(3+7x-5/x^2-x-2-3x/x+1):4/x-2+5/x+3=frac{9x^2+43x+8}{4(x^2+4x+3)} eching

$$( 3 + \frac { 7 x - 5 } { x ^ { 2 } - x - 2 } - \frac { 3 x } { x + 1 } ) : \frac { 4 } { x - 2 } + \frac { 5 } { x + 3 } = \frac { 9 x ^ { 2 } + 43 x + 8 } { 4 ( x ^ { 2 } + 4 x + 3 ) }$$

x uchun yechish

$$x=7$$

Yechim qadamlari

x qiymati -3,-1 qiymatlaridan birortasiga teng bo‘lmaydi, chunki nolga bo‘lish mumkin emas. Tenglamaning ikkala tarafini 4\left(x+1\right)\left(x+3\right) ga, x+3,4\left(x^{2}+4x+3\right) ning eng kichik karralisiga ko‘paytiring.

$$\left(x+1\right)\left(x+3\right)\left(x-2\right)\left(3+\frac{7x-5}{x^{2}-x-2}-\frac{3x}{x+1}\right)+\left(4x+4\right)\times 5=9x^{2}+43x+8$$

x+1 ga x+3 ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

$$\left(x^{2}+4x+3\right)\left(x-2\right)\left(3+\frac{7x-5}{x^{2}-x-2}-\frac{3x}{x+1}\right)+\left(4x+4\right)\times 5=9x^{2}+43x+8$$

x^{2}+4x+3 ga x-2 ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

$$\left(x^{3}+2x^{2}-5x-6\right)\left(3+\frac{7x-5}{x^{2}-x-2}-\frac{3x}{x+1}\right)+\left(4x+4\right)\times 5=9x^{2}+43x+8$$

Faktor: x^{2}-x-2.

$$\left(x^{3}+2x^{2}-5x-6\right)\left(3+\frac{7x-5}{\left(x-2\right)\left(x+1\right)}-\frac{3x}{x+1}\right)+\left(4x+4\right)\times 5=9x^{2}+43x+8$$

Ifodalarni qo‘shish yoki ayirish uchun ularni yoyib, maxrajlarini bir xil qiling. 3 ni \frac{\left(x-2\right)\left(x+1\right)}{\left(x-2\right)\left(x+1\right)} marotabaga ko'paytirish.

$$\left(x^{3}+2x^{2}-5x-6\right)\left(\frac{3\left(x-2\right)\left(x+1\right)}{\left(x-2\right)\left(x+1\right)}+\frac{7x-5}{\left(x-2\right)\left(x+1\right)}-\frac{3x}{x+1}\right)+\left(4x+4\right)\times 5=9x^{2}+43x+8$$

\frac{3\left(x-2\right)\left(x+1\right)}{\left(x-2\right)\left(x+1\right)} va \frac{7x-5}{\left(x-2\right)\left(x+1\right)} da bir xil maxraji bor, ularning suratini qo‘shish orqali qo‘shing.

$$\left(x^{3}+2x^{2}-5x-6\right)\left(\frac{3\left(x-2\right)\left(x+1\right)+7x-5}{\left(x-2\right)\left(x+1\right)}-\frac{3x}{x+1}\right)+\left(4x+4\right)\times 5=9x^{2}+43x+8$$

3\left(x-2\right)\left(x+1\right)+7x-5 ichidagi ko‘paytirishlarni bajaring.

$$\left(x^{3}+2x^{2}-5x-6\right)\left(\frac{3x^{2}+3x-6x-6+7x-5}{\left(x-2\right)\left(x+1\right)}-\frac{3x}{x+1}\right)+\left(4x+4\right)\times 5=9x^{2}+43x+8$$

3x^{2}+3x-6x-6+7x-5 kabi iboralarga o‘xshab birlashtiring.

$$\left(x^{3}+2x^{2}-5x-6\right)\left(\frac{3x^{2}+4x-11}{\left(x-2\right)\left(x+1\right)}-\frac{3x}{x+1}\right)+\left(4x+4\right)\times 5=9x^{2}+43x+8$$

Ifodalarni qo‘shish yoki ayirish uchun ularni yoyib, maxrajlarini bir xil qiling. \left(x-2\right)\left(x+1\right) va x+1 ning eng kichik umumiy karralisi \left(x-2\right)\left(x+1\right). \frac{3x}{x+1} ni \frac{x-2}{x-2} marotabaga ko'paytirish.

$$\left(x^{3}+2x^{2}-5x-6\right)\left(\frac{3x^{2}+4x-11}{\left(x-2\right)\left(x+1\right)}-\frac{3x\left(x-2\right)}{\left(x-2\right)\left(x+1\right)}\right)+\left(4x+4\right)\times 5=9x^{2}+43x+8$$

