30/x^2-1+7-18x/x^3+1=13/x^2-x+1 eching

$$\frac { 30 } { x ^ { 2 } - 1 } + \frac { 7 - 18 x } { x ^ { 3 } + 1 } = \frac { 13 } { x ^ { 2 } - x + 1 }$$

x uchun yechish

$$x=-9$$

$$x=4$$

Guruhlash orqali faktorlash qadamlari

x qiymati -1,1 qiymatlaridan birortasiga teng bo‘lmaydi, chunki nolga bo‘lish mumkin emas. Tenglamaning ikkala tarafini \left(x-1\right)\left(x+1\right)\left(x^{2}-x+1\right) ga, x^{2}-1,x^{3}+1,x^{2}-x+1 ning eng kichik karralisiga ko‘paytiring.

$$\left(x^{2}-x+1\right)\times 30+\left(x-1\right)\left(7-18x\right)=\left(x^{2}-1\right)\times 13$$

x^{2}-x+1 ga 30 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$30x^{2}-30x+30+\left(x-1\right)\left(7-18x\right)=\left(x^{2}-1\right)\times 13$$

x-1 ga 7-18x ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

$$30x^{2}-30x+30+25x-18x^{2}-7=\left(x^{2}-1\right)\times 13$$

-5x ni olish uchun -30x va 25x ni birlashtirish.

$$30x^{2}-5x+30-18x^{2}-7=\left(x^{2}-1\right)\times 13$$

12x^{2} ni olish uchun 30x^{2} va -18x^{2} ni birlashtirish.

$$12x^{2}-5x+30-7=\left(x^{2}-1\right)\times 13$$

23 olish uchun 30 dan 7 ni ayirish.

$$12x^{2}-5x+23=\left(x^{2}-1\right)\times 13$$

x^{2}-1 ga 13 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$12x^{2}-5x+23=13x^{2}-13$$

Ikkala tarafdan 13x^{2} ni ayirish.

$$12x^{2}-5x+23-13x^{2}=-13$$

-x^{2} ni olish uchun 12x^{2} va -13x^{2} ni birlashtirish.

$$-x^{2}-5x+23=-13$$

13 ni ikki tarafga qo’shing.

$$-x^{2}-5x+23+13=0$$

36 olish uchun 23 va 13'ni qo'shing.

$$-x^{2}-5x+36=0$$

Tenglamani yechish uchun guruhlash orqali chap qoʻl tomonni faktorlang. Avvalo, chap qoʻl tomon -x^{2}+ax+bx+36 sifatida qayta yozilishi kerak. a va b ni topish uchun yechiladigan tizimni sozlang.

$$a+b=-5 ab=-36=-36$$

ab manfiy boʻlganda, a va b da qarama-qarshi belgilar bor. a+b manfiy boʻlganda, manfiy sonda musbatga nisbatdan kattaroq mutlaq qiymat bor. -36-mahsulotni beruvchi bunday butun juftliklarni roʻyxat qiling.

$$1,-36 2,-18 3,-12 4,-9 6,-6$$

Har bir juftlik yigʻindisini hisoblang.

$$1-36=-35 2-18=-16 3-12=-9 4-9=-5 6-6=0$$

Yechim – -5 yigʻindisini beruvchi juftlik.

$$a=4 b=-9$$

-x^{2}-5x+36 ni \left(-x^{2}+4x\right)+\left(-9x+36\right) sifatida qaytadan yozish.

$$\left(-x^{2}+4x\right)+\left(-9x+36\right)$$

Birinchi guruhda x ni va ikkinchi guruhda 9 ni faktordan chiqaring.

$$x\left(-x+4\right)+9\left(-x+4\right)$$

Distributiv funktsiyasidan foydalangan holda -x+4 umumiy terminini chiqaring.

$$\left(-x+4\right)\left(x+9\right)$$

Tenglamani yechish uchun -x+4=0 va x+9=0 ni yeching.

$$x=4 x=-9$$

Kvadrat tenglamalaridan foydalanish qoidalari

x qiymati -1,1 qiymatlaridan birortasiga teng bo‘lmaydi, chunki nolga bo‘lish mumkin emas. Tenglamaning ikkala tarafini \left(x-1\right)\left(x+1\right)\left(x^{2}-x+1\right) ga, x^{2}-1,x^{3}+1,x^{2}-x+1 ning eng kichik karralisiga ko‘paytiring.

$$\left(x^{2}-x+1\right)\times 30+\left(x-1\right)\left(7-18x\right)=\left(x^{2}-1\right)\times 13$$

x^{2}-x+1 ga 30 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$30x^{2}-30x+30+\left(x-1\right)\left(7-18x\right)=\left(x^{2}-1\right)\times 13$$

x-1 ga 7-18x ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

$$30x^{2}-30x+30+25x-18x^{2}-7=\left(x^{2}-1\right)\times 13$$

-5x ni olish uchun -30x va 25x ni birlashtirish.

$$30x^{2}-5x+30-18x^{2}-7=\left(x^{2}-1\right)\times 13$$

12x^{2} ni olish uchun 30x^{2} va -18x^{2} ni birlashtirish.

$$12x^{2}-5x+30-7=\left(x^{2}-1\right)\times 13$$

23 olish uchun 30 dan 7 ni ayirish.

$$12x^{2}-5x+23=\left(x^{2}-1\right)\times 13$$

x^{2}-1 ga 13 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$12x^{2}-5x+23=13x^{2}-13$$

