400=x^2/left(284-xright)^2 eching

$$400= \frac{ { x }^{ 2 } }{ { \left(284-x \right) }^{ 2 } }$$

x uchun yechish

$$x = \frac{5680}{19} = 298\frac{18}{19} \approx 298,947368421$$

$$x = \frac{5680}{21} = 270\frac{10}{21} \approx 270,476190476$$

Kvadrat tenglamalaridan foydalanish qoidalari

x qiymati 284 teng bo‘lmaydi, chunki nolga bo‘lish mumkin emas. Tenglamaning ikkala tarafini \left(x-284\right)^{2} ga ko'paytirish.

$$400\left(x-284\right)^{2}=x^{2}$$

\left(a-b\right)^{2}=a^{2}-2ab+b^{2} binom teoremasini \left(x-284\right)^{2} kengaytirilishi uchun ishlating.

$$400\left(x^{2}-568x+80656\right)=x^{2}$$

400 ga x^{2}-568x+80656 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$400x^{2}-227200x+32262400=x^{2}$$

Ikkala tarafdan x^{2} ni ayirish.

$$400x^{2}-227200x+32262400-x^{2}=0$$

399x^{2} ni olish uchun 400x^{2} va -x^{2} ni birlashtirish.

$$399x^{2}-227200x+32262400=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 399 ni a, -227200 ni b va 32262400 ni c bilan almashtiring.

$$x=\frac{-\left(-227200\right)±\sqrt{\left(-227200\right)^{2}-4\times 399\times 32262400}}{2\times 399}$$

-227200 kvadratini chiqarish.

$$x=\frac{-\left(-227200\right)±\sqrt{51619840000-4\times 399\times 32262400}}{2\times 399}$$

-4 ni 399 marotabaga ko'paytirish.

$$x=\frac{-\left(-227200\right)±\sqrt{51619840000-1596\times 32262400}}{2\times 399}$$

-1596 ni 32262400 marotabaga ko'paytirish.

$$x=\frac{-\left(-227200\right)±\sqrt{51619840000-51490790400}}{2\times 399}$$

51619840000 ni -51490790400 ga qo'shish.

$$x=\frac{-\left(-227200\right)±\sqrt{129049600}}{2\times 399}$$

129049600 ning kvadrat ildizini chiqarish.

$$x=\frac{-\left(-227200\right)±11360}{2\times 399}$$

-227200 ning teskarisi 227200 ga teng.

$$x=\frac{227200±11360}{2\times 399}$$

2 ni 399 marotabaga ko'paytirish.

$$x=\frac{227200±11360}{798}$$

x=\frac{227200±11360}{798} tenglamasini yeching, bunda ± musbat. 227200 ni 11360 ga qo'shish.

$$x=\frac{238560}{798}$$

\frac{238560}{798} ulushini 42 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

$$x=\frac{5680}{19}$$

x=\frac{227200±11360}{798} tenglamasini yeching, bunda ± manfiy. 227200 dan 11360 ni ayirish.

$$x=\frac{215840}{798}$$

\frac{215840}{798} ulushini 38 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

$$x=\frac{5680}{21}$$

Tenglama yechildi.

$$x=\frac{5680}{19} x=\frac{5680}{21}$$

Kvadratni yakunlash bo'yicha qadamlar

x qiymati 284 teng bo‘lmaydi, chunki nolga bo‘lish mumkin emas. Tenglamaning ikkala tarafini \left(x-284\right)^{2} ga ko'paytirish.

$$400\left(x-284\right)^{2}=x^{2}$$

\left(a-b\right)^{2}=a^{2}-2ab+b^{2} binom teoremasini \left(x-284\right)^{2} kengaytirilishi uchun ishlating.

$$400\left(x^{2}-568x+80656\right)=x^{2}$$

400 ga x^{2}-568x+80656 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$400x^{2}-227200x+32262400=x^{2}$$

Ikkala tarafdan x^{2} ni ayirish.

$$400x^{2}-227200x+32262400-x^{2}=0$$

399x^{2} ni olish uchun 400x^{2} va -x^{2} ni birlashtirish.

$$399x^{2}-227200x+32262400=0$$

Ikkala tarafdan 32262400 ni ayirish. Har qanday sonni noldan ayirsangiz, o‘zining manfiyi chiqadi.

$$399x^{2}-227200x=-32262400$$

Ikki tarafini 399 ga bo‘ling.

$$\frac{399x^{2}-227200x}{399}=-\frac{32262400}{399}$$

399 ga bo'lish 399 ga ko'paytirishni bekor qiladi.

$$x^{2}-\frac{227200}{399}x=-\frac{32262400}{399}$$

-\frac{227200}{399} ni bo‘lish, x shartining koeffitsienti, 2 ga -\frac{113600}{399} olish uchun. Keyin, -\frac{113600}{399} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}-\frac{227200}{399}x+\left(-\frac{113600}{399}\right)^{2}=-\frac{32262400}{399}+\left(-\frac{113600}{399}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib -\frac{113600}{399} kvadratini chiqarish.

$$x^{2}-\frac{227200}{399}x+\frac{12904960000}{159201}=-\frac{32262400}{399}+\frac{12904960000}{159201}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali -\frac{32262400}{399} ni \frac{12904960000}{159201} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$x^{2}-\frac{227200}{399}x+\frac{12904960000}{159201}=\frac{32262400}{159201}$$

x^{2}-\frac{227200}{399}x+\frac{12904960000}{159201} omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(x-\frac{113600}{399}\right)^{2}=\frac{32262400}{159201}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x-\frac{113600}{399}\right)^{2}}=\sqrt{\frac{32262400}{159201}}$$

Qisqartirish.

$$x-\frac{113600}{399}=\frac{5680}{399} x-\frac{113600}{399}=-\frac{5680}{399}$$

\frac{113600}{399} ni tenglamaning ikkala tarafiga qo'shish.

$$x=\frac{5680}{19} x=\frac{5680}{21}$$