\frac{3x^{2}+4x-11}{\left(x-2\right)\left(x+1\right)} va \frac{3x\left(x-2\right)}{\left(x-2\right)\left(x+1\right)} da bir xil maxraji bor, ularning suratini ayirish orqali ayiring.

$$\left(x^{3}+2x^{2}-5x-6\right)\times \frac{3x^{2}+4x-11-3x\left(x-2\right)}{\left(x-2\right)\left(x+1\right)}+\left(4x+4\right)\times 5=9x^{2}+43x+8$$

3x^{2}+4x-11-3x\left(x-2\right) ichidagi ko‘paytirishlarni bajaring.

$$\left(x^{3}+2x^{2}-5x-6\right)\times \frac{3x^{2}+4x-11-3x^{2}+6x}{\left(x-2\right)\left(x+1\right)}+\left(4x+4\right)\times 5=9x^{2}+43x+8$$

3x^{2}+4x-11-3x^{2}+6x kabi iboralarga o‘xshab birlashtiring.

$$\left(x^{3}+2x^{2}-5x-6\right)\times \frac{10x-11}{\left(x-2\right)\left(x+1\right)}+\left(4x+4\right)\times 5=9x^{2}+43x+8$$

\left(x^{3}+2x^{2}-5x-6\right)\times \frac{10x-11}{\left(x-2\right)\left(x+1\right)} ni yagona kasrga aylantiring.

$$\frac{\left(x^{3}+2x^{2}-5x-6\right)\left(10x-11\right)}{\left(x-2\right)\left(x+1\right)}+\left(4x+4\right)\times 5=9x^{2}+43x+8$$

4x+4 ga 5 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$\frac{\left(x^{3}+2x^{2}-5x-6\right)\left(10x-11\right)}{\left(x-2\right)\left(x+1\right)}+20x+20=9x^{2}+43x+8$$

Ifodalarni qo‘shish yoki ayirish uchun ularni yoyib, maxrajlarini bir xil qiling. 20x+20 ni \frac{\left(x-2\right)\left(x+1\right)}{\left(x-2\right)\left(x+1\right)} marotabaga ko'paytirish.

$$\frac{\left(x^{3}+2x^{2}-5x-6\right)\left(10x-11\right)}{\left(x-2\right)\left(x+1\right)}+\frac{\left(20x+20\right)\left(x-2\right)\left(x+1\right)}{\left(x-2\right)\left(x+1\right)}=9x^{2}+43x+8$$

\frac{\left(x^{3}+2x^{2}-5x-6\right)\left(10x-11\right)}{\left(x-2\right)\left(x+1\right)} va \frac{\left(20x+20\right)\left(x-2\right)\left(x+1\right)}{\left(x-2\right)\left(x+1\right)} da bir xil maxraji bor, ularning suratini qo‘shish orqali qo‘shing.

$$\frac{\left(x^{3}+2x^{2}-5x-6\right)\left(10x-11\right)+\left(20x+20\right)\left(x-2\right)\left(x+1\right)}{\left(x-2\right)\left(x+1\right)}=9x^{2}+43x+8$$

\left(x^{3}+2x^{2}-5x-6\right)\left(10x-11\right)+\left(20x+20\right)\left(x-2\right)\left(x+1\right) ichidagi ko‘paytirishlarni bajaring.

$$\frac{10x^{4}-11x^{3}+20x^{3}-22x^{2}-50x^{2}+55x-60x+66+20x^{3}-20x^{2}-40x+20x^{2}-20x-40}{\left(x-2\right)\left(x+1\right)}=9x^{2}+43x+8$$

10x^{4}-11x^{3}+20x^{3}-22x^{2}-50x^{2}+55x-60x+66+20x^{3}-20x^{2}-40x+20x^{2}-20x-40 kabi iboralarga o‘xshab birlashtiring.

$$\frac{10x^{4}+29x^{3}-72x^{2}-65x+26}{\left(x-2\right)\left(x+1\right)}=9x^{2}+43x+8$$

x-2 ga x+1 ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

$$\frac{10x^{4}+29x^{3}-72x^{2}-65x+26}{x^{2}-x-2}=9x^{2}+43x+8$$

Ikkala tarafdan 9x^{2} ni ayirish.

$$\frac{10x^{4}+29x^{3}-72x^{2}-65x+26}{x^{2}-x-2}-9x^{2}=43x+8$$

Faktor: x^{2}-x-2.

$$\frac{10x^{4}+29x^{3}-72x^{2}-65x+26}{\left(x-2\right)\left(x+1\right)}-9x^{2}=43x+8$$

