81/45-y+15=81/y eching

$$\frac{ 81 }{ 45-y } +15 = \frac{ 81 }{ y }$$

y uchun yechish

$$y = \frac{9 \sqrt{661} + 279}{10} \approx 51,038928238$$

$$y = \frac{279 - 9 \sqrt{661}}{10} \approx 4,761071762$$

Kvadrat tenglamalaridan foydalanish qoidalari

y qiymati 0,45 qiymatlaridan birortasiga teng bo‘lmaydi, chunki nolga bo‘lish mumkin emas. Tenglamaning ikkala tarafini y\left(y-45\right) ga, 45-y,y ning eng kichik karralisiga ko‘paytiring.

$$-y\times 81+y\left(y-45\right)\times 15=\left(y-45\right)\times 81$$

-81 hosil qilish uchun -1 va 81 ni ko'paytirish.

$$-81y+y\left(y-45\right)\times 15=\left(y-45\right)\times 81$$

y ga y-45 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$-81y+\left(y^{2}-45y\right)\times 15=\left(y-45\right)\times 81$$

y^{2}-45y ga 15 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$-81y+15y^{2}-675y=\left(y-45\right)\times 81$$

-756y ni olish uchun -81y va -675y ni birlashtirish.

$$-756y+15y^{2}=\left(y-45\right)\times 81$$

y-45 ga 81 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$-756y+15y^{2}=81y-3645$$

Ikkala tarafdan 81y ni ayirish.

$$-756y+15y^{2}-81y=-3645$$

-837y ni olish uchun -756y va -81y ni birlashtirish.

$$-837y+15y^{2}=-3645$$

3645 ni ikki tarafga qo’shing.

$$-837y+15y^{2}+3645=0$$

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.

$$15y^{2}-837y+3645=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 15 ni a, -837 ni b va 3645 ni c bilan almashtiring.

$$y=\frac{-\left(-837\right)±\sqrt{\left(-837\right)^{2}-4\times 15\times 3645}}{2\times 15}$$

-837 kvadratini chiqarish.

$$y=\frac{-\left(-837\right)±\sqrt{700569-4\times 15\times 3645}}{2\times 15}$$

-4 ni 15 marotabaga ko'paytirish.

$$y=\frac{-\left(-837\right)±\sqrt{700569-60\times 3645}}{2\times 15}$$

-60 ni 3645 marotabaga ko'paytirish.

$$y=\frac{-\left(-837\right)±\sqrt{700569-218700}}{2\times 15}$$

700569 ni -218700 ga qo'shish.

$$y=\frac{-\left(-837\right)±\sqrt{481869}}{2\times 15}$$

481869 ning kvadrat ildizini chiqarish.

$$y=\frac{-\left(-837\right)±27\sqrt{661}}{2\times 15}$$

-837 ning teskarisi 837 ga teng.

$$y=\frac{837±27\sqrt{661}}{2\times 15}$$

2 ni 15 marotabaga ko'paytirish.

$$y=\frac{837±27\sqrt{661}}{30}$$

y=\frac{837±27\sqrt{661}}{30} tenglamasini yeching, bunda ± musbat. 837 ni 27\sqrt{661} ga qo'shish.

$$y=\frac{27\sqrt{661}+837}{30}$$

837+27\sqrt{661} ni 30 ga bo'lish.

$$y=\frac{9\sqrt{661}+279}{10}$$

y=\frac{837±27\sqrt{661}}{30} tenglamasini yeching, bunda ± manfiy. 837 dan 27\sqrt{661} ni ayirish.

$$y=\frac{837-27\sqrt{661}}{30}$$

837-27\sqrt{661} ni 30 ga bo'lish.

$$y=\frac{279-9\sqrt{661}}{10}$$

Tenglama yechildi.

$$y=\frac{9\sqrt{661}+279}{10} y=\frac{279-9\sqrt{661}}{10}$$

Kvadratni yakunlash bo'yicha qadamlar

y qiymati 0,45 qiymatlaridan birortasiga teng bo‘lmaydi, chunki nolga bo‘lish mumkin emas. Tenglamaning ikkala tarafini y\left(y-45\right) ga, 45-y,y ning eng kichik karralisiga ko‘paytiring.

$$-y\times 81+y\left(y-45\right)\times 15=\left(y-45\right)\times 81$$

-81 hosil qilish uchun -1 va 81 ni ko'paytirish.

$$-81y+y\left(y-45\right)\times 15=\left(y-45\right)\times 81$$

y ga y-45 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$-81y+\left(y^{2}-45y\right)\times 15=\left(y-45\right)\times 81$$

y^{2}-45y ga 15 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$-81y+15y^{2}-675y=\left(y-45\right)\times 81$$

-756y ni olish uchun -81y va -675y ni birlashtirish.

$$-756y+15y^{2}=\left(y-45\right)\times 81$$

y-45 ga 81 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$-756y+15y^{2}=81y-3645$$

Ikkala tarafdan 81y ni ayirish.

$$-756y+15y^{2}-81y=-3645$$

-837y ni olish uchun -756y va -81y ni birlashtirish.

$$-837y+15y^{2}=-3645$$

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

$$15y^{2}-837y=-3645$$

Ikki tarafini 15 ga bo‘ling.

$$\frac{15y^{2}-837y}{15}=-\frac{3645}{15}$$

15 ga bo'lish 15 ga ko'paytirishni bekor qiladi.

$$y^{2}+\left(-\frac{837}{15}\right)y=-\frac{3645}{15}$$

\frac{-837}{15} ulushini 3 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

$$y^{2}-\frac{279}{5}y=-\frac{3645}{15}$$

-3645 ni 15 ga bo'lish.

$$y^{2}-\frac{279}{5}y=-243$$

-\frac{279}{5} ni bo‘lish, x shartining koeffitsienti, 2 ga -\frac{279}{10} olish uchun. Keyin, -\frac{279}{10} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$y^{2}-\frac{279}{5}y+\left(-\frac{279}{10}\right)^{2}=-243+\left(-\frac{279}{10}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib -\frac{279}{10} kvadratini chiqarish.

$$y^{2}-\frac{279}{5}y+\frac{77841}{100}=-243+\frac{77841}{100}$$

-243 ni \frac{77841}{100} ga qo'shish.

$$y^{2}-\frac{279}{5}y+\frac{77841}{100}=\frac{53541}{100}$$

y^{2}-\frac{279}{5}y+\frac{77841}{100} omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(y-\frac{279}{10}\right)^{2}=\frac{53541}{100}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(y-\frac{279}{10}\right)^{2}}=\sqrt{\frac{53541}{100}}$$

Qisqartirish.

$$y-\frac{279}{10}=\frac{9\sqrt{661}}{10} y-\frac{279}{10}=-\frac{9\sqrt{661}}{10}$$

\frac{279}{10} ni tenglamaning ikkala tarafiga qo'shish.

$$y=\frac{9\sqrt{661}+279}{10} y=\frac{279-9\sqrt{661}}{10}$$