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

$$9 ( x ) = \frac { x ^ { 2 } + x + 1 } { x - 2 }$$

x uchun yechish

$$x = \frac{\sqrt{393} + 19}{16} \approx 2,426514225$$

$$x=\frac{19-\sqrt{393}}{16}\approx -0,051514225$$

Kvadrat tenglamalaridan foydalanish qoidalari

x qiymati 2 teng bo‘lmaydi, chunki nolga bo‘lish mumkin emas. Tenglamaning ikkala tarafini x-2 ga ko'paytirish.

$$9x\left(x-2\right)=x^{2}+x+1$$

9x ga x-2 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$9x^{2}-18x=x^{2}+x+1$$

Ikkala tarafdan x^{2} ni ayirish.

$$9x^{2}-18x-x^{2}=x+1$$

8x^{2} ni olish uchun 9x^{2} va -x^{2} ni birlashtirish.

$$8x^{2}-18x=x+1$$

Ikkala tarafdan x ni ayirish.

$$8x^{2}-18x-x=1$$

-19x ni olish uchun -18x va -x ni birlashtirish.

$$8x^{2}-19x=1$$

Ikkala tarafdan 1 ni ayirish.

$$8x^{2}-19x-1=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 8 ni a, -19 ni b va -1 ni c bilan almashtiring.

$$x=\frac{-\left(-19\right)±\sqrt{\left(-19\right)^{2}-4\times 8\left(-1\right)}}{2\times 8}$$

-19 kvadratini chiqarish.

$$x=\frac{-\left(-19\right)±\sqrt{361-4\times 8\left(-1\right)}}{2\times 8}$$

-4 ni 8 marotabaga ko'paytirish.

$$x=\frac{-\left(-19\right)±\sqrt{361-32\left(-1\right)}}{2\times 8}$$

-32 ni -1 marotabaga ko'paytirish.

$$x=\frac{-\left(-19\right)±\sqrt{361+32}}{2\times 8}$$

361 ni 32 ga qo'shish.

$$x=\frac{-\left(-19\right)±\sqrt{393}}{2\times 8}$$

-19 ning teskarisi 19 ga teng.

$$x=\frac{19±\sqrt{393}}{2\times 8}$$

2 ni 8 marotabaga ko'paytirish.

$$x=\frac{19±\sqrt{393}}{16}$$

x=\frac{19±\sqrt{393}}{16} tenglamasini yeching, bunda ± musbat. 19 ni \sqrt{393} ga qo'shish.

$$x=\frac{\sqrt{393}+19}{16}$$

x=\frac{19±\sqrt{393}}{16} tenglamasini yeching, bunda ± manfiy. 19 dan \sqrt{393} ni ayirish.

$$x=\frac{19-\sqrt{393}}{16}$$

Tenglama yechildi.

$$x=\frac{\sqrt{393}+19}{16} x=\frac{19-\sqrt{393}}{16}$$

Kvadratni yakunlash bo'yicha qadamlar

x qiymati 2 teng bo‘lmaydi, chunki nolga bo‘lish mumkin emas. Tenglamaning ikkala tarafini x-2 ga ko'paytirish.

$$9x\left(x-2\right)=x^{2}+x+1$$

9x ga x-2 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$9x^{2}-18x=x^{2}+x+1$$

Ikkala tarafdan x^{2} ni ayirish.

$$9x^{2}-18x-x^{2}=x+1$$

8x^{2} ni olish uchun 9x^{2} va -x^{2} ni birlashtirish.

$$8x^{2}-18x=x+1$$

Ikkala tarafdan x ni ayirish.

$$8x^{2}-18x-x=1$$

-19x ni olish uchun -18x va -x ni birlashtirish.

$$8x^{2}-19x=1$$

Ikki tarafini 8 ga bo‘ling.

$$\frac{8x^{2}-19x}{8}=\frac{1}{8}$$

8 ga bo'lish 8 ga ko'paytirishni bekor qiladi.

$$x^{2}-\frac{19}{8}x=\frac{1}{8}$$

-\frac{19}{8} ni bo‘lish, x shartining koeffitsienti, 2 ga -\frac{19}{16} olish uchun. Keyin, -\frac{19}{16} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}-\frac{19}{8}x+\left(-\frac{19}{16}\right)^{2}=\frac{1}{8}+\left(-\frac{19}{16}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib -\frac{19}{16} kvadratini chiqarish.

$$x^{2}-\frac{19}{8}x+\frac{361}{256}=\frac{1}{8}+\frac{361}{256}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali \frac{1}{8} ni \frac{361}{256} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$x^{2}-\frac{19}{8}x+\frac{361}{256}=\frac{393}{256}$$

x^{2}-\frac{19}{8}x+\frac{361}{256} omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(x-\frac{19}{16}\right)^{2}=\frac{393}{256}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x-\frac{19}{16}\right)^{2}}=\sqrt{\frac{393}{256}}$$

Qisqartirish.

$$x-\frac{19}{16}=\frac{\sqrt{393}}{16} x-\frac{19}{16}=-\frac{\sqrt{393}}{16}$$

\frac{19}{16} ni tenglamaning ikkala tarafiga qo'shish.

$$x=\frac{\sqrt{393}+19}{16} x=\frac{19-\sqrt{393}}{16}$$