Ifodalarni qo‘shish yoki ayirish uchun ularni yoyib, maxrajlarini bir xil qiling. -9x^{2} ni \frac{\left(x-2\right)\left(x+1\right)}{\left(x-2\right)\left(x+1\right)} marotabaga ko'paytirish.

$$\frac{10x^{4}+29x^{3}-72x^{2}-65x+26}{\left(x-2\right)\left(x+1\right)}+\frac{-9x^{2}\left(x-2\right)\left(x+1\right)}{\left(x-2\right)\left(x+1\right)}=43x+8$$

\frac{10x^{4}+29x^{3}-72x^{2}-65x+26}{\left(x-2\right)\left(x+1\right)} va \frac{-9x^{2}\left(x-2\right)\left(x+1\right)}{\left(x-2\right)\left(x+1\right)} da bir xil maxraji bor, ularning suratini qo‘shish orqali qo‘shing.

$$\frac{10x^{4}+29x^{3}-72x^{2}-65x+26-9x^{2}\left(x-2\right)\left(x+1\right)}{\left(x-2\right)\left(x+1\right)}=43x+8$$

10x^{4}+29x^{3}-72x^{2}-65x+26-9x^{2}\left(x-2\right)\left(x+1\right) ichidagi ko‘paytirishlarni bajaring.

$$\frac{10x^{4}+29x^{3}-72x^{2}-65x+26-9x^{4}-9x^{3}+18x^{3}+18x^{2}}{\left(x-2\right)\left(x+1\right)}=43x+8$$

10x^{4}+29x^{3}-72x^{2}-65x+26-9x^{4}-9x^{3}+18x^{3}+18x^{2} kabi iboralarga o‘xshab birlashtiring.

$$\frac{x^{4}+38x^{3}-54x^{2}-65x+26}{\left(x-2\right)\left(x+1\right)}=43x+8$$

Ikkala tarafdan 43x ni ayirish.

$$\frac{x^{4}+38x^{3}-54x^{2}-65x+26}{\left(x-2\right)\left(x+1\right)}-43x=8$$

x-2 ga x+1 ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

$$\frac{x^{4}+38x^{3}-54x^{2}-65x+26}{x^{2}-x-2}-43x=8$$

Faktor: x^{2}-x-2.

$$\frac{x^{4}+38x^{3}-54x^{2}-65x+26}{\left(x-2\right)\left(x+1\right)}-43x=8$$

Ifodalarni qo‘shish yoki ayirish uchun ularni yoyib, maxrajlarini bir xil qiling. -43x ni \frac{\left(x-2\right)\left(x+1\right)}{\left(x-2\right)\left(x+1\right)} marotabaga ko'paytirish.

$$\frac{x^{4}+38x^{3}-54x^{2}-65x+26}{\left(x-2\right)\left(x+1\right)}+\frac{-43x\left(x-2\right)\left(x+1\right)}{\left(x-2\right)\left(x+1\right)}=8$$

\frac{x^{4}+38x^{3}-54x^{2}-65x+26}{\left(x-2\right)\left(x+1\right)} va \frac{-43x\left(x-2\right)\left(x+1\right)}{\left(x-2\right)\left(x+1\right)} da bir xil maxraji bor, ularning suratini qo‘shish orqali qo‘shing.

$$\frac{x^{4}+38x^{3}-54x^{2}-65x+26-43x\left(x-2\right)\left(x+1\right)}{\left(x-2\right)\left(x+1\right)}=8$$

x^{4}+38x^{3}-54x^{2}-65x+26-43x\left(x-2\right)\left(x+1\right) ichidagi ko‘paytirishlarni bajaring.

$$\frac{x^{4}+38x^{3}-54x^{2}-65x+26-43x^{3}-43x^{2}+86x^{2}+86x}{\left(x-2\right)\left(x+1\right)}=8$$

x^{4}+38x^{3}-54x^{2}-65x+26-43x^{3}-43x^{2}+86x^{2}+86x kabi iboralarga o‘xshab birlashtiring.

$$\frac{x^{4}-5x^{3}-11x^{2}+21x+26}{\left(x-2\right)\left(x+1\right)}=8$$

Ikkala tarafdan 8 ni ayirish.

$$\frac{x^{4}-5x^{3}-11x^{2}+21x+26}{\left(x-2\right)\left(x+1\right)}-8=0$$

x-2 ga x+1 ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

$$\frac{x^{4}-5x^{3}-11x^{2}+21x+26}{x^{2}-x-2}-8=0$$

Faktor: x^{2}-x-2.

$$\frac{x^{4}-5x^{3}-11x^{2}+21x+26}{\left(x-2\right)\left(x+1\right)}-8=0$$

Ifodalarni qo‘shish yoki ayirish uchun ularni yoyib, maxrajlarini bir xil qiling. 8 ni \frac{\left(x-2\right)\left(x+1\right)}{\left(x-2\right)\left(x+1\right)} marotabaga ko'paytirish.

