1/x^2+x-2+1/x^2+7x+10+1/x^2+13x+40=1/3x-3-frac{1}{21} eching

$$\frac { 1 } { x ^ { 2 } + x - 2 } + \frac { 1 } { x ^ { 2 } + 7 x + 10 } + \frac { 1 } { x ^ { 2 } + 13 x + 40 } = \frac { 1 } { 3 x - 3 } - \frac { 1 } { 21 }$$

x uchun yechish

$$x=-1$$

Yechim qadamlari

x qiymati -8,-5,-2,1 qiymatlaridan birortasiga teng bo‘lmaydi, chunki nolga bo‘lish mumkin emas. Tenglamaning ikkala tarafini 21\left(x-1\right)\left(x+2\right)\left(x+5\right)\left(x+8\right) ga, x^{2}+x-2,x^{2}+7x+10,x^{2}+13x+40,3x-3,21 ning eng kichik karralisiga ko‘paytiring.

$$21\left(x+5\right)\left(x+8\right)+21\left(x-1\right)\left(x+8\right)+21\left(x+2\right)\left(x-1\right)=7\left(x+2\right)\left(x+5\right)\left(x+8\right)+21\left(x-1\right)\left(x+2\right)\left(x+5\right)\left(x+8\right)\left(-\frac{1}{21}\right)$$

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

$$\left(21x+105\right)\left(x+8\right)+21\left(x-1\right)\left(x+8\right)+21\left(x+2\right)\left(x-1\right)=7\left(x+2\right)\left(x+5\right)\left(x+8\right)+21\left(x-1\right)\left(x+2\right)\left(x+5\right)\left(x+8\right)\left(-\frac{1}{21}\right)$$

21x+105 ga x+8 ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

$$21x^{2}+273x+840+21\left(x-1\right)\left(x+8\right)+21\left(x+2\right)\left(x-1\right)=7\left(x+2\right)\left(x+5\right)\left(x+8\right)+21\left(x-1\right)\left(x+2\right)\left(x+5\right)\left(x+8\right)\left(-\frac{1}{21}\right)$$

21 ga x-1 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$21x^{2}+273x+840+\left(21x-21\right)\left(x+8\right)+21\left(x+2\right)\left(x-1\right)=7\left(x+2\right)\left(x+5\right)\left(x+8\right)+21\left(x-1\right)\left(x+2\right)\left(x+5\right)\left(x+8\right)\left(-\frac{1}{21}\right)$$

21x-21 ga x+8 ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

$$21x^{2}+273x+840+21x^{2}+147x-168+21\left(x+2\right)\left(x-1\right)=7\left(x+2\right)\left(x+5\right)\left(x+8\right)+21\left(x-1\right)\left(x+2\right)\left(x+5\right)\left(x+8\right)\left(-\frac{1}{21}\right)$$

42x^{2} ni olish uchun 21x^{2} va 21x^{2} ni birlashtirish.

$$42x^{2}+273x+840+147x-168+21\left(x+2\right)\left(x-1\right)=7\left(x+2\right)\left(x+5\right)\left(x+8\right)+21\left(x-1\right)\left(x+2\right)\left(x+5\right)\left(x+8\right)\left(-\frac{1}{21}\right)$$

420x ni olish uchun 273x va 147x ni birlashtirish.

$$42x^{2}+420x+840-168+21\left(x+2\right)\left(x-1\right)=7\left(x+2\right)\left(x+5\right)\left(x+8\right)+21\left(x-1\right)\left(x+2\right)\left(x+5\right)\left(x+8\right)\left(-\frac{1}{21}\right)$$

672 olish uchun 840 dan 168 ni ayirish.

$$42x^{2}+420x+672+21\left(x+2\right)\left(x-1\right)=7\left(x+2\right)\left(x+5\right)\left(x+8\right)+21\left(x-1\right)\left(x+2\right)\left(x+5\right)\left(x+8\right)\left(-\frac{1}{21}\right)$$