Ikkala tarafdan 13x^{2} ni ayirish.

$$12x^{2}-5x+23-13x^{2}=-13$$

-x^{2} ni olish uchun 12x^{2} va -13x^{2} ni birlashtirish.

$$-x^{2}-5x+23=-13$$

13 ni ikki tarafga qo’shing.

$$-x^{2}-5x+23+13=0$$

36 olish uchun 23 va 13'ni qo'shing.

$$-x^{2}-5x+36=0$$

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 36 ni c bilan almashtiring.

$$x=\frac{-\left(-5\right)±\sqrt{\left(-5\right)^{2}-4\left(-1\right)\times 36}}{2\left(-1\right)}$$

-5 kvadratini chiqarish.

$$x=\frac{-\left(-5\right)±\sqrt{25-4\left(-1\right)\times 36}}{2\left(-1\right)}$$

-4 ni -1 marotabaga ko'paytirish.

$$x=\frac{-\left(-5\right)±\sqrt{25+4\times 36}}{2\left(-1\right)}$$

4 ni 36 marotabaga ko'paytirish.

$$x=\frac{-\left(-5\right)±\sqrt{25+144}}{2\left(-1\right)}$$

25 ni 144 ga qo'shish.

$$x=\frac{-\left(-5\right)±\sqrt{169}}{2\left(-1\right)}$$

169 ning kvadrat ildizini chiqarish.

$$x=\frac{-\left(-5\right)±13}{2\left(-1\right)}$$

-5 ning teskarisi 5 ga teng.

$$x=\frac{5±13}{2\left(-1\right)}$$

2 ni -1 marotabaga ko'paytirish.

$$x=\frac{5±13}{-2}$$

x=\frac{5±13}{-2} tenglamasini yeching, bunda ± musbat. 5 ni 13 ga qo'shish.

$$x=\frac{18}{-2}$$

18 ni -2 ga bo'lish.

$$x=-9$$

x=\frac{5±13}{-2} tenglamasini yeching, bunda ± manfiy. 5 dan 13 ni ayirish.

$$x=-\frac{8}{-2}$$

-8 ni -2 ga bo'lish.

$$x=4$$

Tenglama yechildi.

$$x=-9 x=4$$

Kvadratni yakunlash bo'yicha qadamlar

x qiymati -1,1 qiymatlaridan birortasiga teng bo‘lmaydi, chunki nolga bo‘lish mumkin emas. Tenglamaning ikkala tarafini \left(x-1\right)\left(x+1\right)\left(x^{2}-x+1\right) ga, x^{2}-1,x^{3}+1,x^{2}-x+1 ning eng kichik karralisiga ko‘paytiring.

$$\left(x^{2}-x+1\right)\times 30+\left(x-1\right)\left(7-18x\right)=\left(x^{2}-1\right)\times 13$$

x^{2}-x+1 ga 30 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$30x^{2}-30x+30+\left(x-1\right)\left(7-18x\right)=\left(x^{2}-1\right)\times 13$$

x-1 ga 7-18x ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

$$30x^{2}-30x+30+25x-18x^{2}-7=\left(x^{2}-1\right)\times 13$$

-5x ni olish uchun -30x va 25x ni birlashtirish.

$$30x^{2}-5x+30-18x^{2}-7=\left(x^{2}-1\right)\times 13$$

12x^{2} ni olish uchun 30x^{2} va -18x^{2} ni birlashtirish.

$$12x^{2}-5x+30-7=\left(x^{2}-1\right)\times 13$$

23 olish uchun 30 dan 7 ni ayirish.

$$12x^{2}-5x+23=\left(x^{2}-1\right)\times 13$$

x^{2}-1 ga 13 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$12x^{2}-5x+23=13x^{2}-13$$

Ikkala tarafdan 13x^{2} ni ayirish.

$$12x^{2}-5x+23-13x^{2}=-13$$

-x^{2} ni olish uchun 12x^{2} va -13x^{2} ni birlashtirish.

$$-x^{2}-5x+23=-13$$

Ikkala tarafdan 23 ni ayirish.

$$-x^{2}-5x=-13-23$$

-36 olish uchun -13 dan 23 ni ayirish.

$$-x^{2}-5x=-36$$

Ikki tarafini -1 ga bo‘ling.

$$\frac{-x^{2}-5x}{-1}=-\frac{36}{-1}$$

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

$$x^{2}+\left(-\frac{5}{-1}\right)x=-\frac{36}{-1}$$

-5 ni -1 ga bo'lish.

$$x^{2}+5x=-\frac{36}{-1}$$

-36 ni -1 ga bo'lish.

$$x^{2}+5x=36$$

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.

$$x^{2}+5x+\left(\frac{5}{2}\right)^{2}=36+\left(\frac{5}{2}\right)^{2}$$

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

$$x^{2}+5x+\frac{25}{4}=36+\frac{25}{4}$$

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

$$x^{2}+5x+\frac{25}{4}=\frac{169}{4}$$

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.

$$\left(x+\frac{5}{2}\right)^{2}=\frac{169}{4}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x+\frac{5}{2}\right)^{2}}=\sqrt{\frac{169}{4}}$$

Qisqartirish.

$$x+\frac{5}{2}=\frac{13}{2} x+\frac{5}{2}=-\frac{13}{2}$$

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

$$x=4 x=-9$$