$$\frac{x^{4}-5x^{3}-11x^{2}+21x+26}{\left(x-2\right)\left(x+1\right)}-\frac{8\left(x-2\right)\left(x+1\right)}{\left(x-2\right)\left(x+1\right)}=0$$

\frac{x^{4}-5x^{3}-11x^{2}+21x+26}{\left(x-2\right)\left(x+1\right)} va \frac{8\left(x-2\right)\left(x+1\right)}{\left(x-2\right)\left(x+1\right)} da bir xil maxraji bor, ularning suratini ayirish orqali ayiring.

$$\frac{x^{4}-5x^{3}-11x^{2}+21x+26-8\left(x-2\right)\left(x+1\right)}{\left(x-2\right)\left(x+1\right)}=0$$

x^{4}-5x^{3}-11x^{2}+21x+26-8\left(x-2\right)\left(x+1\right) ichidagi ko‘paytirishlarni bajaring.

$$\frac{x^{4}-5x^{3}-11x^{2}+21x+26-8x^{2}-8x+16x+16}{\left(x-2\right)\left(x+1\right)}=0$$

x^{4}-5x^{3}-11x^{2}+21x+26-8x^{2}-8x+16x+16 kabi iboralarga o‘xshab birlashtiring.

$$\frac{x^{4}-5x^{3}-19x^{2}+29x+42}{\left(x-2\right)\left(x+1\right)}=0$$

x qiymati -1,2 qiymatlaridan birortasiga teng bo‘lmaydi, chunki nolga bo‘lish mumkin emas. Tenglamaning ikkala tarafini \left(x-2\right)\left(x+1\right) ga ko'paytirish.

$$x^{4}-5x^{3}-19x^{2}+29x+42=0$$

Ratsional ildiz teoremasiga koʻra, koʻphadlarning barcha ratsional ildizlari \frac{p}{q} shakli ichida, bu yerda p konstant shart 42 bilan boʻlinadi va q yetakchi koeffisientni 1 boʻladi. Barcha nomzodlarni oching \frac{p}{q}.

$$±42,±21,±14,±7,±6,±3,±2,±1$$

Eng kichigidan boshlab, mutlaq qiymatgacha butun son qiymatlarni sinab koʻrish orqali ana shunday bitta ildizni toping. Agar butun sonlar ildizlari topilmasa, kasrlarni sinab koʻring.

$$x=-1$$

Faktor teoremasiga koʻra, x-k har bir k ildizining faktoridir. x^{3}-6x^{2}-13x+42 ni olish uchun x^{4}-5x^{3}-19x^{2}+29x+42 ni x+1 ga bo‘ling. Natija 0 ga teng boʻlgandagi tenglamani yeching.

$$x^{3}-6x^{2}-13x+42=0$$

Ratsional ildiz teoremasiga koʻra, koʻphadlarning barcha ratsional ildizlari \frac{p}{q} shakli ichida, bu yerda p konstant shart 42 bilan boʻlinadi va q yetakchi koeffisientni 1 boʻladi. Barcha nomzodlarni oching \frac{p}{q}.

$$±42,±21,±14,±7,±6,±3,±2,±1$$

Eng kichigidan boshlab, mutlaq qiymatgacha butun son qiymatlarni sinab koʻrish orqali ana shunday bitta ildizni toping. Agar butun sonlar ildizlari topilmasa, kasrlarni sinab koʻring.

$$x=2$$

Faktor teoremasiga koʻra, x-k har bir k ildizining faktoridir. x^{2}-4x-21 ni olish uchun x^{3}-6x^{2}-13x+42 ni x-2 ga bo‘ling. Natija 0 ga teng boʻlgandagi tenglamani yeching.

$$x^{2}-4x-21=0$$

ax^{2}+bx+c=0 shaklidagi barcha tenglamalarni bu formula bilan yechish mumkin: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. Kvadrat tenglamada a uchun 1 ni, b uchun -4 ni va c uchun -21 ni ayiring.

$$x=\frac{-\left(-4\right)±\sqrt{\left(-4\right)^{2}-4\times 1\left(-21\right)}}{2}$$

Hisoblarni amalga oshiring.

$$x=\frac{4±10}{2}$$

x^{2}-4x-21=0 tenglamasini ± plus va ± minus boʻlgan holatida ishlang.

$$x=-3 x=7$$

Oʻzgaruvchi teng boʻlmagan qiymatlarni olib tashlang.

$$x=7$$

Barcha topilgan yechimlar roʻyxati.

$$x=-1 x=2 x=-3 x=7$$

x qiymati -1,2,-3 qiymatlaridan birortasiga teng bo‘lmaydi.

$$x=7$$