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

$$42x^{2}+420x+672+\left(21x+42\right)\left(x-1\right)=7\left(x+2\right)\left(x+5\right)\left(x+8\right)+21\left(x-1\right)\left(x+2\right)\left(x+5\right)\left(x+8\right)\left(-\frac{1}{21}\right)$$

21x+42 ga x-1 ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

$$42x^{2}+420x+672+21x^{2}+21x-42=7\left(x+2\right)\left(x+5\right)\left(x+8\right)+21\left(x-1\right)\left(x+2\right)\left(x+5\right)\left(x+8\right)\left(-\frac{1}{21}\right)$$

63x^{2} ni olish uchun 42x^{2} va 21x^{2} ni birlashtirish.

$$63x^{2}+420x+672+21x-42=7\left(x+2\right)\left(x+5\right)\left(x+8\right)+21\left(x-1\right)\left(x+2\right)\left(x+5\right)\left(x+8\right)\left(-\frac{1}{21}\right)$$

441x ni olish uchun 420x va 21x ni birlashtirish.

$$63x^{2}+441x+672-42=7\left(x+2\right)\left(x+5\right)\left(x+8\right)+21\left(x-1\right)\left(x+2\right)\left(x+5\right)\left(x+8\right)\left(-\frac{1}{21}\right)$$

630 olish uchun 672 dan 42 ni ayirish.

$$63x^{2}+441x+630=7\left(x+2\right)\left(x+5\right)\left(x+8\right)+21\left(x-1\right)\left(x+2\right)\left(x+5\right)\left(x+8\right)\left(-\frac{1}{21}\right)$$

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

$$63x^{2}+441x+630=\left(7x+14\right)\left(x+5\right)\left(x+8\right)+21\left(x-1\right)\left(x+2\right)\left(x+5\right)\left(x+8\right)\left(-\frac{1}{21}\right)$$

7x+14 ga x+5 ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

$$63x^{2}+441x+630=\left(7x^{2}+49x+70\right)\left(x+8\right)+21\left(x-1\right)\left(x+2\right)\left(x+5\right)\left(x+8\right)\left(-\frac{1}{21}\right)$$

7x^{2}+49x+70 ga x+8 ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

$$63x^{2}+441x+630=7x^{3}+105x^{2}+462x+560+21\left(x-1\right)\left(x+2\right)\left(x+5\right)\left(x+8\right)\left(-\frac{1}{21}\right)$$

-1 hosil qilish uchun 21 va -\frac{1}{21} ni ko'paytirish.

$$63x^{2}+441x+630=7x^{3}+105x^{2}+462x+560-\left(x-1\right)\left(x+2\right)\left(x+5\right)\left(x+8\right)$$

-1 ga x-1 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$63x^{2}+441x+630=7x^{3}+105x^{2}+462x+560+\left(-x+1\right)\left(x+2\right)\left(x+5\right)\left(x+8\right)$$

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

$$63x^{2}+441x+630=7x^{3}+105x^{2}+462x+560+\left(-x^{2}-x+2\right)\left(x+5\right)\left(x+8\right)$$

-x^{2}-x+2 ga x+5 ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

$$63x^{2}+441x+630=7x^{3}+105x^{2}+462x+560+\left(-x^{3}-6x^{2}-3x+10\right)\left(x+8\right)$$

-x^{3}-6x^{2}-3x+10 ga x+8 ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

$$63x^{2}+441x+630=7x^{3}+105x^{2}+462x+560-x^{4}-14x^{3}-51x^{2}-14x+80$$

-7x^{3} ni olish uchun 7x^{3} va -14x^{3} ni birlashtirish.

$$63x^{2}+441x+630=-7x^{3}+105x^{2}+462x+560-x^{4}-51x^{2}-14x+80$$

54x^{2} ni olish uchun 105x^{2} va -51x^{2} ni birlashtirish.

