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

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

x uchun yechish (complex solution)

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

Yechim qadamlari

x qiymati -1,-i,i,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-i\right)\left(x+i\right) ga, x^{2}-1,x^{2}+1,x^{4}-1 ning eng kichik karralisiga ko‘paytiring.

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

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

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

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

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

2x^{2}-2 teskarisini topish uchun har birining teskarisini toping.

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

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

$$2+2=4$$

4 olish uchun 2 va 2'ni qo'shing.

$$4=4$$

4 va 4 ni taqqoslang.

$$\text{true}$$

Bu har qanday x uchun to‘g‘ri.

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

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

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

x uchun yechish

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

Yechim 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}+1\right) ga, x^{2}-1,x^{2}+1,x^{4}-1 ning eng kichik karralisiga ko‘paytiring.

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

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

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

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

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

2x^{2}-2 teskarisini topish uchun har birining teskarisini toping.

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

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

$$2+2=4$$

4 olish uchun 2 va 2'ni qo'shing.

$$4=4$$

4 va 4 ni taqqoslang.

$$\text{true}$$

Bu har qanday x uchun to‘g‘ri.

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

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

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