5x+1/x+2+x-1/x=2 eching

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

x uchun yechish

$$x=-\frac{1}{2}=-0,5$$

$$x=1$$

Guruhlash orqali faktorlash qadamlari

x qiymati -2,0 qiymatlaridan birortasiga teng bo‘lmaydi, chunki nolga bo‘lish mumkin emas. Tenglamaning ikkala tarafini x\left(x+2\right) ga, x+2,x ning eng kichik karralisiga ko‘paytiring.

$$x\left(5x+1\right)+\left(x+2\right)\left(x-1\right)=2x\left(x+2\right)$$

x ga 5x+1 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

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

x+2 ga x-1 ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

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

6x^{2} ni olish uchun 5x^{2} va x^{2} ni birlashtirish.

$$6x^{2}+x+x-2=2x\left(x+2\right)$$

2x ni olish uchun x va x ni birlashtirish.

$$6x^{2}+2x-2=2x\left(x+2\right)$$

2x ga x+2 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$6x^{2}+2x-2=2x^{2}+4x$$

Ikkala tarafdan 2x^{2} ni ayirish.

$$6x^{2}+2x-2-2x^{2}=4x$$

4x^{2} ni olish uchun 6x^{2} va -2x^{2} ni birlashtirish.

$$4x^{2}+2x-2=4x$$

Ikkala tarafdan 4x ni ayirish.

$$4x^{2}+2x-2-4x=0$$

-2x ni olish uchun 2x va -4x ni birlashtirish.

$$4x^{2}-2x-2=0$$

Ikki tarafini 2 ga bo‘ling.

$$2x^{2}-x-1=0$$

Tenglamani yechish uchun guruhlash orqali chap qoʻl tomonni faktorlang. Avvalo, chap qoʻl tomon 2x^{2}+ax+bx-1 sifatida qayta yozilishi kerak. a va b ni topish uchun yechiladigan tizimni sozlang.

$$a+b=-1 ab=2\left(-1\right)=-2$$

ab manfiy boʻlganda, a va b da qarama-qarshi belgilar bor. a+b manfiy boʻlganda, manfiy sonda musbatga nisbatdan kattaroq mutlaq qiymat bor. Faqat bundan juftlik tizim yechimidir.

$$a=-2 b=1$$

2x^{2}-x-1 ni \left(2x^{2}-2x\right)+\left(x-1\right) sifatida qaytadan yozish.

$$\left(2x^{2}-2x\right)+\left(x-1\right)$$

2x^{2}-2x ichida 2x ni ajrating.

$$2x\left(x-1\right)+x-1$$

Distributiv funktsiyasidan foydalangan holda x-1 umumiy terminini chiqaring.

$$\left(x-1\right)\left(2x+1\right)$$

Tenglamani yechish uchun x-1=0 va 2x+1=0 ni yeching.

$$x=1 x=-\frac{1}{2}$$

Kvadrat tenglamalaridan foydalanish qoidalari

x qiymati -2,0 qiymatlaridan birortasiga teng bo‘lmaydi, chunki nolga bo‘lish mumkin emas. Tenglamaning ikkala tarafini x\left(x+2\right) ga, x+2,x ning eng kichik karralisiga ko‘paytiring.

$$x\left(5x+1\right)+\left(x+2\right)\left(x-1\right)=2x\left(x+2\right)$$

x ga 5x+1 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

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

x+2 ga x-1 ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

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

6x^{2} ni olish uchun 5x^{2} va x^{2} ni birlashtirish.

$$6x^{2}+x+x-2=2x\left(x+2\right)$$

2x ni olish uchun x va x ni birlashtirish.

$$6x^{2}+2x-2=2x\left(x+2\right)$$

2x ga x+2 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$6x^{2}+2x-2=2x^{2}+4x$$

Ikkala tarafdan 2x^{2} ni ayirish.

$$6x^{2}+2x-2-2x^{2}=4x$$

4x^{2} ni olish uchun 6x^{2} va -2x^{2} ni birlashtirish.

$$4x^{2}+2x-2=4x$$

Ikkala tarafdan 4x ni ayirish.

$$4x^{2}+2x-2-4x=0$$

-2x ni olish uchun 2x va -4x ni birlashtirish.

$$4x^{2}-2x-2=0$$

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

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

-2 kvadratini chiqarish.

