x^2/(164-2x)^2=400 eching

$$\frac { x ^ { 2 } } { ( 164 - 2 x ) ^ { 2 } } = 400$$

x uchun yechish

$$x = \frac{3280}{39} = 84\frac{4}{39} \approx 84,102564103$$

$$x=80$$

Kvadrat tenglamalaridan foydalanish qoidalari

x qiymati 82 teng bo‘lmaydi, chunki nolga bo‘lish mumkin emas. Tenglamaning ikkala tarafini 4\left(x-82\right)^{2} ga ko'paytirish.

$$x^{2}=1600\left(x-82\right)^{2}$$

\left(a-b\right)^{2}=a^{2}-2ab+b^{2} binom teoremasini \left(x-82\right)^{2} kengaytirilishi uchun ishlating.

$$x^{2}=1600\left(x^{2}-164x+6724\right)$$

1600 ga x^{2}-164x+6724 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$x^{2}=1600x^{2}-262400x+10758400$$

Ikkala tarafdan 1600x^{2} ni ayirish.

$$x^{2}-1600x^{2}=-262400x+10758400$$

-1599x^{2} ni olish uchun x^{2} va -1600x^{2} ni birlashtirish.

$$-1599x^{2}=-262400x+10758400$$

262400x ni ikki tarafga qo’shing.

$$-1599x^{2}+262400x=10758400$$

Ikkala tarafdan 10758400 ni ayirish.

$$-1599x^{2}+262400x-10758400=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} -1599 ni a, 262400 ni b va -10758400 ni c bilan almashtiring.

$$x=\frac{-262400±\sqrt{262400^{2}-4\left(-1599\right)\left(-10758400\right)}}{2\left(-1599\right)}$$

262400 kvadratini chiqarish.

$$x=\frac{-262400±\sqrt{68853760000-4\left(-1599\right)\left(-10758400\right)}}{2\left(-1599\right)}$$

-4 ni -1599 marotabaga ko'paytirish.

$$x=\frac{-262400±\sqrt{68853760000+6396\left(-10758400\right)}}{2\left(-1599\right)}$$

6396 ni -10758400 marotabaga ko'paytirish.

$$x=\frac{-262400±\sqrt{68853760000-68810726400}}{2\left(-1599\right)}$$

68853760000 ni -68810726400 ga qo'shish.

$$x=\frac{-262400±\sqrt{43033600}}{2\left(-1599\right)}$$

43033600 ning kvadrat ildizini chiqarish.

$$x=\frac{-262400±6560}{2\left(-1599\right)}$$

2 ni -1599 marotabaga ko'paytirish.

$$x=\frac{-262400±6560}{-3198}$$

x=\frac{-262400±6560}{-3198} tenglamasini yeching, bunda ± musbat. -262400 ni 6560 ga qo'shish.

$$x=-\frac{255840}{-3198}$$

-255840 ni -3198 ga bo'lish.

$$x=80$$

x=\frac{-262400±6560}{-3198} tenglamasini yeching, bunda ± manfiy. -262400 dan 6560 ni ayirish.

$$x=-\frac{268960}{-3198}$$

\frac{-268960}{-3198} ulushini 82 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

$$x=\frac{3280}{39}$$

Tenglama yechildi.

$$x=80 x=\frac{3280}{39}$$

Kvadratni yakunlash bo'yicha qadamlar

x qiymati 82 teng bo‘lmaydi, chunki nolga bo‘lish mumkin emas. Tenglamaning ikkala tarafini 4\left(x-82\right)^{2} ga ko'paytirish.

$$x^{2}=1600\left(x-82\right)^{2}$$

\left(a-b\right)^{2}=a^{2}-2ab+b^{2} binom teoremasini \left(x-82\right)^{2} kengaytirilishi uchun ishlating.

$$x^{2}=1600\left(x^{2}-164x+6724\right)$$

1600 ga x^{2}-164x+6724 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$x^{2}=1600x^{2}-262400x+10758400$$

Ikkala tarafdan 1600x^{2} ni ayirish.

$$x^{2}-1600x^{2}=-262400x+10758400$$

-1599x^{2} ni olish uchun x^{2} va -1600x^{2} ni birlashtirish.

$$-1599x^{2}=-262400x+10758400$$

262400x ni ikki tarafga qo’shing.

$$-1599x^{2}+262400x=10758400$$

Ikki tarafini -1599 ga bo‘ling.

$$\frac{-1599x^{2}+262400x}{-1599}=\frac{10758400}{-1599}$$

-1599 ga bo'lish -1599 ga ko'paytirishni bekor qiladi.

$$x^{2}+\frac{262400}{-1599}x=\frac{10758400}{-1599}$$

\frac{262400}{-1599} ulushini 41 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

$$x^{2}-\frac{6400}{39}x=\frac{10758400}{-1599}$$

\frac{10758400}{-1599} ulushini 41 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

$$x^{2}-\frac{6400}{39}x=-\frac{262400}{39}$$

-\frac{6400}{39} ni bo‘lish, x shartining koeffitsienti, 2 ga -\frac{3200}{39} olish uchun. Keyin, -\frac{3200}{39} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}-\frac{6400}{39}x+\left(-\frac{3200}{39}\right)^{2}=-\frac{262400}{39}+\left(-\frac{3200}{39}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib -\frac{3200}{39} kvadratini chiqarish.

$$x^{2}-\frac{6400}{39}x+\frac{10240000}{1521}=-\frac{262400}{39}+\frac{10240000}{1521}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali -\frac{262400}{39} ni \frac{10240000}{1521} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$x^{2}-\frac{6400}{39}x+\frac{10240000}{1521}=\frac{6400}{1521}$$

x^{2}-\frac{6400}{39}x+\frac{10240000}{1521} omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(x-\frac{3200}{39}\right)^{2}=\frac{6400}{1521}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x-\frac{3200}{39}\right)^{2}}=\sqrt{\frac{6400}{1521}}$$

Qisqartirish.

$$x-\frac{3200}{39}=\frac{80}{39} x-\frac{3200}{39}=-\frac{80}{39}$$

\frac{3200}{39} ni tenglamaning ikkala tarafiga qo'shish.

$$x=\frac{3280}{39} x=80$$