360/x-360/x+5=1 eching

$$\frac { 360 } { x } - \frac { 360 } { x + 5 } = 1$$

x uchun yechish

$$x=-45$$

$$x=40$$

Guruhlash orqali faktorlash qadamlari

x qiymati -5,0 qiymatlaridan birortasiga teng bo‘lmaydi, chunki nolga bo‘lish mumkin emas. Tenglamaning ikkala tarafini x\left(x+5\right) ga, x,x+5 ning eng kichik karralisiga ko‘paytiring.

$$\left(x+5\right)\times 360-x\times 360=x\left(x+5\right)$$

x+5 ga 360 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$360x+1800-x\times 360=x\left(x+5\right)$$

x ga x+5 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$360x+1800-x\times 360=x^{2}+5x$$

Ikkala tarafdan x^{2} ni ayirish.

$$360x+1800-x\times 360-x^{2}=5x$$

Ikkala tarafdan 5x ni ayirish.

$$360x+1800-x\times 360-x^{2}-5x=0$$

355x ni olish uchun 360x va -5x ni birlashtirish.

$$355x+1800-x\times 360-x^{2}=0$$

-360 hosil qilish uchun -1 va 360 ni ko'paytirish.

$$355x+1800-360x-x^{2}=0$$

-5x ni olish uchun 355x va -360x ni birlashtirish.

$$-5x+1800-x^{2}=0$$

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

$$-x^{2}-5x+1800=0$$

Tenglamani yechish uchun guruhlash orqali chap qoʻl tomonni faktorlang. Avvalo, chap qoʻl tomon -x^{2}+ax+bx+1800 sifatida qayta yozilishi kerak. a va b ni topish uchun yechiladigan tizimni sozlang.

$$a+b=-5 ab=-1800=-1800$$

ab manfiy boʻlganda, a va b da qarama-qarshi belgilar bor. a+b manfiy boʻlganda, manfiy sonda musbatga nisbatdan kattaroq mutlaq qiymat bor. -1800-mahsulotni beruvchi bunday butun juftliklarni roʻyxat qiling.

$$1,-1800 2,-900 3,-600 4,-450 5,-360 6,-300 8,-225 9,-200 10,-180 12,-150 15,-120 18,-100 20,-90 24,-75 25,-72 30,-60 36,-50 40,-45$$

Har bir juftlik yigʻindisini hisoblang.

$$1-1800=-1799 2-900=-898 3-600=-597 4-450=-446 5-360=-355 6-300=-294 8-225=-217 9-200=-191 10-180=-170 12-150=-138 15-120=-105 18-100=-82 20-90=-70 24-75=-51 25-72=-47 30-60=-30 36-50=-14 40-45=-5$$

Yechim – -5 yigʻindisini beruvchi juftlik.

$$a=40 b=-45$$

-x^{2}-5x+1800 ni \left(-x^{2}+40x\right)+\left(-45x+1800\right) sifatida qaytadan yozish.

$$\left(-x^{2}+40x\right)+\left(-45x+1800\right)$$

Birinchi guruhda x ni va ikkinchi guruhda 45 ni faktordan chiqaring.

$$x\left(-x+40\right)+45\left(-x+40\right)$$

Distributiv funktsiyasidan foydalangan holda -x+40 umumiy terminini chiqaring.

$$\left(-x+40\right)\left(x+45\right)$$

Tenglamani yechish uchun -x+40=0 va x+45=0 ni yeching.

$$x=40 x=-45$$

Kvadrat tenglamalaridan foydalanish qoidalari

x qiymati -5,0 qiymatlaridan birortasiga teng bo‘lmaydi, chunki nolga bo‘lish mumkin emas. Tenglamaning ikkala tarafini x\left(x+5\right) ga, x,x+5 ning eng kichik karralisiga ko‘paytiring.

$$\left(x+5\right)\times 360-x\times 360=x\left(x+5\right)$$

x+5 ga 360 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$360x+1800-x\times 360=x\left(x+5\right)$$

x ga x+5 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$360x+1800-x\times 360=x^{2}+5x$$

Ikkala tarafdan x^{2} ni ayirish.

$$360x+1800-x\times 360-x^{2}=5x$$

Ikkala tarafdan 5x ni ayirish.

$$360x+1800-x\times 360-x^{2}-5x=0$$

355x ni olish uchun 360x va -5x ni birlashtirish.

$$355x+1800-x\times 360-x^{2}=0$$

-360 hosil qilish uchun -1 va 360 ni ko'paytirish.

$$355x+1800-360x-x^{2}=0$$

-5x ni olish uchun 355x va -360x ni birlashtirish.

$$-5x+1800-x^{2}=0$$

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.

