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

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

x uchun yechish (complex solution)

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

Yechim qadamlari

\left(x+1\right)^{2} hosil qilish uchun x+1 va x+1 ni ko'paytirish.

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

\left(x-1\right)^{2} hosil qilish uchun x-1 va x-1 ni ko'paytirish.

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

\left(x^{2}+1\right)^{2} hosil qilish uchun x^{2}+1 va x^{2}+1 ni ko'paytirish.

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

\left(a+b\right)^{2}=a^{2}+2ab+b^{2} binom teoremasini \left(x+1\right)^{2} kengaytirilishi uchun ishlating.

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

\left(a-b\right)^{2}=a^{2}-2ab+b^{2} binom teoremasini \left(x-1\right)^{2} kengaytirilishi uchun ishlating.

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

\frac{1}{4} ga x^{2}+2x+1 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

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

\frac{1}{4}x^{2}+\frac{1}{2}x+\frac{1}{4} ga x^{2}-2x+1 ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

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

\frac{1}{2}x^{2} ni olish uchun -\frac{1}{2}x^{2} va x^{2} ni birlashtirish.

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

\left(a+b\right)^{2}=a^{2}+2ab+b^{2} binom teoremasini \left(x^{2}+1\right)^{2} kengaytirilishi uchun ishlating.

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

Daraja ko‘rsatkichini boshqa ko‘rsatkichga oshirish uchun, darajalarini ko‘paytiring. 2 va 2 ni ko‘paytirib, 4 ni oling.

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

\frac{1}{4} ga x^{4}+2x^{2}+1 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

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

Ikkala tarafdan \frac{1}{4}x^{4} ni ayirish.

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

0 ni olish uchun \frac{1}{4}x^{4} va -\frac{1}{4}x^{4} ni birlashtirish.

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

Ikkala tarafdan \frac{1}{2}x^{2} ni ayirish.

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

0 ni olish uchun \frac{1}{2}x^{2} va -\frac{1}{2}x^{2} ni birlashtirish.

$$\frac{1}{4}=\frac{1}{4}$$

\frac{1}{4} va \frac{1}{4} ni taqqoslang.

$$\text{true}$$

Bu har qanday x uchun to‘g‘ri.

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

x uchun yechish

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

Yechim qadamlari

\left(x+1\right)^{2} hosil qilish uchun x+1 va x+1 ni ko'paytirish.

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

\left(x-1\right)^{2} hosil qilish uchun x-1 va x-1 ni ko'paytirish.

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

\left(x^{2}+1\right)^{2} hosil qilish uchun x^{2}+1 va x^{2}+1 ni ko'paytirish.

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

\left(a+b\right)^{2}=a^{2}+2ab+b^{2} binom teoremasini \left(x+1\right)^{2} kengaytirilishi uchun ishlating.

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

\left(a-b\right)^{2}=a^{2}-2ab+b^{2} binom teoremasini \left(x-1\right)^{2} kengaytirilishi uchun ishlating.

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

\frac{1}{4} ga x^{2}+2x+1 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

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

\frac{1}{4}x^{2}+\frac{1}{2}x+\frac{1}{4} ga x^{2}-2x+1 ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

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

\frac{1}{2}x^{2} ni olish uchun -\frac{1}{2}x^{2} va x^{2} ni birlashtirish.

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

\left(a+b\right)^{2}=a^{2}+2ab+b^{2} binom teoremasini \left(x^{2}+1\right)^{2} kengaytirilishi uchun ishlating.

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

Daraja ko‘rsatkichini boshqa ko‘rsatkichga oshirish uchun, darajalarini ko‘paytiring. 2 va 2 ni ko‘paytirib, 4 ni oling.

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

\frac{1}{4} ga x^{4}+2x^{2}+1 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

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

Ikkala tarafdan \frac{1}{4}x^{4} ni ayirish.

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

0 ni olish uchun \frac{1}{4}x^{4} va -\frac{1}{4}x^{4} ni birlashtirish.

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

Ikkala tarafdan \frac{1}{2}x^{2} ni ayirish.

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

0 ni olish uchun \frac{1}{2}x^{2} va -\frac{1}{2}x^{2} ni birlashtirish.

$$\frac{1}{4}=\frac{1}{4}$$

\frac{1}{4} va \frac{1}{4} ni taqqoslang.

$$\text{true}$$

Bu har qanday x uchun to‘g‘ri.

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