(2x+4)^2-5x(7-3x)(7+3x)-(3x-2)^2-40x^2=-205 eching

$$( 2 x + 4 ) ^ { 2 } - 5 x ( 7 - 3 x ) ( 7 + 3 x ) - ( 3 x - 2 ) ^ { 2 } - 40 x ^ { 2 } = - 205$$

x uchun yechish

$$x = \frac{\sqrt{1085}}{15} \approx 2,195955879$$

$$x = -\frac{\sqrt{1085}}{15} \approx -2,195955879$$

$$x=1$$

Yechim qadamlari

\left(a+b\right)^{2}=a^{2}+2ab+b^{2} binom teoremasini \left(2x+4\right)^{2} kengaytirilishi uchun ishlating.

$$4x^{2}+16x+16-5x\left(7-3x\right)\left(7+3x\right)-\left(3x-2\right)^{2}-40x^{2}=-205$$

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

$$4x^{2}+16x+16-5x\left(7-3x\right)\left(7+3x\right)-\left(9x^{2}-12x+4\right)-40x^{2}=-205$$

9x^{2}-12x+4 teskarisini topish uchun har birining teskarisini toping.

$$4x^{2}+16x+16-5x\left(7-3x\right)\left(7+3x\right)-9x^{2}+12x-4-40x^{2}=-205$$

-49x^{2} ni olish uchun -9x^{2} va -40x^{2} ni birlashtirish.

$$4x^{2}+16x+16-5x\left(7-3x\right)\left(7+3x\right)-49x^{2}+12x-4=-205$$

205 ni ikki tarafga qo’shing.

$$4x^{2}+16x+16-5x\left(7-3x\right)\left(7+3x\right)-49x^{2}+12x-4+205=0$$

201 olish uchun -4 va 205'ni qo'shing.

$$4x^{2}+16x+16-5x\left(7-3x\right)\left(7+3x\right)-49x^{2}+12x+201=0$$

-5x ga 7-3x ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$4x^{2}+16x+16+\left(-35x+15x^{2}\right)\left(7+3x\right)-49x^{2}+12x+201=0$$

-35x+15x^{2} ga 7+3x ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

$$4x^{2}+16x+16-245x+45x^{3}-49x^{2}+12x+201=0$$

-229x ni olish uchun 16x va -245x ni birlashtirish.

$$4x^{2}-229x+16+45x^{3}-49x^{2}+12x+201=0$$

-45x^{2} ni olish uchun 4x^{2} va -49x^{2} ni birlashtirish.

$$-45x^{2}-229x+16+45x^{3}+12x+201=0$$

-217x ni olish uchun -229x va 12x ni birlashtirish.

$$-45x^{2}-217x+16+45x^{3}+201=0$$

217 olish uchun 16 va 201'ni qo'shing.

$$-45x^{2}-217x+217+45x^{3}=0$$

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

$$45x^{3}-45x^{2}-217x+217=0$$

Ratsional ildiz teoremasiga koʻra, koʻphadlarning barcha ratsional ildizlari \frac{p}{q} shakli ichida, bu yerda p konstant shart 217 bilan boʻlinadi va q yetakchi koeffisientni 45 boʻladi. Barcha nomzodlarni oching \frac{p}{q}.

$$±\frac{217}{45},±\frac{217}{15},±\frac{217}{9},±\frac{217}{5},±\frac{217}{3},±217,±\frac{31}{45},±\frac{31}{15},±\frac{31}{9},±\frac{31}{5},±\frac{31}{3},±31,±\frac{7}{45},±\frac{7}{15},±\frac{7}{9},±\frac{7}{5},±\frac{7}{3},±7,±\frac{1}{45},±\frac{1}{15},±\frac{1}{9},±\frac{1}{5},±\frac{1}{3},±1$$

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. 45x^{2}-217 ni olish uchun 45x^{3}-45x^{2}-217x+217 ni x-1 ga bo‘ling. Natija 0 ga teng boʻlgandagi tenglamani yeching.

$$45x^{2}-217=0$$

ax^{2}+bx+c=0 shaklidagi barcha tenglamalarni bu formula bilan yechish mumkin: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. Kvadrat tenglamada a uchun 45 ni, b uchun 0 ni va c uchun -217 ni ayiring.

$$x=\frac{0±\sqrt{0^{2}-4\times 45\left(-217\right)}}{2\times 45}$$

Hisoblarni amalga oshiring.

$$x=\frac{0±6\sqrt{1085}}{90}$$

45x^{2}-217=0 tenglamasini ± plus va ± minus boʻlgan holatida ishlang.

$$x=-\frac{\sqrt{1085}}{15} x=\frac{\sqrt{1085}}{15}$$

Barcha topilgan yechimlar roʻyxati.

$$x=1 x=-\frac{\sqrt{1085}}{15} x=\frac{\sqrt{1085}}{15}$$