P:P=(2+x/2-x+frac{4x^{2}}{x^2-4}-2-x/2+x):x^2-3x/2x^2-x^3 eching

$$P : P = ( \frac { 2 + x } { 2 - x } + \frac { 4 x ^ { 2 } } { x ^ { 2 } - 4 } - \frac { 2 - x } { 2 + x } ) : \frac { x ^ { 2 } - 3 x } { 2 x ^ { 2 } - x ^ { 3 } }$$

P uchun yechish

$$P\neq 0$$

$$x = \frac{\sqrt[3]{6 \sqrt{80229} + 1765} + \sqrt[3]{1765 - 6 \sqrt{80229}} + 7}{12} = 2,1802301552804595$$

Chiziqli tenglamalarni yechish bo'yicha qadamlar

P qiymati 0 teng bo‘lmaydi, chunki nolga bo‘lish mumkin emas. Tenglamaning ikkala tarafini P ga ko'paytirish.

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

Faktor: x^{2}-4.

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

Ifodalarni qo‘shish yoki ayirish uchun ularni yoyib, maxrajlarini bir xil qiling. 2-x va \left(x-2\right)\left(x+2\right) ning eng kichik umumiy karralisi \left(x-2\right)\left(x+2\right). \frac{2+x}{2-x} ni \frac{-\left(x+2\right)}{-\left(x+2\right)} marotabaga ko'paytirish.

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

\frac{\left(2+x\right)\left(-1\right)\left(x+2\right)}{\left(x-2\right)\left(x+2\right)} va \frac{4x^{2}}{\left(x-2\right)\left(x+2\right)} da bir xil maxraji bor, ularning suratini qo‘shish orqali qo‘shing.

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

\left(2+x\right)\left(-1\right)\left(x+2\right)+4x^{2} ichidagi ko‘paytirishlarni bajaring.

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

-2x-4-x^{2}-2x+4x^{2} kabi iboralarga o‘xshab birlashtiring.

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

\frac{-4x-4+3x^{2}}{\left(x-2\right)\left(x+2\right)} ichida hali faktorlanmagan ifodalarni faktorlang.

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

Surat va maxrajdagi ikkala x-2 ni qisqartiring.

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

\frac{3x+2}{x+2} va \frac{2-x}{2+x} da bir xil maxraji bor, ularning suratini ayirish orqali ayiring.

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

3x+2-\left(2-x\right) ichidagi ko‘paytirishlarni bajaring.

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

3x+2-2+x kabi iboralarga o‘xshab birlashtiring.

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

P\times \frac{4x}{x+2} ni yagona kasrga aylantiring.

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

\frac{P\times 4x}{x+2}x\left(-3+x\right)^{-1} ga 2-x ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

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

2\times \frac{P\times 4x}{x+2} ni yagona kasrga aylantiring.

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

\frac{2P\times 4x}{x+2}x ni yagona kasrga aylantiring.

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

\frac{2P\times 4xx}{x+2}\left(-3+x\right)^{-1} ni yagona kasrga aylantiring.

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

\frac{4Px}{x+2}\left(-3+x\right)^{-1} ni yagona kasrga aylantiring.

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

\frac{4Px\left(-3+x\right)^{-1}}{x+2}x^{2} ni yagona kasrga aylantiring.

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

\frac{2P\times 4xx\left(-3+x\right)^{-1}}{x+2} va \frac{4Px\left(-3+x\right)^{-1}x^{2}}{x+2} da bir xil maxraji bor, ularning suratini ayirish orqali ayiring.

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

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

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

Ayni asosning daraja ko‘rsatkichlarini ko‘paytirish uchun ularning darajalarini qo‘shing. 1 va 2 ni qo‘shib, 3 ni oling.

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

8 hosil qilish uchun 2 va 4 ni ko'paytirish.

$$P=\frac{8Px^{2}\left(-3+x\right)^{-1}-4Px^{3}\left(-3+x\right)^{-1}}{x+2}$$

Ikkala tarafdan \frac{8Px^{2}\left(-3+x\right)^{-1}-4Px^{3}\left(-3+x\right)^{-1}}{x+2} ni ayirish.

$$P-\frac{8Px^{2}\left(-3+x\right)^{-1}-4Px^{3}\left(-3+x\right)^{-1}}{x+2}=0$$

Tenglamaning ikkala tarafini x+2 ga ko'paytirish.

