6x-36/x^2-36div12x^2-36x/x^2+9x-36=x+12/2x^2+12x eching

$$\frac { 6 x - 36 } { x ^ { 2 } - 36 } \div \frac { 12 x ^ { 2 } - 36 x } { x ^ { 2 } + 9 x - 36 } = \frac { x + 12 } { 2 x ^ { 2 } + 12 x }$$

x uchun yechish (complex solution)

$$x\in \mathrm{C}\setminus -6,6,0,-12,3$$

Qisqa yechim qadamlari

x qiymati -6,0 qiymatlaridan birortasiga teng bo‘lmaydi, chunki nolga bo‘lish mumkin emas. Tenglamaning ikkala tarafini 2x\left(x+6\right) ga ko'paytirish.

$$\frac{1}{6}\left(x+6\right)\left(12+x\right)\times \frac{6x-36}{x^{2}-36}=x+12$$

\frac{1}{6} ga x+6 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$\left(\frac{1}{6}x+1\right)\left(12+x\right)\times \frac{6x-36}{x^{2}-36}=x+12$$

\frac{1}{6}x+1 ga 12+x ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

$$\left(3x+\frac{1}{6}x^{2}+12\right)\times \frac{6x-36}{x^{2}-36}=x+12$$

3x+\frac{1}{6}x^{2}+12 ga \frac{6x-36}{x^{2}-36} ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$3x\times \frac{6x-36}{x^{2}-36}+\frac{1}{6}x^{2}\times \frac{6x-36}{x^{2}-36}+12\times \frac{6x-36}{x^{2}-36}=x+12$$

3\times \frac{6x-36}{x^{2}-36} ni yagona kasrga aylantiring.

$$\frac{3\left(6x-36\right)}{x^{2}-36}x+\frac{1}{6}x^{2}\times \frac{6x-36}{x^{2}-36}+12\times \frac{6x-36}{x^{2}-36}=x+12$$

Suratni maxrajga va maxrajini suratga ko‘paytirish orqali \frac{1}{6} ni \frac{6x-36}{x^{2}-36} ga ko‘paytiring.

$$\frac{3\left(6x-36\right)}{x^{2}-36}x+\frac{6x-36}{6\left(x^{2}-36\right)}x^{2}+12\times \frac{6x-36}{x^{2}-36}=x+12$$

12\times \frac{6x-36}{x^{2}-36} ni yagona kasrga aylantiring.

$$\frac{3\left(6x-36\right)}{x^{2}-36}x+\frac{6x-36}{6\left(x^{2}-36\right)}x^{2}+\frac{12\left(6x-36\right)}{x^{2}-36}=x+12$$

3 ga 6x-36 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$\frac{18x-108}{x^{2}-36}x+\frac{6x-36}{6\left(x^{2}-36\right)}x^{2}+\frac{12\left(6x-36\right)}{x^{2}-36}=x+12$$

\frac{18x-108}{x^{2}-36}x ni yagona kasrga aylantiring.

$$\frac{\left(18x-108\right)x}{x^{2}-36}+\frac{6x-36}{6\left(x^{2}-36\right)}x^{2}+\frac{12\left(6x-36\right)}{x^{2}-36}=x+12$$

\frac{6x-36}{6\left(x^{2}-36\right)} ichida hali faktorlanmagan ifodalarni faktorlang.

$$\frac{\left(18x-108\right)x}{x^{2}-36}+\frac{6\left(x-6\right)}{6\left(x-6\right)\left(x+6\right)}x^{2}+\frac{12\left(6x-36\right)}{x^{2}-36}=x+12$$

Surat va maxrajdagi ikkala 6 ni qisqartiring.

$$\frac{\left(18x-108\right)x}{x^{2}-36}+\frac{x-6}{\left(x-6\right)\left(x+6\right)}x^{2}+\frac{12\left(6x-36\right)}{x^{2}-36}=x+12$$

\frac{x-6}{\left(x-6\right)\left(x+6\right)}x^{2} ni yagona kasrga aylantiring.

$$\frac{\left(18x-108\right)x}{x^{2}-36}+\frac{\left(x-6\right)x^{2}}{\left(x-6\right)\left(x+6\right)}+\frac{12\left(6x-36\right)}{x^{2}-36}=x+12$$

