1-2/x+1-4/x^2-1=x/1-x eching

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

x uchun yechish

$$x = \frac{3}{2} = 1\frac{1}{2} = 1,5$$

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) ga, x+1,x^{2}-1,1-x ning eng kichik karralisiga ko‘paytiring.

$$\left(x-1\right)\left(x+1\right)-\left(x-1\right)\times 2-4=-\left(1+x\right)x$$

Hisoblang: \left(x-1\right)\left(x+1\right). Ko‘paytirish qoida yordamida turli kvadratlarga aylantirilishi mumkin: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}. 1 kvadratini chiqarish.

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

x-1 ga 2 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

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

2x-2 teskarisini topish uchun har birining teskarisini toping.

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

1 olish uchun -1 va 2'ni qo'shing.

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

-3 olish uchun 1 dan 4 ni ayirish.

$$x^{2}-3-2x=-\left(1+x\right)x$$

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

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

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

$$x^{2}-3-2x=-x-x^{2}$$

x ni ikki tarafga qo’shing.

$$x^{2}-3-2x+x=-x^{2}$$

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

$$x^{2}-3-x=-x^{2}$$

x^{2} ni ikki tarafga qo’shing.

$$x^{2}-3-x+x^{2}=0$$

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

$$2x^{2}-3-x=0$$

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

$$2x^{2}-x-3=0$$

Tenglamani yechish uchun guruhlash orqali chap qoʻl tomonni faktorlang. Avvalo, chap qoʻl tomon 2x^{2}+ax+bx-3 sifatida qayta yozilishi kerak. a va b ni topish uchun yechiladigan tizimni sozlang.

$$a+b=-1 ab=2\left(-3\right)=-6$$

ab manfiy boʻlganda, a va b da qarama-qarshi belgilar bor. a+b manfiy boʻlganda, manfiy sonda musbatga nisbatdan kattaroq mutlaq qiymat bor. -6-mahsulotni beruvchi bunday butun juftliklarni roʻyxat qiling.

$$1,-6 2,-3$$

Har bir juftlik yigʻindisini hisoblang.

$$1-6=-5 2-3=-1$$

Yechim – -1 yigʻindisini beruvchi juftlik.

$$a=-3 b=2$$

2x^{2}-x-3 ni \left(2x^{2}-3x\right)+\left(2x-3\right) sifatida qaytadan yozish.

$$\left(2x^{2}-3x\right)+\left(2x-3\right)$$

2x^{2}-3x ichida x ni ajrating.

$$x\left(2x-3\right)+2x-3$$

Distributiv funktsiyasidan foydalangan holda 2x-3 umumiy terminini chiqaring.

$$\left(2x-3\right)\left(x+1\right)$$

Tenglamani yechish uchun 2x-3=0 va x+1=0 ni yeching.

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

x qiymati -1 teng bo‘lmaydi.

$$x=\frac{3}{2}$$

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) ga, x+1,x^{2}-1,1-x ning eng kichik karralisiga ko‘paytiring.

$$\left(x-1\right)\left(x+1\right)-\left(x-1\right)\times 2-4=-\left(1+x\right)x$$

Hisoblang: \left(x-1\right)\left(x+1\right). Ko‘paytirish qoida yordamida turli kvadratlarga aylantirilishi mumkin: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}. 1 kvadratini chiqarish.

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

x-1 ga 2 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

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

2x-2 teskarisini topish uchun har birining teskarisini toping.

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

1 olish uchun -1 va 2'ni qo'shing.

