f(x)/g(x)=f(x)(g(x))^-1 eching

$$\frac { f ( x ) } { g ( x ) } = f ( x ) ( g ( x ) ) ^ { - 1 }$$

f uchun yechish (complex solution)

$$f\in \mathrm{C}$$

$$g\neq 0\text{ and }x\neq 0$$

Yechim qadamlari

Tenglamaning ikkala tarafini gx ga ko'paytirish.

$$fx=fx\left(gx\right)^{-1}gx$$

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

$$fx=fx^{2}\left(gx\right)^{-1}g$$

\left(gx\right)^{-1} ni kengaytirish.

$$fx=fx^{2}g^{-1}x^{-1}g$$

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

$$fx=fx^{1}g^{-1}g$$

1 daraja ko‘rsatkichini x ga hisoblang va x ni qiymatni oling.

$$fx=fxg^{-1}g$$

Ikkala tarafdan fxg^{-1}g ni ayirish.

$$fx-fxg^{-1}g=0$$

Shartlarni qayta saralash.

$$fx-\frac{1}{g}fgx=0$$

Tenglamaning ikkala tarafini g ga ko'paytirish.

$$fxg-\frac{1}{g}fgxg=0$$

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

$$fxg-\frac{1}{g}fg^{2}x=0$$

\frac{1}{g}f ni yagona kasrga aylantiring.

$$fxg-\frac{f}{g}g^{2}x=0$$

\frac{f}{g}g^{2} ni yagona kasrga aylantiring.

$$fxg-\frac{fg^{2}}{g}x=0$$

Surat va maxrajdagi ikkala g ni qisqartiring.

$$fxg-fgx=0$$

0 ni olish uchun fxg va -fgx ni birlashtirish.

$$0=0$$

0 va 0 ni taqqoslang.

$$\text{true}$$

Bu har qanday f uchun to‘g‘ri.

$$f\in \mathrm{C}$$

f uchun yechish

$$f\in \mathrm{R}$$

$$g\neq 0\text{ and }x\neq 0$$

Yechim qadamlari

Tenglamaning ikkala tarafini gx ga ko'paytirish.

$$fx=fx\left(gx\right)^{-1}gx$$

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

$$fx=fx^{2}\left(gx\right)^{-1}g$$

\left(gx\right)^{-1} ni kengaytirish.

$$fx=fx^{2}g^{-1}x^{-1}g$$

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

$$fx=fx^{1}g^{-1}g$$

1 daraja ko‘rsatkichini x ga hisoblang va x ni qiymatni oling.

$$fx=fxg^{-1}g$$

Ikkala tarafdan fxg^{-1}g ni ayirish.

$$fx-fxg^{-1}g=0$$

Shartlarni qayta saralash.

$$fx-\frac{1}{g}fgx=0$$

Tenglamaning ikkala tarafini g ga ko'paytirish.

$$fxg-\frac{1}{g}fgxg=0$$

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

$$fxg-\frac{1}{g}fg^{2}x=0$$

\frac{1}{g}f ni yagona kasrga aylantiring.

$$fxg-\frac{f}{g}g^{2}x=0$$

\frac{f}{g}g^{2} ni yagona kasrga aylantiring.

$$fxg-\frac{fg^{2}}{g}x=0$$

Surat va maxrajdagi ikkala g ni qisqartiring.

$$fxg-fgx=0$$

0 ni olish uchun fxg va -fgx ni birlashtirish.

$$0=0$$

0 va 0 ni taqqoslang.

$$\text{true}$$

Bu har qanday f uchun to‘g‘ri.

$$f\in \mathrm{R}$$

g uchun yechish

$$g\neq 0$$

$$x\neq 0$$

Yechim qadamlari

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

$$fx=fx\left(gx\right)^{-1}gx$$

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

$$fx=fx^{2}\left(gx\right)^{-1}g$$

\left(gx\right)^{-1} ni kengaytirish.

$$fx=fx^{2}g^{-1}x^{-1}g$$

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

$$fx=fx^{1}g^{-1}g$$

1 daraja ko‘rsatkichini x ga hisoblang va x ni qiymatni oling.

$$fx=fxg^{-1}g$$

Tomonlarni almashtirib, barcha oʻzgaruvchi shartlar chap tomonga oʻtkazing.

$$fxg^{-1}g=fx$$

Shartlarni qayta saralash.

$$\frac{1}{g}fgx=fx$$

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

$$1fgx=fxg$$

Ikkala tarafdan fxg ni ayirish.

$$1fgx-fxg=0$$

0 ni olish uchun 1fgx va -fxg ni birlashtirish.

$$0=0$$

0 va 0 ni taqqoslang.

$$\text{true}$$

Bu har qanday g uchun to‘g‘ri.

$$g\in \mathrm{R}$$

g qiymati 0 teng bo‘lmaydi.

$$g\in \mathrm{R}\setminus 0$$