12 ga 6x-36 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$\frac{\left(18x-108\right)x}{x^{2}-36}+\frac{\left(x-6\right)x^{2}}{\left(x-6\right)\left(x+6\right)}+\frac{72x-432}{x^{2}-36}=x+12$$

Faktor: x^{2}-36.

$$\frac{\left(18x-108\right)x}{\left(x-6\right)\left(x+6\right)}+\frac{\left(x-6\right)x^{2}}{\left(x-6\right)\left(x+6\right)}+\frac{72x-432}{x^{2}-36}=x+12$$

\frac{\left(18x-108\right)x}{\left(x-6\right)\left(x+6\right)} va \frac{\left(x-6\right)x^{2}}{\left(x-6\right)\left(x+6\right)} da bir xil maxraji bor, ularning suratini qo‘shish orqali qo‘shing.

$$\frac{\left(18x-108\right)x+\left(x-6\right)x^{2}}{\left(x-6\right)\left(x+6\right)}+\frac{72x-432}{x^{2}-36}=x+12$$

\left(18x-108\right)x+\left(x-6\right)x^{2} ichidagi ko‘paytirishlarni bajaring.

$$\frac{18x^{2}-108x+x^{3}-6x^{2}}{\left(x-6\right)\left(x+6\right)}+\frac{72x-432}{x^{2}-36}=x+12$$

18x^{2}-108x+x^{3}-6x^{2} kabi iboralarga o‘xshab birlashtiring.

$$\frac{12x^{2}-108x+x^{3}}{\left(x-6\right)\left(x+6\right)}+\frac{72x-432}{x^{2}-36}=x+12$$

Faktor: x^{2}-36.

$$\frac{12x^{2}-108x+x^{3}}{\left(x-6\right)\left(x+6\right)}+\frac{72x-432}{\left(x-6\right)\left(x+6\right)}=x+12$$

\frac{12x^{2}-108x+x^{3}}{\left(x-6\right)\left(x+6\right)} va \frac{72x-432}{\left(x-6\right)\left(x+6\right)} da bir xil maxraji bor, ularning suratini qo‘shish orqali qo‘shing.

$$\frac{12x^{2}-108x+x^{3}+72x-432}{\left(x-6\right)\left(x+6\right)}=x+12$$

12x^{2}-108x+x^{3}+72x-432 kabi iboralarga o‘xshab birlashtiring.

$$\frac{12x^{2}-36x+x^{3}-432}{\left(x-6\right)\left(x+6\right)}=x+12$$

Hisoblang: \left(x-6\right)\left(x+6\right). Ko‘paytirish qoida yordamida turli kvadratlarga aylantirilishi mumkin: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}. 6 kvadratini chiqarish.

$$\frac{12x^{2}-36x+x^{3}-432}{x^{2}-36}=x+12$$

Ikkala tarafdan x ni ayirish.

$$\frac{12x^{2}-36x+x^{3}-432}{x^{2}-36}-x=12$$

Faktor: x^{2}-36.

$$\frac{12x^{2}-36x+x^{3}-432}{\left(x-6\right)\left(x+6\right)}-x=12$$

Ifodalarni qo‘shish yoki ayirish uchun ularni yoyib, maxrajlarini bir xil qiling. x ni \frac{\left(x-6\right)\left(x+6\right)}{\left(x-6\right)\left(x+6\right)} marotabaga ko'paytirish.

$$\frac{12x^{2}-36x+x^{3}-432}{\left(x-6\right)\left(x+6\right)}-\frac{x\left(x-6\right)\left(x+6\right)}{\left(x-6\right)\left(x+6\right)}=12$$

\frac{12x^{2}-36x+x^{3}-432}{\left(x-6\right)\left(x+6\right)} va \frac{x\left(x-6\right)\left(x+6\right)}{\left(x-6\right)\left(x+6\right)} da bir xil maxraji bor, ularning suratini ayirish orqali ayiring.

$$\frac{12x^{2}-36x+x^{3}-432-x\left(x-6\right)\left(x+6\right)}{\left(x-6\right)\left(x+6\right)}=12$$

12x^{2}-36x+x^{3}-432-x\left(x-6\right)\left(x+6\right) ichidagi ko‘paytirishlarni bajaring.

$$\frac{12x^{2}-36x+x^{3}-432-x^{3}-6x^{2}+6x^{2}+36x}{\left(x-6\right)\left(x+6\right)}=12$$

