75/x=75/x+15+1/4 eching

$$\frac { 75 } { x } = \frac { 75 } { x + 15 } + \frac { 1 } { 4 }$$

x uchun yechish

$$x=-75$$

$$x=60$$

Guruhlash orqali faktorlash qadamlari

x qiymati -15,0 qiymatlaridan birortasiga teng bo‘lmaydi, chunki nolga bo‘lish mumkin emas. Tenglamaning ikkala tarafini 4x\left(x+15\right) ga, x,x+15,4 ning eng kichik karralisiga ko‘paytiring.

$$\left(4x+60\right)\times 75=4x\times 75+4x\left(x+15\right)\times \frac{1}{4}$$

4x+60 ga 75 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$300x+4500=4x\times 75+4x\left(x+15\right)\times \frac{1}{4}$$

300 hosil qilish uchun 4 va 75 ni ko'paytirish.

$$300x+4500=300x+4x\left(x+15\right)\times \frac{1}{4}$$

1 hosil qilish uchun 4 va \frac{1}{4} ni ko'paytirish.

$$300x+4500=300x+x\left(x+15\right)$$

x ga x+15 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$300x+4500=300x+x^{2}+15x$$

315x ni olish uchun 300x va 15x ni birlashtirish.

$$300x+4500=315x+x^{2}$$

Ikkala tarafdan 315x ni ayirish.

$$300x+4500-315x=x^{2}$$

-15x ni olish uchun 300x va -315x ni birlashtirish.

$$-15x+4500=x^{2}$$

Ikkala tarafdan x^{2} ni ayirish.

$$-15x+4500-x^{2}=0$$

Polinomni standart shaklga keltirish uchun uni qayta tartiblang. Shartlarni eng yuqoridan eng pastki qiymat ko'rsatgichiga joylashtirish.

$$-x^{2}-15x+4500=0$$

Tenglamani yechish uchun guruhlash orqali chap qoʻl tomonni faktorlang. Avvalo, chap qoʻl tomon -x^{2}+ax+bx+4500 sifatida qayta yozilishi kerak. a va b ni topish uchun yechiladigan tizimni sozlang.

$$a+b=-15 ab=-4500=-4500$$

ab manfiy boʻlganda, a va b da qarama-qarshi belgilar bor. a+b manfiy boʻlganda, manfiy sonda musbatga nisbatdan kattaroq mutlaq qiymat bor. -4500-mahsulotni beruvchi bunday butun juftliklarni roʻyxat qiling.

$$1,-4500 2,-2250 3,-1500 4,-1125 5,-900 6,-750 9,-500 10,-450 12,-375 15,-300 18,-250 20,-225 25,-180 30,-150 36,-125 45,-100 50,-90 60,-75$$

Har bir juftlik yigʻindisini hisoblang.

$$1-4500=-4499 2-2250=-2248 3-1500=-1497 4-1125=-1121 5-900=-895 6-750=-744 9-500=-491 10-450=-440 12-375=-363 15-300=-285 18-250=-232 20-225=-205 25-180=-155 30-150=-120 36-125=-89 45-100=-55 50-90=-40 60-75=-15$$

Yechim – -15 yigʻindisini beruvchi juftlik.

$$a=60 b=-75$$

-x^{2}-15x+4500 ni \left(-x^{2}+60x\right)+\left(-75x+4500\right) sifatida qaytadan yozish.

$$\left(-x^{2}+60x\right)+\left(-75x+4500\right)$$

Birinchi guruhda x ni va ikkinchi guruhda 75 ni faktordan chiqaring.

$$x\left(-x+60\right)+75\left(-x+60\right)$$

Distributiv funktsiyasidan foydalangan holda -x+60 umumiy terminini chiqaring.

$$\left(-x+60\right)\left(x+75\right)$$

Tenglamani yechish uchun -x+60=0 va x+75=0 ni yeching.

$$x=60 x=-75$$

Kvadrat tenglamalaridan foydalanish qoidalari

x qiymati -15,0 qiymatlaridan birortasiga teng bo‘lmaydi, chunki nolga bo‘lish mumkin emas. Tenglamaning ikkala tarafini 4x\left(x+15\right) ga, x,x+15,4 ning eng kichik karralisiga ko‘paytiring.

$$\left(4x+60\right)\times 75=4x\times 75+4x\left(x+15\right)\times \frac{1}{4}$$

4x+60 ga 75 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$300x+4500=4x\times 75+4x\left(x+15\right)\times \frac{1}{4}$$

300 hosil qilish uchun 4 va 75 ni ko'paytirish.

$$300x+4500=300x+4x\left(x+15\right)\times \frac{1}{4}$$

1 hosil qilish uchun 4 va \frac{1}{4} ni ko'paytirish.

$$300x+4500=300x+x\left(x+15\right)$$

x ga x+15 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$300x+4500=300x+x^{2}+15x$$

315x ni olish uchun 300x va 15x ni birlashtirish.

$$300x+4500=315x+x^{2}$$

Ikkala tarafdan 315x ni ayirish.

$$300x+4500-315x=x^{2}$$

-15x ni olish uchun 300x va -315x ni birlashtirish.

