3+4x/x^2+x-1=3/x-x/x+1 eching

$$\frac { 3 + 4 x } { x ^ { 2 } + x } - 1 = \frac { 3 } { x } - \frac { x } { x + 1 }$$

x uchun yechish (complex solution)

$$x\in \mathrm{C}\setminus -1,0$$

Yechim qadamlari

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

$$3+4x+x\left(x+1\right)\left(-1\right)=\left(x+1\right)\times 3-xx$$

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

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

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

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

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

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

3x ni olish uchun 4x va -x ni birlashtirish.

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

x+1 ga 3 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

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

Ikkala tarafdan 3x ni ayirish.

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

0 ni olish uchun 3x va -3x ni birlashtirish.

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

Ikkala tarafdan 3 ni ayirish.

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

0 olish uchun 3 dan 3 ni ayirish.

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

x^{2} ni ikki tarafga qo’shing.

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

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

$$0=0$$

0 va 0 ni taqqoslang.

$$\text{true}$$

Bu har qanday x uchun to‘g‘ri.

$$x\in \mathrm{C}$$

x qiymati -1,0 qiymatlaridan birortasiga teng bo‘lmaydi.

$$x\in \mathrm{C}\setminus -1,0$$

x uchun yechish

$$x\in \mathrm{R}\setminus -1,0$$

Yechim qadamlari

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

$$3+4x+x\left(x+1\right)\left(-1\right)=\left(x+1\right)\times 3-xx$$

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

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

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

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

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

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

3x ni olish uchun 4x va -x ni birlashtirish.

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

x+1 ga 3 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

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

Ikkala tarafdan 3x ni ayirish.

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

0 ni olish uchun 3x va -3x ni birlashtirish.

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

Ikkala tarafdan 3 ni ayirish.

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

0 olish uchun 3 dan 3 ni ayirish.

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

x^{2} ni ikki tarafga qo’shing.

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

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

$$0=0$$

0 va 0 ni taqqoslang.

$$\text{true}$$

Bu har qanday x uchun to‘g‘ri.

$$x\in \mathrm{R}$$

x qiymati -1,0 qiymatlaridan birortasiga teng bo‘lmaydi.

$$x\in \mathrm{R}\setminus -1,0$$