x^2/x+100=50 eching

$$\frac { x ^ { 2 } } { x + 100 } = 50$$

x uchun yechish

$$x=-50$$

$$x=100$$

Faktorlash qadamlari

x qiymati -100 teng bo‘lmaydi, chunki nolga bo‘lish mumkin emas. Tenglamaning ikkala tarafini x+100 ga ko'paytirish.

$$x^{2}=50\left(x+100\right)$$

50 ga x+100 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$x^{2}=50x+5000$$

Ikkala tarafdan 50x ni ayirish.

$$x^{2}-50x=5000$$

Ikkala tarafdan 5000 ni ayirish.

$$x^{2}-50x-5000=0$$

Bu tenglamani yechish uchun x^{2}+\left(a+b\right)x+ab=\left(x+a\right)\left(x+b\right) formulasi yordamida x^{2}-50x-5000 ni faktorlang. a va b ni topish uchun yechiladigan tizimni sozlang.

$$a+b=-50 ab=-5000$$

ab manfiy boʻlganda, a va b da qarama-qarshi belgilar bor. a+b manfiy boʻlganda, manfiy sonda musbatga nisbatdan kattaroq mutlaq qiymat bor. -5000-mahsulotni beruvchi bunday butun juftliklarni roʻyxat qiling.

$$1,-5000 2,-2500 4,-1250 5,-1000 8,-625 10,-500 20,-250 25,-200 40,-125 50,-100$$

Har bir juftlik yigʻindisini hisoblang.

$$1-5000=-4999 2-2500=-2498 4-1250=-1246 5-1000=-995 8-625=-617 10-500=-490 20-250=-230 25-200=-175 40-125=-85 50-100=-50$$

Yechim – -50 yigʻindisini beruvchi juftlik.

$$a=-100 b=50$$

Faktorlangan \left(x+a\right)\left(x+b\right) ifodani olingan qiymatlar bilan qaytadan yozing.

$$\left(x-100\right)\left(x+50\right)$$

Tenglamani yechish uchun x-100=0 va x+50=0 ni yeching.

$$x=100 x=-50$$

Guruhlash orqali faktorlash qadamlari

x qiymati -100 teng bo‘lmaydi, chunki nolga bo‘lish mumkin emas. Tenglamaning ikkala tarafini x+100 ga ko'paytirish.

$$x^{2}=50\left(x+100\right)$$

50 ga x+100 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$x^{2}=50x+5000$$

Ikkala tarafdan 50x ni ayirish.

$$x^{2}-50x=5000$$

Ikkala tarafdan 5000 ni ayirish.

$$x^{2}-50x-5000=0$$

Tenglamani yechish uchun guruhlash orqali chap qoʻl tomonni faktorlang. Avvalo, chap qoʻl tomon x^{2}+ax+bx-5000 sifatida qayta yozilishi kerak. a va b ni topish uchun yechiladigan tizimni sozlang.

$$a+b=-50 ab=1\left(-5000\right)=-5000$$

ab manfiy boʻlganda, a va b da qarama-qarshi belgilar bor. a+b manfiy boʻlganda, manfiy sonda musbatga nisbatdan kattaroq mutlaq qiymat bor. -5000-mahsulotni beruvchi bunday butun juftliklarni roʻyxat qiling.

$$1,-5000 2,-2500 4,-1250 5,-1000 8,-625 10,-500 20,-250 25,-200 40,-125 50,-100$$

Har bir juftlik yigʻindisini hisoblang.

$$1-5000=-4999 2-2500=-2498 4-1250=-1246 5-1000=-995 8-625=-617 10-500=-490 20-250=-230 25-200=-175 40-125=-85 50-100=-50$$

Yechim – -50 yigʻindisini beruvchi juftlik.

$$a=-100 b=50$$

x^{2}-50x-5000 ni \left(x^{2}-100x\right)+\left(50x-5000\right) sifatida qaytadan yozish.

$$\left(x^{2}-100x\right)+\left(50x-5000\right)$$

Birinchi guruhda x ni va ikkinchi guruhda 50 ni faktordan chiqaring.

$$x\left(x-100\right)+50\left(x-100\right)$$

Distributiv funktsiyasidan foydalangan holda x-100 umumiy terminini chiqaring.

$$\left(x-100\right)\left(x+50\right)$$

Tenglamani yechish uchun x-100=0 va x+50=0 ni yeching.

$$x=100 x=-50$$

Kvadrat tenglamalaridan foydalanish qoidalari

x qiymati -100 teng bo‘lmaydi, chunki nolga bo‘lish mumkin emas. Tenglamaning ikkala tarafini x+100 ga ko'paytirish.

$$x^{2}=50\left(x+100\right)$$

50 ga x+100 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$x^{2}=50x+5000$$

Ikkala tarafdan 50x ni ayirish.

$$x^{2}-50x=5000$$

Ikkala tarafdan 5000 ni ayirish.

$$x^{2}-50x-5000=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 1 ni a, -50 ni b va -5000 ni c bilan almashtiring.

$$x=\frac{-\left(-50\right)±\sqrt{\left(-50\right)^{2}-4\left(-5000\right)}}{2}$$

-50 kvadratini chiqarish.

$$x=\frac{-\left(-50\right)±\sqrt{2500-4\left(-5000\right)}}{2}$$

-4 ni -5000 marotabaga ko'paytirish.

$$x=\frac{-\left(-50\right)±\sqrt{2500+20000}}{2}$$

2500 ni 20000 ga qo'shish.

$$x=\frac{-\left(-50\right)±\sqrt{22500}}{2}$$

22500 ning kvadrat ildizini chiqarish.

$$x=\frac{-\left(-50\right)±150}{2}$$

-50 ning teskarisi 50 ga teng.

$$x=\frac{50±150}{2}$$

x=\frac{50±150}{2} tenglamasini yeching, bunda ± musbat. 50 ni 150 ga qo'shish.

$$x=\frac{200}{2}$$

200 ni 2 ga bo'lish.

$$x=100$$

x=\frac{50±150}{2} tenglamasini yeching, bunda ± manfiy. 50 dan 150 ni ayirish.

$$x=-\frac{100}{2}$$

-100 ni 2 ga bo'lish.

$$x=-50$$

Tenglama yechildi.

$$x=100 x=-50$$

Kvadratni yakunlash bo'yicha qadamlar

x qiymati -100 teng bo‘lmaydi, chunki nolga bo‘lish mumkin emas. Tenglamaning ikkala tarafini x+100 ga ko'paytirish.

$$x^{2}=50\left(x+100\right)$$

50 ga x+100 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$x^{2}=50x+5000$$

Ikkala tarafdan 50x ni ayirish.

$$x^{2}-50x=5000$$

-50 ni bo‘lish, x shartining koeffitsienti, 2 ga -25 olish uchun. Keyin, -25 ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}-50x+\left(-25\right)^{2}=5000+\left(-25\right)^{2}$$

-25 kvadratini chiqarish.

$$x^{2}-50x+625=5000+625$$

5000 ni 625 ga qo'shish.

$$x^{2}-50x+625=5625$$

x^{2}-50x+625 omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(x-25\right)^{2}=5625$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x-25\right)^{2}}=\sqrt{5625}$$

Qisqartirish.

$$x-25=75 x-25=-75$$

25 ni tenglamaning ikkala tarafiga qo'shish.

$$x=100 x=-50$$