2(x^2-1)^2-7(x^2-1)+6=0 eching

$$2 ( x ^ { 2 } - 1 ) ^ { 2 } - 7 ( x ^ { 2 } - 1 ) + 6 = 0$$

x uchun yechish

$$x = -\frac{\sqrt{10}}{2} \approx -1,58113883$$

$$x = \frac{\sqrt{10}}{2} \approx 1,58113883$$

$$x=\sqrt{3}\approx 1,732050808$$

$$x=-\sqrt{3}\approx -1,732050808$$

Kvadrat tenglamalaridan foydalanish qoidalari

\left(a-b\right)^{2}=a^{2}-2ab+b^{2} binom teoremasini \left(x^{2}-1\right)^{2} kengaytirilishi uchun ishlating.

$$2\left(\left(x^{2}\right)^{2}-2x^{2}+1\right)-7\left(x^{2}-1\right)+6=0$$

Daraja ko‘rsatkichini boshqa ko‘rsatkichga oshirish uchun, darajalarini ko‘paytiring. 2 va 2 ni ko‘paytirib, 4 ni oling.

$$2\left(x^{4}-2x^{2}+1\right)-7\left(x^{2}-1\right)+6=0$$

2 ga x^{4}-2x^{2}+1 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$2x^{4}-4x^{2}+2-7\left(x^{2}-1\right)+6=0$$

-7 ga x^{2}-1 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$2x^{4}-4x^{2}+2-7x^{2}+7+6=0$$

-11x^{2} ni olish uchun -4x^{2} va -7x^{2} ni birlashtirish.

$$2x^{4}-11x^{2}+2+7+6=0$$

9 olish uchun 2 va 7'ni qo'shing.

$$2x^{4}-11x^{2}+9+6=0$$

15 olish uchun 9 va 6'ni qo'shing.

$$2x^{4}-11x^{2}+15=0$$

x^{2} uchun t ni almashtiring.

$$2t^{2}-11t+15=0$$

ax^{2}+bx+c=0 shaklidagi barcha tenglamalarni bu formula bilan yechish mumkin: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. Kvadrat tenglamada a uchun 2 ni, b uchun -11 ni va c uchun 15 ni ayiring.

$$t=\frac{-\left(-11\right)±\sqrt{\left(-11\right)^{2}-4\times 2\times 15}}{2\times 2}$$

Hisoblarni amalga oshiring.

$$t=\frac{11±1}{4}$$

t=\frac{11±1}{4} tenglamasini ± plus va ± minus boʻlgan holatida ishlang.

$$t=3 t=\frac{5}{2}$$

x=t^{2} boʻlganda, yechimlar har bir t uchun x=±\sqrt{t} hisoblanishi orqali olinadi.

$$x=\sqrt{3} x=-\sqrt{3} x=\frac{\sqrt{10}}{2} x=-\frac{\sqrt{10}}{2}$$