240 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$560x+240x+4800=11x\left(x+20\right)$$

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

$$800x+4800=11x\left(x+20\right)$$

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

$$800x+4800=11x^{2}+220x$$

Ikkala tarafdan 11x^{2} ni ayirish.

$$800x+4800-11x^{2}=220x$$

Ikkala tarafdan 220x ni ayirish.

$$800x+4800-11x^{2}-220x=0$$

580x ni olish uchun 800x va -220x ni birlashtirish.

$$580x+4800-11x^{2}=0$$

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

$$-11x^{2}+580x+4800=0$$

Tenglamani yechish uchun guruhlash orqali chap qoʻl tomonni faktorlang. Avvalo, chap qoʻl tomon -11x^{2}+ax+bx+4800 sifatida qayta yozilishi kerak. a va b ni topish uchun yechiladigan tizimni sozlang.

$$a+b=580 ab=-11\times 4800=-52800$$

ab manfiy boʻlganda, a va b da qarama-qarshi belgilar bor. a+b musbat boʻlganda, musbat sonda manfiyga nisbatdan kattaroq mutlaq qiymat bor. -52800-mahsulotni beruvchi bunday butun juftliklarni roʻyxat qiling.

$$-1,52800 -2,26400 -3,17600 -4,13200 -5,10560 -6,8800 -8,6600 -10,5280 -11,4800 -12,4400 -15,3520 -16,3300 -20,2640 -22,2400 -24,2200 -25,2112 -30,1760 -32,1650 -33,1600 -40,1320 -44,1200 -48,1100 -50,1056 -55,960 -60,880 -64,825 -66,800 -75,704 -80,660 -88,600 -96,550 -100,528 -110,480 -120,440 -132,400 -150,352 -160,330 -165,320 -176,300 -192,275 -200,264 -220,240$$

Har bir juftlik yigʻindisini hisoblang.

$$-1+52800=52799 -2+26400=26398 -3+17600=17597 -4+13200=13196 -5+10560=10555 -6+8800=8794 -8+6600=6592 -10+5280=5270 -11+4800=4789 -12+4400=4388 -15+3520=3505 -16+3300=3284 -20+2640=2620 -22+2400=2378 -24+2200=2176 -25+2112=2087 -30+1760=1730 -32+1650=1618 -33+1600=1567 -40+1320=1280 -44+1200=1156 -48+1100=1052 -50+1056=1006 -55+960=905 -60+880=820 -64+825=761 -66+800=734 -75+704=629 -80+660=580 -88+600=512 -96+550=454 -100+528=428 -110+480=370 -120+440=320 -132+400=268 -150+352=202 -160+330=170 -165+320=155 -176+300=124 -192+275=83 -200+264=64 -220+240=20$$

Yechim – 580 yigʻindisini beruvchi juftlik.

$$a=660 b=-80$$

-11x^{2}+580x+4800 ni \left(-11x^{2}+660x\right)+\left(-80x+4800\right) sifatida qaytadan yozish.

$$\left(-11x^{2}+660x\right)+\left(-80x+4800\right)$$

Birinchi guruhda 11x ni va ikkinchi guruhda 80 ni faktordan chiqaring.

$$11x\left(-x+60\right)+80\left(-x+60\right)$$

Distributiv funktsiyasidan foydalangan holda -x+60 umumiy terminini chiqaring.

$$\left(-x+60\right)\left(11x+80\right)$$

Tenglamani yechish uchun -x+60=0 va 11x+80=0 ni yeching.

$$x=60 x=-\frac{80}{11}$$

Kvadrat tenglamalaridan foydalanish qoidalari

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

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

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

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

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

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

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

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

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

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

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

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

x+20 ga 240 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$560x+240x+4800=11x\left(x+20\right)$$

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

$$800x+4800=11x\left(x+20\right)$$

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

$$800x+4800=11x^{2}+220x$$

Ikkala tarafdan 11x^{2} ni ayirish.

$$800x+4800-11x^{2}=220x$$

Ikkala tarafdan 220x ni ayirish.

$$800x+4800-11x^{2}-220x=0$$

580x ni olish uchun 800x va -220x ni birlashtirish.

$$580x+4800-11x^{2}=0$$

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.