$$\left(x+2\right)P-\left(8Px^{2}\left(-3+x\right)^{-1}-4Px^{3}\left(-3+x\right)^{-1}\right)=0$$

Shartlarni qayta saralash.

$$-\left(-4\times \frac{1}{x-3}Px^{3}+8\times \frac{1}{x-3}Px^{2}\right)+P\left(x+2\right)=0$$

Tenglamaning ikkala tarafini x-3 ga ko'paytirish.

$$-\left(-4\times \frac{1}{x-3}Px^{3}+8\times \frac{1}{x-3}Px^{2}\right)\left(x-3\right)+P\left(x+2\right)\left(x-3\right)=0$$

-4\times \frac{1}{x-3} ni yagona kasrga aylantiring.

$$-\left(\frac{-4}{x-3}Px^{3}+8\times \frac{1}{x-3}Px^{2}\right)\left(x-3\right)+P\left(x+2\right)\left(x-3\right)=0$$

\frac{-4}{x-3}P ni yagona kasrga aylantiring.

$$-\left(\frac{-4P}{x-3}x^{3}+8\times \frac{1}{x-3}Px^{2}\right)\left(x-3\right)+P\left(x+2\right)\left(x-3\right)=0$$

\frac{-4P}{x-3}x^{3} ni yagona kasrga aylantiring.

$$-\left(\frac{-4Px^{3}}{x-3}+8\times \frac{1}{x-3}Px^{2}\right)\left(x-3\right)+P\left(x+2\right)\left(x-3\right)=0$$

8\times \frac{1}{x-3} ni yagona kasrga aylantiring.

$$-\left(\frac{-4Px^{3}}{x-3}+\frac{8}{x-3}Px^{2}\right)\left(x-3\right)+P\left(x+2\right)\left(x-3\right)=0$$

\frac{8}{x-3}P ni yagona kasrga aylantiring.

$$-\left(\frac{-4Px^{3}}{x-3}+\frac{8P}{x-3}x^{2}\right)\left(x-3\right)+P\left(x+2\right)\left(x-3\right)=0$$

\frac{8P}{x-3}x^{2} ni yagona kasrga aylantiring.

$$-\left(\frac{-4Px^{3}}{x-3}+\frac{8Px^{2}}{x-3}\right)\left(x-3\right)+P\left(x+2\right)\left(x-3\right)=0$$

\frac{-4Px^{3}}{x-3} va \frac{8Px^{2}}{x-3} da bir xil maxraji bor, ularning suratini qo‘shish orqali qo‘shing.

$$-\frac{-4Px^{3}+8Px^{2}}{x-3}\left(x-3\right)+P\left(x+2\right)\left(x-3\right)=0$$

\frac{-4Px^{3}+8Px^{2}}{x-3}\left(x-3\right) ni yagona kasrga aylantiring.

$$-\frac{\left(-4Px^{3}+8Px^{2}\right)\left(x-3\right)}{x-3}+P\left(x+2\right)\left(x-3\right)=0$$

Surat va maxrajdagi ikkala x-3 ni qisqartiring.

$$-\left(-4Px^{3}+8Px^{2}\right)+P\left(x+2\right)\left(x-3\right)=0$$

-4Px^{3}+8Px^{2} teskarisini topish uchun har birining teskarisini toping.

$$4Px^{3}-8Px^{2}+P\left(x+2\right)\left(x-3\right)=0$$

P ga x+2 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$4Px^{3}-8Px^{2}+\left(Px+2P\right)\left(x-3\right)=0$$

Px+2P ga x-3 ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

$$4Px^{3}-8Px^{2}+Px^{2}-Px-6P=0$$

-7Px^{2} ni olish uchun -8Px^{2} va Px^{2} ni birlashtirish.

$$4Px^{3}-7Px^{2}-Px-6P=0$$

P'ga ega bo'lgan barcha shartlarni birlashtirish.

$$\left(4x^{3}-7x^{2}-x-6\right)P=0$$

0 ni -x-7x^{2}-6+4x^{3} ga bo'lish.

$$P=0$$

P qiymati 0 teng bo‘lmaydi.

$$P\in \emptyset$$

x uchun yechish

$$x = \frac{\sqrt[3]{6 \sqrt{80229} + 1765} + \sqrt[3]{1765 - 6 \sqrt{80229}} + 7}{12} = 2,1802301552804595$$

$$P\neq 0$$