2x_{0}=frac{x_{0}+1}{x_{0}-1} eching

$$2 x _ { 0 } = \frac { x _ { 0 } + 1 } { x _ { 0 } - 1 }$$

x_0 uchun yechish

$$x_{0} = \frac{\sqrt{17} + 3}{4} \approx 1,780776406$$

$$x_{0}=\frac{3-\sqrt{17}}{4}\approx -0,280776406$$

Kvadrat tenglamalaridan foydalanish qoidalari

x_{0} qiymati 1 teng bo‘lmaydi, chunki nolga bo‘lish mumkin emas. Tenglamaning ikkala tarafini x_{0}-1 ga ko'paytirish.

$$2x_{0}\left(x_{0}-1\right)=x_{0}+1$$

2x_{0} ga x_{0}-1 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$2x_{0}^{2}-2x_{0}=x_{0}+1$$

Ikkala tarafdan x_{0} ni ayirish.

$$2x_{0}^{2}-2x_{0}-x_{0}=1$$

-3x_{0} ni olish uchun -2x_{0} va -x_{0} ni birlashtirish.

$$2x_{0}^{2}-3x_{0}=1$$

Ikkala tarafdan 1 ni ayirish.

$$2x_{0}^{2}-3x_{0}-1=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 2 ni a, -3 ni b va -1 ni c bilan almashtiring.

$$x_{0}=\frac{-\left(-3\right)±\sqrt{\left(-3\right)^{2}-4\times 2\left(-1\right)}}{2\times 2}$$

-3 kvadratini chiqarish.

$$x_{0}=\frac{-\left(-3\right)±\sqrt{9-4\times 2\left(-1\right)}}{2\times 2}$$

-4 ni 2 marotabaga ko'paytirish.

$$x_{0}=\frac{-\left(-3\right)±\sqrt{9-8\left(-1\right)}}{2\times 2}$$

-8 ni -1 marotabaga ko'paytirish.

$$x_{0}=\frac{-\left(-3\right)±\sqrt{9+8}}{2\times 2}$$

9 ni 8 ga qo'shish.

$$x_{0}=\frac{-\left(-3\right)±\sqrt{17}}{2\times 2}$$

-3 ning teskarisi 3 ga teng.

$$x_{0}=\frac{3±\sqrt{17}}{2\times 2}$$

2 ni 2 marotabaga ko'paytirish.

$$x_{0}=\frac{3±\sqrt{17}}{4}$$

x_{0}=\frac{3±\sqrt{17}}{4} tenglamasini yeching, bunda ± musbat. 3 ni \sqrt{17} ga qo'shish.

$$x_{0}=\frac{\sqrt{17}+3}{4}$$

x_{0}=\frac{3±\sqrt{17}}{4} tenglamasini yeching, bunda ± manfiy. 3 dan \sqrt{17} ni ayirish.

$$x_{0}=\frac{3-\sqrt{17}}{4}$$

Tenglama yechildi.

$$x_{0}=\frac{\sqrt{17}+3}{4} x_{0}=\frac{3-\sqrt{17}}{4}$$

Kvadratni yakunlash bo'yicha qadamlar

x_{0} qiymati 1 teng bo‘lmaydi, chunki nolga bo‘lish mumkin emas. Tenglamaning ikkala tarafini x_{0}-1 ga ko'paytirish.

$$2x_{0}\left(x_{0}-1\right)=x_{0}+1$$

2x_{0} ga x_{0}-1 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$2x_{0}^{2}-2x_{0}=x_{0}+1$$

Ikkala tarafdan x_{0} ni ayirish.

$$2x_{0}^{2}-2x_{0}-x_{0}=1$$

-3x_{0} ni olish uchun -2x_{0} va -x_{0} ni birlashtirish.

$$2x_{0}^{2}-3x_{0}=1$$

Ikki tarafini 2 ga bo‘ling.

$$\frac{2x_{0}^{2}-3x_{0}}{2}=\frac{1}{2}$$

2 ga bo'lish 2 ga ko'paytirishni bekor qiladi.

$$x_{0}^{2}-\frac{3}{2}x_{0}=\frac{1}{2}$$

-\frac{3}{2} ni bo‘lish, x shartining koeffitsienti, 2 ga -\frac{3}{4} olish uchun. Keyin, -\frac{3}{4} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x_{0}^{2}-\frac{3}{2}x_{0}+\left(-\frac{3}{4}\right)^{2}=\frac{1}{2}+\left(-\frac{3}{4}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib -\frac{3}{4} kvadratini chiqarish.

$$x_{0}^{2}-\frac{3}{2}x_{0}+\frac{9}{16}=\frac{1}{2}+\frac{9}{16}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali \frac{1}{2} ni \frac{9}{16} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$x_{0}^{2}-\frac{3}{2}x_{0}+\frac{9}{16}=\frac{17}{16}$$

x_{0}^{2}-\frac{3}{2}x_{0}+\frac{9}{16} omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(x_{0}-\frac{3}{4}\right)^{2}=\frac{17}{16}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x_{0}-\frac{3}{4}\right)^{2}}=\sqrt{\frac{17}{16}}$$

Qisqartirish.

$$x_{0}-\frac{3}{4}=\frac{\sqrt{17}}{4} x_{0}-\frac{3}{4}=-\frac{\sqrt{17}}{4}$$

\frac{3}{4} ni tenglamaning ikkala tarafiga qo'shish.

$$x_{0}=\frac{\sqrt{17}+3}{4} x_{0}=\frac{3-\sqrt{17}}{4}$$