$$-15x+4500=x^{2}$$

Ikkala tarafdan x^{2} ni ayirish.

$$-15x+4500-x^{2}=0$$

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.

$$-x^{2}-15x+4500=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} -1 ni a, -15 ni b va 4500 ni c bilan almashtiring.

$$x=\frac{-\left(-15\right)±\sqrt{\left(-15\right)^{2}-4\left(-1\right)\times 4500}}{2\left(-1\right)}$$

-15 kvadratini chiqarish.

$$x=\frac{-\left(-15\right)±\sqrt{225-4\left(-1\right)\times 4500}}{2\left(-1\right)}$$

-4 ni -1 marotabaga ko'paytirish.

$$x=\frac{-\left(-15\right)±\sqrt{225+4\times 4500}}{2\left(-1\right)}$$

4 ni 4500 marotabaga ko'paytirish.

$$x=\frac{-\left(-15\right)±\sqrt{225+18000}}{2\left(-1\right)}$$

225 ni 18000 ga qo'shish.

$$x=\frac{-\left(-15\right)±\sqrt{18225}}{2\left(-1\right)}$$

18225 ning kvadrat ildizini chiqarish.

$$x=\frac{-\left(-15\right)±135}{2\left(-1\right)}$$

-15 ning teskarisi 15 ga teng.

$$x=\frac{15±135}{2\left(-1\right)}$$

2 ni -1 marotabaga ko'paytirish.

$$x=\frac{15±135}{-2}$$

x=\frac{15±135}{-2} tenglamasini yeching, bunda ± musbat. 15 ni 135 ga qo'shish.

$$x=\frac{150}{-2}$$

150 ni -2 ga bo'lish.

$$x=-75$$

x=\frac{15±135}{-2} tenglamasini yeching, bunda ± manfiy. 15 dan 135 ni ayirish.

$$x=-\frac{120}{-2}$$

-120 ni -2 ga bo'lish.

$$x=60$$

Tenglama yechildi.

$$x=-75 x=60$$

Kvadratni yakunlash bo'yicha qadamlar

x qiymati -15,0 qiymatlaridan birortasiga teng bo‘lmaydi, chunki nolga bo‘lish mumkin emas. Tenglamaning ikkala tarafini 4x\left(x+15\right) ga, x,x+15,4 ning eng kichik karralisiga ko‘paytiring.

$$\left(4x+60\right)\times 75=4x\times 75+4x\left(x+15\right)\times \frac{1}{4}$$

4x+60 ga 75 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$300x+4500=4x\times 75+4x\left(x+15\right)\times \frac{1}{4}$$

300 hosil qilish uchun 4 va 75 ni ko'paytirish.

$$300x+4500=300x+4x\left(x+15\right)\times \frac{1}{4}$$

1 hosil qilish uchun 4 va \frac{1}{4} ni ko'paytirish.

$$300x+4500=300x+x\left(x+15\right)$$

x ga x+15 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$300x+4500=300x+x^{2}+15x$$

315x ni olish uchun 300x va 15x ni birlashtirish.

$$300x+4500=315x+x^{2}$$

Ikkala tarafdan 315x ni ayirish.

$$300x+4500-315x=x^{2}$$

-15x ni olish uchun 300x va -315x ni birlashtirish.

$$-15x+4500=x^{2}$$

Ikkala tarafdan x^{2} ni ayirish.

$$-15x+4500-x^{2}=0$$

Ikkala tarafdan 4500 ni ayirish. Har qanday sonni noldan ayirsangiz, o‘zining manfiyi chiqadi.

$$-15x-x^{2}=-4500$$

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

$$-x^{2}-15x=-4500$$

Ikki tarafini -1 ga bo‘ling.

$$\frac{-x^{2}-15x}{-1}=-\frac{4500}{-1}$$

-1 ga bo'lish -1 ga ko'paytirishni bekor qiladi.

$$x^{2}+\left(-\frac{15}{-1}\right)x=-\frac{4500}{-1}$$

-15 ni -1 ga bo'lish.

$$x^{2}+15x=-\frac{4500}{-1}$$

-4500 ni -1 ga bo'lish.

$$x^{2}+15x=4500$$

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

$$x^{2}+15x+\left(\frac{15}{2}\right)^{2}=4500+\left(\frac{15}{2}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib \frac{15}{2} kvadratini chiqarish.

$$x^{2}+15x+\frac{225}{4}=4500+\frac{225}{4}$$

4500 ni \frac{225}{4} ga qo'shish.

$$x^{2}+15x+\frac{225}{4}=\frac{18225}{4}$$

x^{2}+15x+\frac{225}{4} omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(x+\frac{15}{2}\right)^{2}=\frac{18225}{4}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x+\frac{15}{2}\right)^{2}}=\sqrt{\frac{18225}{4}}$$

Qisqartirish.

$$x+\frac{15}{2}=\frac{135}{2} x+\frac{15}{2}=-\frac{135}{2}$$

Tenglamaning ikkala tarafidan \frac{15}{2} ni ayirish.

$$x=60 x=-75$$