72=8/(y-3)^2 eching

$$72 = \frac { 8 } { ( y - 3 ) ^ { 2 } }$$

y uchun yechish

$$y = \frac{8}{3} = 2\frac{2}{3} \approx 2,666666667$$

$$y = \frac{10}{3} = 3\frac{1}{3} \approx 3,333333333$$

Kvadrat tenglamalaridan foydalanish qoidalari

y qiymati 3 teng bo‘lmaydi, chunki nolga bo‘lish mumkin emas. Tenglamaning ikkala tarafini \left(y-3\right)^{2} ga ko'paytirish.

$$72\left(y-3\right)^{2}=8$$

\left(a-b\right)^{2}=a^{2}-2ab+b^{2} binom teoremasini \left(y-3\right)^{2} kengaytirilishi uchun ishlating.

$$72\left(y^{2}-6y+9\right)=8$$

72 ga y^{2}-6y+9 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$72y^{2}-432y+648=8$$

Ikkala tarafdan 8 ni ayirish.

$$72y^{2}-432y+648-8=0$$

640 olish uchun 648 dan 8 ni ayirish.

$$72y^{2}-432y+640=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 72 ni a, -432 ni b va 640 ni c bilan almashtiring.

$$y=\frac{-\left(-432\right)±\sqrt{\left(-432\right)^{2}-4\times 72\times 640}}{2\times 72}$$

-432 kvadratini chiqarish.

$$y=\frac{-\left(-432\right)±\sqrt{186624-4\times 72\times 640}}{2\times 72}$$

-4 ni 72 marotabaga ko'paytirish.

$$y=\frac{-\left(-432\right)±\sqrt{186624-288\times 640}}{2\times 72}$$

-288 ni 640 marotabaga ko'paytirish.

$$y=\frac{-\left(-432\right)±\sqrt{186624-184320}}{2\times 72}$$

186624 ni -184320 ga qo'shish.

$$y=\frac{-\left(-432\right)±\sqrt{2304}}{2\times 72}$$

2304 ning kvadrat ildizini chiqarish.

$$y=\frac{-\left(-432\right)±48}{2\times 72}$$

-432 ning teskarisi 432 ga teng.

$$y=\frac{432±48}{2\times 72}$$

2 ni 72 marotabaga ko'paytirish.

$$y=\frac{432±48}{144}$$

y=\frac{432±48}{144} tenglamasini yeching, bunda ± musbat. 432 ni 48 ga qo'shish.

$$y=\frac{480}{144}$$

\frac{480}{144} ulushini 48 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

$$y=\frac{10}{3}$$

y=\frac{432±48}{144} tenglamasini yeching, bunda ± manfiy. 432 dan 48 ni ayirish.

$$y=\frac{384}{144}$$

\frac{384}{144} ulushini 48 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

$$y=\frac{8}{3}$$

Tenglama yechildi.

$$y=\frac{10}{3} y=\frac{8}{3}$$

Kvadratni yakunlash bo'yicha qadamlar

y qiymati 3 teng bo‘lmaydi, chunki nolga bo‘lish mumkin emas. Tenglamaning ikkala tarafini \left(y-3\right)^{2} ga ko'paytirish.

$$72\left(y-3\right)^{2}=8$$

\left(a-b\right)^{2}=a^{2}-2ab+b^{2} binom teoremasini \left(y-3\right)^{2} kengaytirilishi uchun ishlating.

$$72\left(y^{2}-6y+9\right)=8$$

72 ga y^{2}-6y+9 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$72y^{2}-432y+648=8$$

Ikkala tarafdan 648 ni ayirish.

$$72y^{2}-432y=8-648$$

-640 olish uchun 8 dan 648 ni ayirish.

$$72y^{2}-432y=-640$$

Ikki tarafini 72 ga bo‘ling.

$$\frac{72y^{2}-432y}{72}=-\frac{640}{72}$$

72 ga bo'lish 72 ga ko'paytirishni bekor qiladi.

$$y^{2}+\left(-\frac{432}{72}\right)y=-\frac{640}{72}$$

-432 ni 72 ga bo'lish.

$$y^{2}-6y=-\frac{640}{72}$$

\frac{-640}{72} ulushini 8 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

$$y^{2}-6y=-\frac{80}{9}$$

-6 ni bo‘lish, x shartining koeffitsienti, 2 ga -3 olish uchun. Keyin, -3 ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$y^{2}-6y+\left(-3\right)^{2}=-\frac{80}{9}+\left(-3\right)^{2}$$

-3 kvadratini chiqarish.

$$y^{2}-6y+9=-\frac{80}{9}+9$$

-\frac{80}{9} ni 9 ga qo'shish.

$$y^{2}-6y+9=\frac{1}{9}$$

y^{2}-6y+9 omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(y-3\right)^{2}=\frac{1}{9}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(y-3\right)^{2}}=\sqrt{\frac{1}{9}}$$

Qisqartirish.

$$y-3=\frac{1}{3} y-3=-\frac{1}{3}$$

3 ni tenglamaning ikkala tarafiga qo'shish.

$$y=\frac{10}{3} y=\frac{8}{3}$$