$$63x^{2}+441x+630=-7x^{3}+54x^{2}+462x+560-x^{4}-14x+80$$

448x ni olish uchun 462x va -14x ni birlashtirish.

$$63x^{2}+441x+630=-7x^{3}+54x^{2}+448x+560-x^{4}+80$$

640 olish uchun 560 va 80'ni qo'shing.

$$63x^{2}+441x+630=-7x^{3}+54x^{2}+448x+640-x^{4}$$

7x^{3} ni ikki tarafga qo’shing.

$$63x^{2}+441x+630+7x^{3}=54x^{2}+448x+640-x^{4}$$

Ikkala tarafdan 54x^{2} ni ayirish.

$$63x^{2}+441x+630+7x^{3}-54x^{2}=448x+640-x^{4}$$

9x^{2} ni olish uchun 63x^{2} va -54x^{2} ni birlashtirish.

$$9x^{2}+441x+630+7x^{3}=448x+640-x^{4}$$

Ikkala tarafdan 448x ni ayirish.

$$9x^{2}+441x+630+7x^{3}-448x=640-x^{4}$$

-7x ni olish uchun 441x va -448x ni birlashtirish.

$$9x^{2}-7x+630+7x^{3}=640-x^{4}$$

Ikkala tarafdan 640 ni ayirish.

$$9x^{2}-7x+630+7x^{3}-640=-x^{4}$$

-10 olish uchun 630 dan 640 ni ayirish.

$$9x^{2}-7x-10+7x^{3}=-x^{4}$$

x^{4} ni ikki tarafga qo’shing.

$$9x^{2}-7x-10+7x^{3}+x^{4}=0$$

Tenglamani standart shaklga keltirish uchun uni qayta tartiblash. Shartlarni eng yuqoridan eng pastki qiymat ko'rsatgichiga joylashtirish.

$$x^{4}+7x^{3}+9x^{2}-7x-10=0$$

Ratsional ildiz teoremasiga koʻra, koʻphadlarning barcha ratsional ildizlari \frac{p}{q} shakli ichida, bu yerda p konstant shart -10 bilan boʻlinadi va q yetakchi koeffisientni 1 boʻladi. Barcha nomzodlarni oching \frac{p}{q}.

$$±10,±5,±2,±1$$

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}+8x^{2}+17x+10 ni olish uchun x^{4}+7x^{3}+9x^{2}-7x-10 ni x-1 ga bo‘ling. Natija 0 ga teng boʻlgandagi tenglamani yeching.

$$x^{3}+8x^{2}+17x+10=0$$

Ratsional ildiz teoremasiga koʻra, koʻphadlarning barcha ratsional ildizlari \frac{p}{q} shakli ichida, bu yerda p konstant shart 10 bilan boʻlinadi va q yetakchi koeffisientni 1 boʻladi. Barcha nomzodlarni oching \frac{p}{q}.

$$±10,±5,±2,±1$$

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^{2}+7x+10 ni olish uchun x^{3}+8x^{2}+17x+10 ni x+1 ga bo‘ling. Natija 0 ga teng boʻlgandagi tenglamani yeching.

$$x^{2}+7x+10=0$$

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 7 ni va c uchun 10 ni ayiring.

$$x=\frac{-7±\sqrt{7^{2}-4\times 1\times 10}}{2}$$

Hisoblarni amalga oshiring.

$$x=\frac{-7±3}{2}$$

x^{2}+7x+10=0 tenglamasini ± plus va ± minus boʻlgan holatida ishlang.

$$x=-5 x=-2$$

Oʻzgaruvchi teng boʻlmagan qiymatlarni olib tashlang.

$$x=-1$$

Barcha topilgan yechimlar roʻyxati.

$$x=1 x=-1 x=-5 x=-2$$

x qiymati 1,-5,-2 qiymatlaridan birortasiga teng bo‘lmaydi.

$$x=-1$$