81/41-y+15=71/y eching

$$\frac { 81 } { 41 - y } + 15 = \frac { 71 } { y }$$

y uchun yechish

$$y = \frac{\sqrt{413629} + 767}{30} \approx 47,004665122$$

$$y = \frac{767 - \sqrt{413629}}{30} \approx 4,128668211$$

Kvadrat tenglamalaridan foydalanish qoidalari

y qiymati 0,41 qiymatlaridan birortasiga teng bo‘lmaydi, chunki nolga bo‘lish mumkin emas. Tenglamaning ikkala tarafini y\left(y-41\right) ga, 41-y,y ning eng kichik karralisiga ko‘paytiring.

$$-y\times 81+y\left(y-41\right)\times 15=\left(y-41\right)\times 71$$

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

$$-81y+y\left(y-41\right)\times 15=\left(y-41\right)\times 71$$

y ga y-41 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$-81y+\left(y^{2}-41y\right)\times 15=\left(y-41\right)\times 71$$

y^{2}-41y ga 15 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$-81y+15y^{2}-615y=\left(y-41\right)\times 71$$

-696y ni olish uchun -81y va -615y ni birlashtirish.

$$-696y+15y^{2}=\left(y-41\right)\times 71$$

y-41 ga 71 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$-696y+15y^{2}=71y-2911$$

Ikkala tarafdan 71y ni ayirish.

$$-696y+15y^{2}-71y=-2911$$

-767y ni olish uchun -696y va -71y ni birlashtirish.

$$-767y+15y^{2}=-2911$$

2911 ni ikki tarafga qo’shing.

$$-767y+15y^{2}+2911=0$$

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.

$$15y^{2}-767y+2911=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 15 ni a, -767 ni b va 2911 ni c bilan almashtiring.

$$y=\frac{-\left(-767\right)±\sqrt{\left(-767\right)^{2}-4\times 15\times 2911}}{2\times 15}$$

-767 kvadratini chiqarish.

$$y=\frac{-\left(-767\right)±\sqrt{588289-4\times 15\times 2911}}{2\times 15}$$

-4 ni 15 marotabaga ko'paytirish.

$$y=\frac{-\left(-767\right)±\sqrt{588289-60\times 2911}}{2\times 15}$$

-60 ni 2911 marotabaga ko'paytirish.

$$y=\frac{-\left(-767\right)±\sqrt{588289-174660}}{2\times 15}$$

588289 ni -174660 ga qo'shish.

$$y=\frac{-\left(-767\right)±\sqrt{413629}}{2\times 15}$$

-767 ning teskarisi 767 ga teng.

$$y=\frac{767±\sqrt{413629}}{2\times 15}$$

2 ni 15 marotabaga ko'paytirish.

$$y=\frac{767±\sqrt{413629}}{30}$$

y=\frac{767±\sqrt{413629}}{30} tenglamasini yeching, bunda ± musbat. 767 ni \sqrt{413629} ga qo'shish.

$$y=\frac{\sqrt{413629}+767}{30}$$

y=\frac{767±\sqrt{413629}}{30} tenglamasini yeching, bunda ± manfiy. 767 dan \sqrt{413629} ni ayirish.

$$y=\frac{767-\sqrt{413629}}{30}$$

Tenglama yechildi.

$$y=\frac{\sqrt{413629}+767}{30} y=\frac{767-\sqrt{413629}}{30}$$

Kvadratni yakunlash bo'yicha qadamlar

y qiymati 0,41 qiymatlaridan birortasiga teng bo‘lmaydi, chunki nolga bo‘lish mumkin emas. Tenglamaning ikkala tarafini y\left(y-41\right) ga, 41-y,y ning eng kichik karralisiga ko‘paytiring.

$$-y\times 81+y\left(y-41\right)\times 15=\left(y-41\right)\times 71$$

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

$$-81y+y\left(y-41\right)\times 15=\left(y-41\right)\times 71$$

y ga y-41 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$-81y+\left(y^{2}-41y\right)\times 15=\left(y-41\right)\times 71$$

y^{2}-41y ga 15 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$-81y+15y^{2}-615y=\left(y-41\right)\times 71$$

-696y ni olish uchun -81y va -615y ni birlashtirish.

$$-696y+15y^{2}=\left(y-41\right)\times 71$$

y-41 ga 71 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$-696y+15y^{2}=71y-2911$$

Ikkala tarafdan 71y ni ayirish.

$$-696y+15y^{2}-71y=-2911$$

-767y ni olish uchun -696y va -71y ni birlashtirish.

$$-767y+15y^{2}=-2911$$

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

$$15y^{2}-767y=-2911$$

Ikki tarafini 15 ga bo‘ling.

$$\frac{15y^{2}-767y}{15}=-\frac{2911}{15}$$

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

$$y^{2}-\frac{767}{15}y=-\frac{2911}{15}$$

-\frac{767}{15} ni bo‘lish, x shartining koeffitsienti, 2 ga -\frac{767}{30} olish uchun. Keyin, -\frac{767}{30} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$y^{2}-\frac{767}{15}y+\left(-\frac{767}{30}\right)^{2}=-\frac{2911}{15}+\left(-\frac{767}{30}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib -\frac{767}{30} kvadratini chiqarish.

$$y^{2}-\frac{767}{15}y+\frac{588289}{900}=-\frac{2911}{15}+\frac{588289}{900}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali -\frac{2911}{15} ni \frac{588289}{900} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$y^{2}-\frac{767}{15}y+\frac{588289}{900}=\frac{413629}{900}$$

y^{2}-\frac{767}{15}y+\frac{588289}{900} omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(y-\frac{767}{30}\right)^{2}=\frac{413629}{900}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(y-\frac{767}{30}\right)^{2}}=\sqrt{\frac{413629}{900}}$$

Qisqartirish.

$$y-\frac{767}{30}=\frac{\sqrt{413629}}{30} y-\frac{767}{30}=-\frac{\sqrt{413629}}{30}$$

\frac{767}{30} ni tenglamaning ikkala tarafiga qo'shish.

$$y=\frac{\sqrt{413629}+767}{30} y=\frac{767-\sqrt{413629}}{30}$$