12x^{2}-36x+x^{3}-432-x^{3}-6x^{2}+6x^{2}+36x kabi iboralarga o‘xshab birlashtiring.

$$\frac{12x^{2}-432}{\left(x-6\right)\left(x+6\right)}=12$$

Ikkala tarafdan 12 ni ayirish.

$$\frac{12x^{2}-432}{\left(x-6\right)\left(x+6\right)}-12=0$$

Ifodalarni qo‘shish yoki ayirish uchun ularni yoyib, maxrajlarini bir xil qiling. 12 ni \frac{\left(x-6\right)\left(x+6\right)}{\left(x-6\right)\left(x+6\right)} marotabaga ko'paytirish.

$$\frac{12x^{2}-432}{\left(x-6\right)\left(x+6\right)}-\frac{12\left(x-6\right)\left(x+6\right)}{\left(x-6\right)\left(x+6\right)}=0$$

\frac{12x^{2}-432}{\left(x-6\right)\left(x+6\right)} va \frac{12\left(x-6\right)\left(x+6\right)}{\left(x-6\right)\left(x+6\right)} da bir xil maxraji bor, ularning suratini ayirish orqali ayiring.

$$\frac{12x^{2}-432-12\left(x-6\right)\left(x+6\right)}{\left(x-6\right)\left(x+6\right)}=0$$

12x^{2}-432-12\left(x-6\right)\left(x+6\right) ichidagi ko‘paytirishlarni bajaring.

$$\frac{12x^{2}-432-12x^{2}-72x+72x+432}{\left(x-6\right)\left(x+6\right)}=0$$

12x^{2}-432-12x^{2}-72x+72x+432 kabi iboralarga o‘xshab birlashtiring.

$$\frac{0}{\left(x-6\right)\left(x+6\right)}=0$$

x qiymati -6,6 qiymatlaridan birortasiga teng bo‘lmaydi, chunki nolga bo‘lish mumkin emas. Tenglamaning ikkala tarafini \left(x-6\right)\left(x+6\right) ga ko'paytirish.

$$0=0$$

Bu har qanday x uchun to‘g‘ri.

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

x qiymati -6,6,0 qiymatlaridan birortasiga teng bo‘lmaydi.

$$x\in \mathrm{C}\setminus -6,0,6$$

x uchun yechish

$$x\in \mathrm{R}\setminus 6,-6,0,3,-12$$

Qisqa yechim qadamlari

x qiymati -6,0 qiymatlaridan birortasiga teng bo‘lmaydi, chunki nolga bo‘lish mumkin emas. Tenglamaning ikkala tarafini 2x\left(x+6\right) ga ko'paytirish.

$$\frac{1}{6}\left(x+6\right)\left(12+x\right)\times \frac{6x-36}{x^{2}-36}=x+12$$

\frac{1}{6} ga x+6 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$\left(\frac{1}{6}x+1\right)\left(12+x\right)\times \frac{6x-36}{x^{2}-36}=x+12$$

\frac{1}{6}x+1 ga 12+x ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

$$\left(3x+\frac{1}{6}x^{2}+12\right)\times \frac{6x-36}{x^{2}-36}=x+12$$

3x+\frac{1}{6}x^{2}+12 ga \frac{6x-36}{x^{2}-36} ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$3x\times \frac{6x-36}{x^{2}-36}+\frac{1}{6}x^{2}\times \frac{6x-36}{x^{2}-36}+12\times \frac{6x-36}{x^{2}-36}=x+12$$

3\times \frac{6x-36}{x^{2}-36} ni yagona kasrga aylantiring.

$$\frac{3\left(6x-36\right)}{x^{2}-36}x+\frac{1}{6}x^{2}\times \frac{6x-36}{x^{2}-36}+12\times \frac{6x-36}{x^{2}-36}=x+12$$

Suratni maxrajga va maxrajini suratga ko‘paytirish orqali \frac{1}{6} ni \frac{6x-36}{x^{2}-36} ga ko‘paytiring.

$$\frac{3\left(6x-36\right)}{x^{2}-36}x+\frac{6x-36}{6\left(x^{2}-36\right)}x^{2}+12\times \frac{6x-36}{x^{2}-36}=x+12$$

