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

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

x uchun yechish

$$x=1$$

$$x=-2$$

Guruhlash orqali faktorlash qadamlari

x qiymati -1,0 qiymatlaridan birortasiga teng bo‘lmaydi, chunki nolga bo‘lish mumkin emas. Tenglamaning ikkala tarafini 2x\left(x+1\right) ga, x,x+1,2 ning eng kichik karralisiga ko‘paytiring.

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

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

$$2x^{2}+4x+2+2xx=5x\left(x+1\right)$$

x^{2} hosil qilish uchun x va x ni ko'paytirish.

$$2x^{2}+4x+2+2x^{2}=5x\left(x+1\right)$$

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

$$4x^{2}+4x+2=5x\left(x+1\right)$$

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

$$4x^{2}+4x+2=5x^{2}+5x$$

Ikkala tarafdan 5x^{2} ni ayirish.

$$4x^{2}+4x+2-5x^{2}=5x$$

-x^{2} ni olish uchun 4x^{2} va -5x^{2} ni birlashtirish.

$$-x^{2}+4x+2=5x$$

Ikkala tarafdan 5x ni ayirish.

$$-x^{2}+4x+2-5x=0$$

-x ni olish uchun 4x va -5x ni birlashtirish.

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

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

$$a+b=-1 ab=-2=-2$$

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.

$$a=1 b=-2$$

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

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

Birinchi guruhda x ni va ikkinchi guruhda 2 ni faktordan chiqaring.

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

Distributiv funktsiyasidan foydalangan holda -x+1 umumiy terminini chiqaring.

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

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

$$x=1 x=-2$$

Kvadrat tenglamalaridan foydalanish qoidalari

x qiymati -1,0 qiymatlaridan birortasiga teng bo‘lmaydi, chunki nolga bo‘lish mumkin emas. Tenglamaning ikkala tarafini 2x\left(x+1\right) ga, x,x+1,2 ning eng kichik karralisiga ko‘paytiring.

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

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

$$2x^{2}+4x+2+2xx=5x\left(x+1\right)$$

x^{2} hosil qilish uchun x va x ni ko'paytirish.

$$2x^{2}+4x+2+2x^{2}=5x\left(x+1\right)$$

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

$$4x^{2}+4x+2=5x\left(x+1\right)$$

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

$$4x^{2}+4x+2=5x^{2}+5x$$

Ikkala tarafdan 5x^{2} ni ayirish.

$$4x^{2}+4x+2-5x^{2}=5x$$

-x^{2} ni olish uchun 4x^{2} va -5x^{2} ni birlashtirish.

$$-x^{2}+4x+2=5x$$

Ikkala tarafdan 5x ni ayirish.

$$-x^{2}+4x+2-5x=0$$

-x ni olish uchun 4x va -5x ni birlashtirish.

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

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

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

-4 ni -1 marotabaga ko'paytirish.

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

4 ni 2 marotabaga ko'paytirish.

$$x=\frac{-\left(-1\right)±\sqrt{1+8}}{2\left(-1\right)}$$

1 ni 8 ga qo'shish.

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

9 ning kvadrat ildizini chiqarish.

$$x=\frac{-\left(-1\right)±3}{2\left(-1\right)}$$

-1 ning teskarisi 1 ga teng.

$$x=\frac{1±3}{2\left(-1\right)}$$

2 ni -1 marotabaga ko'paytirish.

$$x=\frac{1±3}{-2}$$

x=\frac{1±3}{-2} tenglamasini yeching, bunda ± musbat. 1 ni 3 ga qo'shish.

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

4 ni -2 ga bo'lish.

$$x=-2$$

x=\frac{1±3}{-2} tenglamasini yeching, bunda ± manfiy. 1 dan 3 ni ayirish.

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

-2 ni -2 ga bo'lish.

$$x=1$$

Tenglama yechildi.

$$x=-2 x=1$$

Kvadratni yakunlash bo'yicha qadamlar

x qiymati -1,0 qiymatlaridan birortasiga teng bo‘lmaydi, chunki nolga bo‘lish mumkin emas. Tenglamaning ikkala tarafini 2x\left(x+1\right) ga, x,x+1,2 ning eng kichik karralisiga ko‘paytiring.

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

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

$$2x^{2}+4x+2+2xx=5x\left(x+1\right)$$

x^{2} hosil qilish uchun x va x ni ko'paytirish.

$$2x^{2}+4x+2+2x^{2}=5x\left(x+1\right)$$

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

$$4x^{2}+4x+2=5x\left(x+1\right)$$

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

$$4x^{2}+4x+2=5x^{2}+5x$$

Ikkala tarafdan 5x^{2} ni ayirish.

$$4x^{2}+4x+2-5x^{2}=5x$$

-x^{2} ni olish uchun 4x^{2} va -5x^{2} ni birlashtirish.

$$-x^{2}+4x+2=5x$$

Ikkala tarafdan 5x ni ayirish.

$$-x^{2}+4x+2-5x=0$$

-x ni olish uchun 4x va -5x ni birlashtirish.

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

Ikkala tarafdan 2 ni ayirish. Har qanday sonni noldan ayirsangiz, o‘zining manfiyi chiqadi.

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

Ikki tarafini -1 ga bo‘ling.

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

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

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

-1 ni -1 ga bo'lish.

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

-2 ni -1 ga bo'lish.

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

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

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

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib \frac{1}{2} kvadratini chiqarish.

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

2 ni \frac{1}{4} ga qo'shish.

$$x^{2}+x+\frac{1}{4}=\frac{9}{4}$$

x^{2}+x+\frac{1}{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.

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

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

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

Qisqartirish.

$$x+\frac{1}{2}=\frac{3}{2} x+\frac{1}{2}=-\frac{3}{2}$$

Tenglamaning ikkala tarafidan \frac{1}{2} ni ayirish.

$$x=1 x=-2$$