x/x-1=8x+1/x-1 eching

$$\frac { x } { x - 1 } = 8 x + \frac { 1 } { x - 1 }$$

x uchun yechish

$$x=\frac{1}{8}=0,125$$

Kvadrat tenglamalaridan foydalanish qoidalari

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

$$x=8x\left(x-1\right)+1$$

8x ga x-1 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

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

Ikkala tarafdan 8x^{2} ni ayirish.

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

8x ni ikki tarafga qo’shing.

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

9x ni olish uchun x va 8x ni birlashtirish.

$$9x-8x^{2}=1$$

Ikkala tarafdan 1 ni ayirish.

$$9x-8x^{2}-1=0$$

ax^{2}+bx+c=0 shaklidagi barcha tenglamalarni kvadrat formulasi bilan yechish mumkin: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. Kvadrat formula ikki yechmni taqdim qiladi, biri ± qo'shish bo'lganda, va ikkinchisi ayiruv bo'lganda.

$$-8x^{2}+9x-1=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} -8 ni a, 9 ni b va -1 ni c bilan almashtiring.

$$x=\frac{-9±\sqrt{9^{2}-4\left(-8\right)\left(-1\right)}}{2\left(-8\right)}$$

9 kvadratini chiqarish.

$$x=\frac{-9±\sqrt{81-4\left(-8\right)\left(-1\right)}}{2\left(-8\right)}$$

-4 ni -8 marotabaga ko'paytirish.

$$x=\frac{-9±\sqrt{81+32\left(-1\right)}}{2\left(-8\right)}$$

32 ni -1 marotabaga ko'paytirish.

$$x=\frac{-9±\sqrt{81-32}}{2\left(-8\right)}$$

81 ni -32 ga qo'shish.

$$x=\frac{-9±\sqrt{49}}{2\left(-8\right)}$$

49 ning kvadrat ildizini chiqarish.

$$x=\frac{-9±7}{2\left(-8\right)}$$

2 ni -8 marotabaga ko'paytirish.

$$x=\frac{-9±7}{-16}$$

x=\frac{-9±7}{-16} tenglamasini yeching, bunda ± musbat. -9 ni 7 ga qo'shish.

$$x=-\frac{2}{-16}$$

\frac{-2}{-16} ulushini 2 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

$$x=\frac{1}{8}$$

x=\frac{-9±7}{-16} tenglamasini yeching, bunda ± manfiy. -9 dan 7 ni ayirish.

$$x=-\frac{16}{-16}$$

-16 ni -16 ga bo'lish.

$$x=1$$

Tenglama yechildi.

$$x=\frac{1}{8} x=1$$

x qiymati 1 teng bo‘lmaydi.

$$x=\frac{1}{8}$$

Kvadratni yakunlash bo'yicha qadamlar

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

$$x=8x\left(x-1\right)+1$$

8x ga x-1 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

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

Ikkala tarafdan 8x^{2} ni ayirish.

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

8x ni ikki tarafga qo’shing.

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

9x ni olish uchun x va 8x ni birlashtirish.

$$9x-8x^{2}=1$$

Bu kabi kvadrat tenglamalarni kvadratni yakunlab yechish mumkin. Kvadratni yechish uchun tenglama avval ushbu shaklda bo'lishi shart: x^{2}+bx=c.

$$-8x^{2}+9x=1$$

Ikki tarafini -8 ga bo‘ling.

$$\frac{-8x^{2}+9x}{-8}=\frac{1}{-8}$$

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

$$x^{2}+\frac{9}{-8}x=\frac{1}{-8}$$

9 ni -8 ga bo'lish.

$$x^{2}-\frac{9}{8}x=\frac{1}{-8}$$

1 ni -8 ga bo'lish.

$$x^{2}-\frac{9}{8}x=-\frac{1}{8}$$

-\frac{9}{8} ni bo‘lish, x shartining koeffitsienti, 2 ga -\frac{9}{16} olish uchun. Keyin, -\frac{9}{16} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}-\frac{9}{8}x+\left(-\frac{9}{16}\right)^{2}=-\frac{1}{8}+\left(-\frac{9}{16}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib -\frac{9}{16} kvadratini chiqarish.

$$x^{2}-\frac{9}{8}x+\frac{81}{256}=-\frac{1}{8}+\frac{81}{256}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali -\frac{1}{8} ni \frac{81}{256} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$x^{2}-\frac{9}{8}x+\frac{81}{256}=\frac{49}{256}$$

x^{2}-\frac{9}{8}x+\frac{81}{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.

$$\left(x-\frac{9}{16}\right)^{2}=\frac{49}{256}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x-\frac{9}{16}\right)^{2}}=\sqrt{\frac{49}{256}}$$

Qisqartirish.

$$x-\frac{9}{16}=\frac{7}{16} x-\frac{9}{16}=-\frac{7}{16}$$

\frac{9}{16} ni tenglamaning ikkala tarafiga qo'shish.

$$x=1 x=\frac{1}{8}$$

x qiymati 1 teng bo‘lmaydi.

$$x=\frac{1}{8}$$