12\times \frac{6x-36}{x^{2}-36} ni yagona kasrga aylantiring.

$$\frac{3\left(6x-36\right)}{x^{2}-36}x+\frac{6x-36}{6\left(x^{2}-36\right)}x^{2}+\frac{12\left(6x-36\right)}{x^{2}-36}=x+12$$

3 ga 6x-36 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$\frac{18x-108}{x^{2}-36}x+\frac{6x-36}{6\left(x^{2}-36\right)}x^{2}+\frac{12\left(6x-36\right)}{x^{2}-36}=x+12$$

\frac{18x-108}{x^{2}-36}x ni yagona kasrga aylantiring.

$$\frac{\left(18x-108\right)x}{x^{2}-36}+\frac{6x-36}{6\left(x^{2}-36\right)}x^{2}+\frac{12\left(6x-36\right)}{x^{2}-36}=x+12$$

\frac{6x-36}{6\left(x^{2}-36\right)} ichida hali faktorlanmagan ifodalarni faktorlang.

$$\frac{\left(18x-108\right)x}{x^{2}-36}+\frac{6\left(x-6\right)}{6\left(x-6\right)\left(x+6\right)}x^{2}+\frac{12\left(6x-36\right)}{x^{2}-36}=x+12$$

Surat va maxrajdagi ikkala 6 ni qisqartiring.

$$\frac{\left(18x-108\right)x}{x^{2}-36}+\frac{x-6}{\left(x-6\right)\left(x+6\right)}x^{2}+\frac{12\left(6x-36\right)}{x^{2}-36}=x+12$$

\frac{x-6}{\left(x-6\right)\left(x+6\right)}x^{2} ni yagona kasrga aylantiring.

$$\frac{\left(18x-108\right)x}{x^{2}-36}+\frac{\left(x-6\right)x^{2}}{\left(x-6\right)\left(x+6\right)}+\frac{12\left(6x-36\right)}{x^{2}-36}=x+12$$

12 ga 6x-36 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$\frac{\left(18x-108\right)x}{x^{2}-36}+\frac{\left(x-6\right)x^{2}}{\left(x-6\right)\left(x+6\right)}+\frac{72x-432}{x^{2}-36}=x+12$$

Faktor: x^{2}-36.

$$\frac{\left(18x-108\right)x}{\left(x-6\right)\left(x+6\right)}+\frac{\left(x-6\right)x^{2}}{\left(x-6\right)\left(x+6\right)}+\frac{72x-432}{x^{2}-36}=x+12$$

\frac{\left(18x-108\right)x}{\left(x-6\right)\left(x+6\right)} va \frac{\left(x-6\right)x^{2}}{\left(x-6\right)\left(x+6\right)} da bir xil maxraji bor, ularning suratini qo‘shish orqali qo‘shing.

$$\frac{\left(18x-108\right)x+\left(x-6\right)x^{2}}{\left(x-6\right)\left(x+6\right)}+\frac{72x-432}{x^{2}-36}=x+12$$

\left(18x-108\right)x+\left(x-6\right)x^{2} ichidagi ko‘paytirishlarni bajaring.

$$\frac{18x^{2}-108x+x^{3}-6x^{2}}{\left(x-6\right)\left(x+6\right)}+\frac{72x-432}{x^{2}-36}=x+12$$

18x^{2}-108x+x^{3}-6x^{2} kabi iboralarga o‘xshab birlashtiring.

$$\frac{12x^{2}-108x+x^{3}}{\left(x-6\right)\left(x+6\right)}+\frac{72x-432}{x^{2}-36}=x+12$$

Faktor: x^{2}-36.

$$\frac{12x^{2}-108x+x^{3}}{\left(x-6\right)\left(x+6\right)}+\frac{72x-432}{\left(x-6\right)\left(x+6\right)}=x+12$$

\frac{12x^{2}-108x+x^{3}}{\left(x-6\right)\left(x+6\right)} va \frac{72x-432}{\left(x-6\right)\left(x+6\right)} da bir xil maxraji bor, ularning suratini qo‘shish orqali qo‘shing.

$$\frac{12x^{2}-108x+x^{3}+72x-432}{\left(x-6\right)\left(x+6\right)}=x+12$$

12x^{2}-108x+x^{3}+72x-432 kabi iboralarga o‘xshab birlashtiring.