$$-x^{2}-5x+1800=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} -1 ni a, -5 ni b va 1800 ni c bilan almashtiring.

$$x=\frac{-\left(-5\right)±\sqrt{\left(-5\right)^{2}-4\left(-1\right)\times 1800}}{2\left(-1\right)}$$

-5 kvadratini chiqarish.

$$x=\frac{-\left(-5\right)±\sqrt{25-4\left(-1\right)\times 1800}}{2\left(-1\right)}$$

-4 ni -1 marotabaga ko'paytirish.

$$x=\frac{-\left(-5\right)±\sqrt{25+4\times 1800}}{2\left(-1\right)}$$

4 ni 1800 marotabaga ko'paytirish.

$$x=\frac{-\left(-5\right)±\sqrt{25+7200}}{2\left(-1\right)}$$

25 ni 7200 ga qo'shish.

$$x=\frac{-\left(-5\right)±\sqrt{7225}}{2\left(-1\right)}$$

7225 ning kvadrat ildizini chiqarish.

$$x=\frac{-\left(-5\right)±85}{2\left(-1\right)}$$

-5 ning teskarisi 5 ga teng.

$$x=\frac{5±85}{2\left(-1\right)}$$

2 ni -1 marotabaga ko'paytirish.

$$x=\frac{5±85}{-2}$$

x=\frac{5±85}{-2} tenglamasini yeching, bunda ± musbat. 5 ni 85 ga qo'shish.

$$x=\frac{90}{-2}$$

90 ni -2 ga bo'lish.

$$x=-45$$

x=\frac{5±85}{-2} tenglamasini yeching, bunda ± manfiy. 5 dan 85 ni ayirish.

$$x=-\frac{80}{-2}$$

-80 ni -2 ga bo'lish.

$$x=40$$

Tenglama yechildi.

$$x=-45 x=40$$

Kvadratni yakunlash bo'yicha qadamlar

x qiymati -5,0 qiymatlaridan birortasiga teng bo‘lmaydi, chunki nolga bo‘lish mumkin emas. Tenglamaning ikkala tarafini x\left(x+5\right) ga, x,x+5 ning eng kichik karralisiga ko‘paytiring.

$$\left(x+5\right)\times 360-x\times 360=x\left(x+5\right)$$

x+5 ga 360 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$360x+1800-x\times 360=x\left(x+5\right)$$

x ga x+5 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$360x+1800-x\times 360=x^{2}+5x$$

Ikkala tarafdan x^{2} ni ayirish.

$$360x+1800-x\times 360-x^{2}=5x$$

Ikkala tarafdan 5x ni ayirish.

$$360x+1800-x\times 360-x^{2}-5x=0$$

355x ni olish uchun 360x va -5x ni birlashtirish.

$$355x+1800-x\times 360-x^{2}=0$$

Ikkala tarafdan 1800 ni ayirish. Har qanday sonni noldan ayirsangiz, o‘zining manfiyi chiqadi.

$$355x-x\times 360-x^{2}=-1800$$

-360 hosil qilish uchun -1 va 360 ni ko'paytirish.

$$355x-360x-x^{2}=-1800$$

-5x ni olish uchun 355x va -360x ni birlashtirish.

$$-5x-x^{2}=-1800$$

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

$$-x^{2}-5x=-1800$$

Ikki tarafini -1 ga bo‘ling.

$$\frac{-x^{2}-5x}{-1}=-\frac{1800}{-1}$$

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

$$x^{2}+\left(-\frac{5}{-1}\right)x=-\frac{1800}{-1}$$

-5 ni -1 ga bo'lish.

$$x^{2}+5x=-\frac{1800}{-1}$$

-1800 ni -1 ga bo'lish.

$$x^{2}+5x=1800$$

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

$$x^{2}+5x+\left(\frac{5}{2}\right)^{2}=1800+\left(\frac{5}{2}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib \frac{5}{2} kvadratini chiqarish.

$$x^{2}+5x+\frac{25}{4}=1800+\frac{25}{4}$$

1800 ni \frac{25}{4} ga qo'shish.

$$x^{2}+5x+\frac{25}{4}=\frac{7225}{4}$$

x^{2}+5x+\frac{25}{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.

$$\left(x+\frac{5}{2}\right)^{2}=\frac{7225}{4}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x+\frac{5}{2}\right)^{2}}=\sqrt{\frac{7225}{4}}$$

Qisqartirish.

$$x+\frac{5}{2}=\frac{85}{2} x+\frac{5}{2}=-\frac{85}{2}$$

Tenglamaning ikkala tarafidan \frac{5}{2} ni ayirish.

$$x=40 x=-45$$