168/x-168/x+14=1 eching

$$\frac { 168 } { x } - \frac { 168 } { x + 14 } = 1$$

x uchun yechish

$$x=-56$$

$$x=42$$

Guruhlash orqali faktorlash qadamlari

x qiymati -14,0 qiymatlaridan birortasiga teng bo‘lmaydi, chunki nolga bo‘lish mumkin emas. Tenglamaning ikkala tarafini x\left(x+14\right) ga, x,x+14 ning eng kichik karralisiga ko‘paytiring.

$$\left(x+14\right)\times 168-x\times 168=x\left(x+14\right)$$

x+14 ga 168 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$168x+2352-x\times 168=x\left(x+14\right)$$

x ga x+14 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$168x+2352-x\times 168=x^{2}+14x$$

Ikkala tarafdan x^{2} ni ayirish.

$$168x+2352-x\times 168-x^{2}=14x$$

Ikkala tarafdan 14x ni ayirish.

$$168x+2352-x\times 168-x^{2}-14x=0$$

154x ni olish uchun 168x va -14x ni birlashtirish.

$$154x+2352-x\times 168-x^{2}=0$$

-168 hosil qilish uchun -1 va 168 ni ko'paytirish.

$$154x+2352-168x-x^{2}=0$$

-14x ni olish uchun 154x va -168x ni birlashtirish.

$$-14x+2352-x^{2}=0$$

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

$$-x^{2}-14x+2352=0$$

Tenglamani yechish uchun guruhlash orqali chap qoʻl tomonni faktorlang. Avvalo, chap qoʻl tomon -x^{2}+ax+bx+2352 sifatida qayta yozilishi kerak. a va b ni topish uchun yechiladigan tizimni sozlang.

$$a+b=-14 ab=-2352=-2352$$

ab manfiy boʻlganda, a va b da qarama-qarshi belgilar bor. a+b manfiy boʻlganda, manfiy sonda musbatga nisbatdan kattaroq mutlaq qiymat bor. -2352-mahsulotni beruvchi bunday butun juftliklarni roʻyxat qiling.

$$1,-2352 2,-1176 3,-784 4,-588 6,-392 7,-336 8,-294 12,-196 14,-168 16,-147 21,-112 24,-98 28,-84 42,-56 48,-49$$

Har bir juftlik yigʻindisini hisoblang.

$$1-2352=-2351 2-1176=-1174 3-784=-781 4-588=-584 6-392=-386 7-336=-329 8-294=-286 12-196=-184 14-168=-154 16-147=-131 21-112=-91 24-98=-74 28-84=-56 42-56=-14 48-49=-1$$

Yechim – -14 yigʻindisini beruvchi juftlik.

$$a=42 b=-56$$

-x^{2}-14x+2352 ni \left(-x^{2}+42x\right)+\left(-56x+2352\right) sifatida qaytadan yozish.

$$\left(-x^{2}+42x\right)+\left(-56x+2352\right)$$

Birinchi guruhda x ni va ikkinchi guruhda 56 ni faktordan chiqaring.

$$x\left(-x+42\right)+56\left(-x+42\right)$$

Distributiv funktsiyasidan foydalangan holda -x+42 umumiy terminini chiqaring.

$$\left(-x+42\right)\left(x+56\right)$$

Tenglamani yechish uchun -x+42=0 va x+56=0 ni yeching.

$$x=42 x=-56$$

Kvadrat tenglamalaridan foydalanish qoidalari

x qiymati -14,0 qiymatlaridan birortasiga teng bo‘lmaydi, chunki nolga bo‘lish mumkin emas. Tenglamaning ikkala tarafini x\left(x+14\right) ga, x,x+14 ning eng kichik karralisiga ko‘paytiring.

$$\left(x+14\right)\times 168-x\times 168=x\left(x+14\right)$$

x+14 ga 168 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$168x+2352-x\times 168=x\left(x+14\right)$$

x ga x+14 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$168x+2352-x\times 168=x^{2}+14x$$

Ikkala tarafdan x^{2} ni ayirish.

$$168x+2352-x\times 168-x^{2}=14x$$

Ikkala tarafdan 14x ni ayirish.

$$168x+2352-x\times 168-x^{2}-14x=0$$

154x ni olish uchun 168x va -14x ni birlashtirish.

$$154x+2352-x\times 168-x^{2}=0$$

-168 hosil qilish uchun -1 va 168 ni ko'paytirish.

$$154x+2352-168x-x^{2}=0$$

-14x ni olish uchun 154x va -168x ni birlashtirish.