$$-11x^{2}+580x+4800=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} -11 ni a, 580 ni b va 4800 ni c bilan almashtiring.

$$x=\frac{-580±\sqrt{580^{2}-4\left(-11\right)\times 4800}}{2\left(-11\right)}$$

580 kvadratini chiqarish.

$$x=\frac{-580±\sqrt{336400-4\left(-11\right)\times 4800}}{2\left(-11\right)}$$

-4 ni -11 marotabaga ko'paytirish.

$$x=\frac{-580±\sqrt{336400+44\times 4800}}{2\left(-11\right)}$$

44 ni 4800 marotabaga ko'paytirish.

$$x=\frac{-580±\sqrt{336400+211200}}{2\left(-11\right)}$$

336400 ni 211200 ga qo'shish.

$$x=\frac{-580±\sqrt{547600}}{2\left(-11\right)}$$

547600 ning kvadrat ildizini chiqarish.

$$x=\frac{-580±740}{2\left(-11\right)}$$

2 ni -11 marotabaga ko'paytirish.

$$x=\frac{-580±740}{-22}$$

x=\frac{-580±740}{-22} tenglamasini yeching, bunda ± musbat. -580 ni 740 ga qo'shish.

$$x=\frac{160}{-22}$$

\frac{160}{-22} ulushini 2 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

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

x=\frac{-580±740}{-22} tenglamasini yeching, bunda ± manfiy. -580 dan 740 ni ayirish.

$$x=-\frac{1320}{-22}$$

-1320 ni -22 ga bo'lish.

$$x=60$$

Tenglama yechildi.

$$x=-\frac{80}{11} x=60$$

Kvadratni yakunlash bo'yicha qadamlar

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

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

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

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

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

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

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

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

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

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

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

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

x+20 ga 240 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$560x+240x+4800=11x\left(x+20\right)$$

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

$$800x+4800=11x\left(x+20\right)$$

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

$$800x+4800=11x^{2}+220x$$

Ikkala tarafdan 11x^{2} ni ayirish.

$$800x+4800-11x^{2}=220x$$

Ikkala tarafdan 220x ni ayirish.

$$800x+4800-11x^{2}-220x=0$$

580x ni olish uchun 800x va -220x ni birlashtirish.

$$580x+4800-11x^{2}=0$$

Ikkala tarafdan 4800 ni ayirish. Har qanday sonni noldan ayirsangiz, o‘zining manfiyi chiqadi.

$$580x-11x^{2}=-4800$$

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

$$-11x^{2}+580x=-4800$$

Ikki tarafini -11 ga bo‘ling.

$$\frac{-11x^{2}+580x}{-11}=-\frac{4800}{-11}$$

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

$$x^{2}+\frac{580}{-11}x=-\frac{4800}{-11}$$

580 ni -11 ga bo'lish.

$$x^{2}-\frac{580}{11}x=-\frac{4800}{-11}$$

-4800 ni -11 ga bo'lish.

$$x^{2}-\frac{580}{11}x=\frac{4800}{11}$$

-\frac{580}{11} ni bo‘lish, x shartining koeffitsienti, 2 ga -\frac{290}{11} olish uchun. Keyin, -\frac{290}{11} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}-\frac{580}{11}x+\left(-\frac{290}{11}\right)^{2}=\frac{4800}{11}+\left(-\frac{290}{11}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib -\frac{290}{11} kvadratini chiqarish.

$$x^{2}-\frac{580}{11}x+\frac{84100}{121}=\frac{4800}{11}+\frac{84100}{121}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali \frac{4800}{11} ni \frac{84100}{121} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$x^{2}-\frac{580}{11}x+\frac{84100}{121}=\frac{136900}{121}$$

x^{2}-\frac{580}{11}x+\frac{84100}{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.

$$\left(x-\frac{290}{11}\right)^{2}=\frac{136900}{121}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x-\frac{290}{11}\right)^{2}}=\sqrt{\frac{136900}{121}}$$

Qisqartirish.

$$x-\frac{290}{11}=\frac{370}{11} x-\frac{290}{11}=-\frac{370}{11}$$

\frac{290}{11} ni tenglamaning ikkala tarafiga qo'shish.

$$x=60 x=-\frac{80}{11}$$