$$\frac{12x^{2}-36x+x^{3}-432}{\left(x-6\right)\left(x+6\right)}=x+12$$

Hisoblang: \left(x-6\right)\left(x+6\right). Ko‘paytirish qoida yordamida turli kvadratlarga aylantirilishi mumkin: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}. 6 kvadratini chiqarish.

$$\frac{12x^{2}-36x+x^{3}-432}{x^{2}-36}=x+12$$

Ikkala tarafdan x ni ayirish.

$$\frac{12x^{2}-36x+x^{3}-432}{x^{2}-36}-x=12$$

Faktor: x^{2}-36.

$$\frac{12x^{2}-36x+x^{3}-432}{\left(x-6\right)\left(x+6\right)}-x=12$$

Ifodalarni qo‘shish yoki ayirish uchun ularni yoyib, maxrajlarini bir xil qiling. x ni \frac{\left(x-6\right)\left(x+6\right)}{\left(x-6\right)\left(x+6\right)} marotabaga ko'paytirish.

$$\frac{12x^{2}-36x+x^{3}-432}{\left(x-6\right)\left(x+6\right)}-\frac{x\left(x-6\right)\left(x+6\right)}{\left(x-6\right)\left(x+6\right)}=12$$

\frac{12x^{2}-36x+x^{3}-432}{\left(x-6\right)\left(x+6\right)} va \frac{x\left(x-6\right)\left(x+6\right)}{\left(x-6\right)\left(x+6\right)} da bir xil maxraji bor, ularning suratini ayirish orqali ayiring.

$$\frac{12x^{2}-36x+x^{3}-432-x\left(x-6\right)\left(x+6\right)}{\left(x-6\right)\left(x+6\right)}=12$$

12x^{2}-36x+x^{3}-432-x\left(x-6\right)\left(x+6\right) ichidagi ko‘paytirishlarni bajaring.

$$\frac{12x^{2}-36x+x^{3}-432-x^{3}-6x^{2}+6x^{2}+36x}{\left(x-6\right)\left(x+6\right)}=12$$

12x^{2}-36x+x^{3}-432-x^{3}-6x^{2}+6x^{2}+36x kabi iboralarga o‘xshab birlashtiring.

$$\frac{12x^{2}-432}{\left(x-6\right)\left(x+6\right)}=12$$

Ikkala tarafdan 12 ni ayirish.

$$\frac{12x^{2}-432}{\left(x-6\right)\left(x+6\right)}-12=0$$

Ifodalarni qo‘shish yoki ayirish uchun ularni yoyib, maxrajlarini bir xil qiling. 12 ni \frac{\left(x-6\right)\left(x+6\right)}{\left(x-6\right)\left(x+6\right)} marotabaga ko'paytirish.

$$\frac{12x^{2}-432}{\left(x-6\right)\left(x+6\right)}-\frac{12\left(x-6\right)\left(x+6\right)}{\left(x-6\right)\left(x+6\right)}=0$$

\frac{12x^{2}-432}{\left(x-6\right)\left(x+6\right)} va \frac{12\left(x-6\right)\left(x+6\right)}{\left(x-6\right)\left(x+6\right)} da bir xil maxraji bor, ularning suratini ayirish orqali ayiring.

$$\frac{12x^{2}-432-12\left(x-6\right)\left(x+6\right)}{\left(x-6\right)\left(x+6\right)}=0$$

12x^{2}-432-12\left(x-6\right)\left(x+6\right) ichidagi ko‘paytirishlarni bajaring.

$$\frac{12x^{2}-432-12x^{2}-72x+72x+432}{\left(x-6\right)\left(x+6\right)}=0$$

12x^{2}-432-12x^{2}-72x+72x+432 kabi iboralarga o‘xshab birlashtiring.

$$\frac{0}{\left(x-6\right)\left(x+6\right)}=0$$

x qiymati -6,6 qiymatlaridan birortasiga teng bo‘lmaydi, chunki nolga bo‘lish mumkin emas. Tenglamaning ikkala tarafini \left(x-6\right)\left(x+6\right) ga ko'paytirish.

$$0=0$$

Bu har qanday x uchun to‘g‘ri.

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

x qiymati -6,6,0 qiymatlaridan birortasiga teng bo‘lmaydi.

$$x\in \mathrm{R}\setminus -6,0,6$$