$$-14x+2352-x^{2}=0$$

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.

$$-x^{2}-14x+2352=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} -1 ni a, -14 ni b va 2352 ni c bilan almashtiring.

$$x=\frac{-\left(-14\right)±\sqrt{\left(-14\right)^{2}-4\left(-1\right)\times 2352}}{2\left(-1\right)}$$

-14 kvadratini chiqarish.

$$x=\frac{-\left(-14\right)±\sqrt{196-4\left(-1\right)\times 2352}}{2\left(-1\right)}$$

-4 ni -1 marotabaga ko'paytirish.

$$x=\frac{-\left(-14\right)±\sqrt{196+4\times 2352}}{2\left(-1\right)}$$

4 ni 2352 marotabaga ko'paytirish.

$$x=\frac{-\left(-14\right)±\sqrt{196+9408}}{2\left(-1\right)}$$

196 ni 9408 ga qo'shish.

$$x=\frac{-\left(-14\right)±\sqrt{9604}}{2\left(-1\right)}$$

9604 ning kvadrat ildizini chiqarish.

$$x=\frac{-\left(-14\right)±98}{2\left(-1\right)}$$

-14 ning teskarisi 14 ga teng.

$$x=\frac{14±98}{2\left(-1\right)}$$

2 ni -1 marotabaga ko'paytirish.

$$x=\frac{14±98}{-2}$$

x=\frac{14±98}{-2} tenglamasini yeching, bunda ± musbat. 14 ni 98 ga qo'shish.

$$x=\frac{112}{-2}$$

112 ni -2 ga bo'lish.

$$x=-56$$

x=\frac{14±98}{-2} tenglamasini yeching, bunda ± manfiy. 14 dan 98 ni ayirish.

$$x=-\frac{84}{-2}$$

-84 ni -2 ga bo'lish.

$$x=42$$

Tenglama yechildi.

$$x=-56 x=42$$

Kvadratni yakunlash bo'yicha qadamlar

x qiymati -14,0 qiymatlaridan birortasiga teng bo‘lmaydi, chunki nolga bo‘lish mumkin emas. Tenglamaning ikkala tarafini x\left(x+14\right) ga, x,x+14 ning eng kichik karralisiga ko‘paytiring.

$$\left(x+14\right)\times 168-x\times 168=x\left(x+14\right)$$

x+14 ga 168 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$168x+2352-x\times 168=x\left(x+14\right)$$

x ga x+14 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$168x+2352-x\times 168=x^{2}+14x$$

Ikkala tarafdan x^{2} ni ayirish.

$$168x+2352-x\times 168-x^{2}=14x$$

Ikkala tarafdan 14x ni ayirish.

$$168x+2352-x\times 168-x^{2}-14x=0$$

154x ni olish uchun 168x va -14x ni birlashtirish.

$$154x+2352-x\times 168-x^{2}=0$$

Ikkala tarafdan 2352 ni ayirish. Har qanday sonni noldan ayirsangiz, o‘zining manfiyi chiqadi.

$$154x-x\times 168-x^{2}=-2352$$

-168 hosil qilish uchun -1 va 168 ni ko'paytirish.

$$154x-168x-x^{2}=-2352$$

-14x ni olish uchun 154x va -168x ni birlashtirish.

$$-14x-x^{2}=-2352$$

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

$$-x^{2}-14x=-2352$$

Ikki tarafini -1 ga bo‘ling.

$$\frac{-x^{2}-14x}{-1}=-\frac{2352}{-1}$$

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

$$x^{2}+\left(-\frac{14}{-1}\right)x=-\frac{2352}{-1}$$

-14 ni -1 ga bo'lish.

$$x^{2}+14x=-\frac{2352}{-1}$$

-2352 ni -1 ga bo'lish.

$$x^{2}+14x=2352$$

14 ni bo‘lish, x shartining koeffitsienti, 2 ga 7 olish uchun. Keyin, 7 ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}+14x+7^{2}=2352+7^{2}$$

7 kvadratini chiqarish.

$$x^{2}+14x+49=2352+49$$

2352 ni 49 ga qo'shish.

$$x^{2}+14x+49=2401$$

x^{2}+14x+49 omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(x+7\right)^{2}=2401$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x+7\right)^{2}}=\sqrt{2401}$$

Qisqartirish.

$$x+7=49 x+7=-49$$

Tenglamaning ikkala tarafidan 7 ni ayirish.

$$x=42 x=-56$$