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

-3 olish uchun 1 dan 4 ni ayirish.

$$x^{2}-3-2x=-\left(1+x\right)x$$

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

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

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

$$x^{2}-3-2x=-x-x^{2}$$

x ni ikki tarafga qo’shing.

$$x^{2}-3-2x+x=-x^{2}$$

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

$$x^{2}-3-x=-x^{2}$$

x^{2} ni ikki tarafga qo’shing.

$$x^{2}-3-x+x^{2}=0$$

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

$$2x^{2}-3-x=0$$

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.

$$2x^{2}-x-3=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 2 ni a, -1 ni b va -3 ni c bilan almashtiring.

$$x=\frac{-\left(-1\right)±\sqrt{1-4\times 2\left(-3\right)}}{2\times 2}$$

-4 ni 2 marotabaga ko'paytirish.

$$x=\frac{-\left(-1\right)±\sqrt{1-8\left(-3\right)}}{2\times 2}$$

-8 ni -3 marotabaga ko'paytirish.

$$x=\frac{-\left(-1\right)±\sqrt{1+24}}{2\times 2}$$

1 ni 24 ga qo'shish.

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

25 ning kvadrat ildizini chiqarish.

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

-1 ning teskarisi 1 ga teng.

$$x=\frac{1±5}{2\times 2}$$

2 ni 2 marotabaga ko'paytirish.

$$x=\frac{1±5}{4}$$

x=\frac{1±5}{4} tenglamasini yeching, bunda ± musbat. 1 ni 5 ga qo'shish.

$$x=\frac{6}{4}$$

\frac{6}{4} ulushini 2 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

$$x=\frac{3}{2}$$

x=\frac{1±5}{4} tenglamasini yeching, bunda ± manfiy. 1 dan 5 ni ayirish.

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

-4 ni 4 ga bo'lish.

$$x=-1$$

Tenglama yechildi.

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

x qiymati -1 teng bo‘lmaydi.

$$x=\frac{3}{2}$$

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) ga, x+1,x^{2}-1,1-x ning eng kichik karralisiga ko‘paytiring.

$$\left(x-1\right)\left(x+1\right)-\left(x-1\right)\times 2-4=-\left(1+x\right)x$$

Hisoblang: \left(x-1\right)\left(x+1\right). Ko‘paytirish qoida yordamida turli kvadratlarga aylantirilishi mumkin: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}. 1 kvadratini chiqarish.

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

x-1 ga 2 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

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

2x-2 teskarisini topish uchun har birining teskarisini toping.

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

1 olish uchun -1 va 2'ni qo'shing.

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

-3 olish uchun 1 dan 4 ni ayirish.

$$x^{2}-3-2x=-\left(1+x\right)x$$

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

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

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

$$x^{2}-3-2x=-x-x^{2}$$

x ni ikki tarafga qo’shing.

$$x^{2}-3-2x+x=-x^{2}$$

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

$$x^{2}-3-x=-x^{2}$$

x^{2} ni ikki tarafga qo’shing.

$$x^{2}-3-x+x^{2}=0$$

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

$$2x^{2}-3-x=0$$

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

$$2x^{2}-x=3$$

Ikki tarafini 2 ga bo‘ling.

$$\frac{2x^{2}-x}{2}=\frac{3}{2}$$

2 ga bo'lish 2 ga ko'paytirishni bekor qiladi.

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

-\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.

$$x^{2}-\frac{1}{2}x+\left(-\frac{1}{4}\right)^{2}=\frac{3}{2}+\left(-\frac{1}{4}\right)^{2}$$

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

$$x^{2}-\frac{1}{2}x+\frac{1}{16}=\frac{3}{2}+\frac{1}{16}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali \frac{3}{2} ni \frac{1}{16} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$x^{2}-\frac{1}{2}x+\frac{1}{16}=\frac{25}{16}$$

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.

$$\left(x-\frac{1}{4}\right)^{2}=\frac{25}{16}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x-\frac{1}{4}\right)^{2}}=\sqrt{\frac{25}{16}}$$

Qisqartirish.

$$x-\frac{1}{4}=\frac{5}{4} x-\frac{1}{4}=-\frac{5}{4}$$

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

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

x qiymati -1 teng bo‘lmaydi.

$$x=\frac{3}{2}$$