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

-4 ni 4 marotabaga ko'paytirish.

$$x=\frac{-\left(-2\right)±\sqrt{4-16\left(-2\right)}}{2\times 4}$$

-16 ni -2 marotabaga ko'paytirish.

$$x=\frac{-\left(-2\right)±\sqrt{4+32}}{2\times 4}$$

4 ni 32 ga qo'shish.

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

36 ning kvadrat ildizini chiqarish.

$$x=\frac{-\left(-2\right)±6}{2\times 4}$$

-2 ning teskarisi 2 ga teng.

$$x=\frac{2±6}{2\times 4}$$

2 ni 4 marotabaga ko'paytirish.

$$x=\frac{2±6}{8}$$

x=\frac{2±6}{8} tenglamasini yeching, bunda ± musbat. 2 ni 6 ga qo'shish.

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

8 ni 8 ga bo'lish.

$$x=1$$

x=\frac{2±6}{8} tenglamasini yeching, bunda ± manfiy. 2 dan 6 ni ayirish.

$$x=-\frac{4}{8}$$

\frac{-4}{8} ulushini 4 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

$$x=-\frac{1}{2}$$

Tenglama yechildi.

$$x=1 x=-\frac{1}{2}$$

Kvadratni yakunlash bo'yicha qadamlar

x qiymati -2,0 qiymatlaridan birortasiga teng bo‘lmaydi, chunki nolga bo‘lish mumkin emas. Tenglamaning ikkala tarafini x\left(x+2\right) ga, x+2,x ning eng kichik karralisiga ko‘paytiring.

$$x\left(5x+1\right)+\left(x+2\right)\left(x-1\right)=2x\left(x+2\right)$$

x ga 5x+1 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

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

x+2 ga x-1 ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

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

6x^{2} ni olish uchun 5x^{2} va x^{2} ni birlashtirish.

$$6x^{2}+x+x-2=2x\left(x+2\right)$$

2x ni olish uchun x va x ni birlashtirish.

$$6x^{2}+2x-2=2x\left(x+2\right)$$

2x ga x+2 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$6x^{2}+2x-2=2x^{2}+4x$$

Ikkala tarafdan 2x^{2} ni ayirish.

$$6x^{2}+2x-2-2x^{2}=4x$$

4x^{2} ni olish uchun 6x^{2} va -2x^{2} ni birlashtirish.

$$4x^{2}+2x-2=4x$$

Ikkala tarafdan 4x ni ayirish.

$$4x^{2}+2x-2-4x=0$$

-2x ni olish uchun 2x va -4x ni birlashtirish.

$$4x^{2}-2x-2=0$$

2 ni ikki tarafga qo’shing. Har qanday songa nolni qo‘shsangiz, o‘zi chiqadi.

$$4x^{2}-2x=2$$

Ikki tarafini 4 ga bo‘ling.

$$\frac{4x^{2}-2x}{4}=\frac{2}{4}$$

4 ga bo'lish 4 ga ko'paytirishni bekor qiladi.

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

\frac{-2}{4} ulushini 2 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

$$x^{2}-\frac{1}{2}x=\frac{2}{4}$$

\frac{2}{4} ulushini 2 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

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

-\frac{1}{2} ni bo‘lish, x shartining koeffitsienti, 2 ga -\frac{1}{4} olish uchun. Keyin, -\frac{1}{4} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}-\frac{1}{2}x+\left(-\frac{1}{4}\right)^{2}=\frac{1}{2}+\left(-\frac{1}{4}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib -\frac{1}{4} kvadratini chiqarish.

$$x^{2}-\frac{1}{2}x+\frac{1}{16}=\frac{1}{2}+\frac{1}{16}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali \frac{1}{2} ni \frac{1}{16} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$x^{2}-\frac{1}{2}x+\frac{1}{16}=\frac{9}{16}$$

x^{2}-\frac{1}{2}x+\frac{1}{16} omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(x-\frac{1}{4}\right)^{2}=\frac{9}{16}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x-\frac{1}{4}\right)^{2}}=\sqrt{\frac{9}{16}}$$

Qisqartirish.

$$x-\frac{1}{4}=\frac{3}{4} x-\frac{1}{4}=-\frac{3}{4}$$

\frac{1}{4} ni tenglamaning ikkala tarafiga qo'shish.

$$x=1 x=-\frac{1}{2}$$