4(1+1/x)=x^2-1 eching

$$4 ( 1 + \frac { 1 } { x } ) = x ^ { 2 } - 1$$

x uchun yechish

$$x=\frac{1-\sqrt{17}}{2}\approx -1,561552813$$

$$x=-1$$

$$x = \frac{\sqrt{17} + 1}{2} \approx 2,561552813$$

Kvadrat tenglamalaridan foydalanish qoidalari

x qiymati 0 teng bo‘lmaydi, chunki nolga bo‘lish mumkin emas. Tenglamaning ikkala tarafini x ga ko'paytirish.

$$4\left(1+\frac{1}{x}\right)x=xx^{2}+x\left(-1\right)$$

Ayni asosning daraja ko‘rsatkichlarini ko‘paytirish uchun ularning darajalarini qo‘shing. 1 va 2 ni qo‘shib, 3 ni oling.

$$4\left(1+\frac{1}{x}\right)x=x^{3}+x\left(-1\right)$$

Ifodalarni qo‘shish yoki ayirish uchun ularni yoyib, maxrajlarini bir xil qiling. 1 ni \frac{x}{x} marotabaga ko'paytirish.

$$4\left(\frac{x}{x}+\frac{1}{x}\right)x=x^{3}+x\left(-1\right)$$

\frac{x}{x} va \frac{1}{x} da bir xil maxraji bor, ularning suratini qo‘shish orqali qo‘shing.

$$4\times \frac{x+1}{x}x=x^{3}+x\left(-1\right)$$

4\times \frac{x+1}{x} ni yagona kasrga aylantiring.

$$\frac{4\left(x+1\right)}{x}x=x^{3}+x\left(-1\right)$$

\frac{4\left(x+1\right)}{x}x ni yagona kasrga aylantiring.

$$\frac{4\left(x+1\right)x}{x}=x^{3}+x\left(-1\right)$$

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

$$\frac{\left(4x+4\right)x}{x}=x^{3}+x\left(-1\right)$$

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

$$\frac{4x^{2}+4x}{x}=x^{3}+x\left(-1\right)$$

Ikkala tarafdan x^{3} ni ayirish.

$$\frac{4x^{2}+4x}{x}-x^{3}=x\left(-1\right)$$

Ifodalarni qo‘shish yoki ayirish uchun ularni yoyib, maxrajlarini bir xil qiling. x^{3} ni \frac{x}{x} marotabaga ko'paytirish.

$$\frac{4x^{2}+4x}{x}-\frac{x^{3}x}{x}=x\left(-1\right)$$

\frac{4x^{2}+4x}{x} va \frac{x^{3}x}{x} da bir xil maxraji bor, ularning suratini ayirish orqali ayiring.

$$\frac{4x^{2}+4x-x^{3}x}{x}=x\left(-1\right)$$

4x^{2}+4x-x^{3}x ichidagi ko‘paytirishlarni bajaring.

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

Ikkala tarafdan x\left(-1\right) ni ayirish.

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

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

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

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

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

4x^{2}+4x-x^{4}-x\left(-1\right)x ichidagi ko‘paytirishlarni bajaring.

$$\frac{4x^{2}+4x-x^{4}+x^{2}}{x}=0$$

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

$$\frac{5x^{2}+4x-x^{4}}{x}=0$$

x qiymati 0 teng bo‘lmaydi, chunki nolga bo‘lish mumkin emas. Tenglamaning ikkala tarafini x ga ko'paytirish.

$$5x^{2}+4x-x^{4}=0$$

x^{2} uchun t ni almashtiring.

$$-t^{2}+5t+4=0$$

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 5 ni va c uchun 4 ni ayiring.

$$t=\frac{-5±\sqrt{5^{2}-4\left(-1\right)\times 4}}{-2}$$

Hisoblarni amalga oshiring.

$$t=\frac{-5±\sqrt{41}}{-2}$$

t=\frac{-5±\sqrt{41}}{-2} tenglamasini ± plus va ± minus boʻlgan holatida ishlang.

$$t=\frac{5-\sqrt{41}}{2} t=\frac{\sqrt{41}+5}{2}$$

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

$$x=\frac{\sqrt{2\sqrt{41}+10}}{2} x=-\frac{\sqrt{2\sqrt{41}+10}}{2}$$