left(x-3right)^2left(10-17xright)^2=0 eching

$${ \left(x-3 \right) }^{ 2 } { \left(10-17x \right) }^{ 2 } = 0$$

x uchun yechish

$$x=\frac{10}{17}\approx 0,588235294$$

$$x=3$$

Yechim qadamlari

\left(a-b\right)^{2}=a^{2}-2ab+b^{2} binom teoremasini \left(x-3\right)^{2} kengaytirilishi uchun ishlating.

$$\left(x^{2}-6x+9\right)\left(10-17x\right)^{2}=0$$

\left(a-b\right)^{2}=a^{2}-2ab+b^{2} binom teoremasini \left(10-17x\right)^{2} kengaytirilishi uchun ishlating.

$$\left(x^{2}-6x+9\right)\left(100-340x+289x^{2}\right)=0$$

x^{2}-6x+9 ga 100-340x+289x^{2} ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

$$4741x^{2}-2074x^{3}+289x^{4}-3660x+900=0$$

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

$$289x^{4}-2074x^{3}+4741x^{2}-3660x+900=0$$

Ratsional ildiz teoremasiga koʻra, koʻphadlarning barcha ratsional ildizlari \frac{p}{q} shakli ichida, bu yerda p konstant shart 900 bilan boʻlinadi va q yetakchi koeffisientni 289 boʻladi. Barcha nomzodlarni oching \frac{p}{q}.

$$±\frac{900}{289},±\frac{900}{17},±900,±\frac{450}{289},±\frac{450}{17},±450,±\frac{300}{289},±\frac{300}{17},±300,±\frac{225}{289},±\frac{225}{17},±225,±\frac{180}{289},±\frac{180}{17},±180,±\frac{150}{289},±\frac{150}{17},±150,±\frac{100}{289},±\frac{100}{17},±100,±\frac{90}{289},±\frac{90}{17},±90,±\frac{75}{289},±\frac{75}{17},±75,±\frac{60}{289},±\frac{60}{17},±60,±\frac{50}{289},±\frac{50}{17},±50,±\frac{45}{289},±\frac{45}{17},±45,±\frac{36}{289},±\frac{36}{17},±36,±\frac{30}{289},±\frac{30}{17},±30,±\frac{25}{289},±\frac{25}{17},±25,±\frac{20}{289},±\frac{20}{17},±20,±\frac{18}{289},±\frac{18}{17},±18,±\frac{15}{289},±\frac{15}{17},±15,±\frac{12}{289},±\frac{12}{17},±12,±\frac{10}{289},±\frac{10}{17},±10,±\frac{9}{289},±\frac{9}{17},±9,±\frac{6}{289},±\frac{6}{17},±6,±\frac{5}{289},±\frac{5}{17},±5,±\frac{4}{289},±\frac{4}{17},±4,±\frac{3}{289},±\frac{3}{17},±3,±\frac{2}{289},±\frac{2}{17},±2,±\frac{1}{289},±\frac{1}{17},±1$$

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

$$x=3$$

Faktor teoremasiga koʻra, x-k har bir k ildizining faktoridir. 289x^{3}-1207x^{2}+1120x-300 ni olish uchun 289x^{4}-2074x^{3}+4741x^{2}-3660x+900 ni x-3 ga bo‘ling. Natija 0 ga teng boʻlgandagi tenglamani yeching.

$$289x^{3}-1207x^{2}+1120x-300=0$$

Ratsional ildiz teoremasiga koʻra, koʻphadlarning barcha ratsional ildizlari \frac{p}{q} shakli ichida, bu yerda p konstant shart -300 bilan boʻlinadi va q yetakchi koeffisientni 289 boʻladi. Barcha nomzodlarni oching \frac{p}{q}.

$$±\frac{300}{289},±\frac{300}{17},±300,±\frac{150}{289},±\frac{150}{17},±150,±\frac{100}{289},±\frac{100}{17},±100,±\frac{75}{289},±\frac{75}{17},±75,±\frac{60}{289},±\frac{60}{17},±60,±\frac{50}{289},±\frac{50}{17},±50,±\frac{30}{289},±\frac{30}{17},±30,±\frac{25}{289},±\frac{25}{17},±25,±\frac{20}{289},±\frac{20}{17},±20,±\frac{15}{289},±\frac{15}{17},±15,±\frac{12}{289},±\frac{12}{17},±12,±\frac{10}{289},±\frac{10}{17},±10,±\frac{6}{289},±\frac{6}{17},±6,±\frac{5}{289},±\frac{5}{17},±5,±\frac{4}{289},±\frac{4}{17},±4,±\frac{3}{289},±\frac{3}{17},±3,±\frac{2}{289},±\frac{2}{17},±2,±\frac{1}{289},±\frac{1}{17},±1$$

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

$$x=3$$

Faktor teoremasiga koʻra, x-k har bir k ildizining faktoridir. 289x^{2}-340x+100 ni olish uchun 289x^{3}-1207x^{2}+1120x-300 ni x-3 ga bo‘ling. Natija 0 ga teng boʻlgandagi tenglamani yeching.

$$289x^{2}-340x+100=0$$

ax^{2}+bx+c=0 shaklidagi barcha tenglamalarni bu formula bilan yechish mumkin: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. Kvadrat tenglamada a uchun 289 ni, b uchun -340 ni va c uchun 100 ni ayiring.

$$x=\frac{-\left(-340\right)±\sqrt{\left(-340\right)^{2}-4\times 289\times 100}}{2\times 289}$$

Hisoblarni amalga oshiring.

$$x=\frac{340±0}{578}$$

Yechimlar bir xil.

$$x=\frac{10}{17}$$

Barcha topilgan yechimlar roʻyxati.

$$x=3 x=\frac{10}{17}$$