x/x-1=3x+1/x-1 eching

$$\frac { x } { x - 1 } = 3 x + \frac { 1 } { x - 1 }$$

x uchun yechish

$$x=\frac{1}{3}\approx 0,333333333$$

Kvadrat tenglamalaridan foydalanish qoidalari

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

$$x=3x\left(x-1\right)+1$$

3x ga x-1 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$x=3x^{2}-3x+1$$

Ikkala tarafdan 3x^{2} ni ayirish.

$$x-3x^{2}=-3x+1$$

3x ni ikki tarafga qo’shing.

$$x-3x^{2}+3x=1$$

4x ni olish uchun x va 3x ni birlashtirish.

$$4x-3x^{2}=1$$

Ikkala tarafdan 1 ni ayirish.

$$4x-3x^{2}-1=0$$

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.

$$-3x^{2}+4x-1=0$$

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

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

4 kvadratini chiqarish.

$$x=\frac{-4±\sqrt{16-4\left(-3\right)\left(-1\right)}}{2\left(-3\right)}$$

-4 ni -3 marotabaga ko'paytirish.

$$x=\frac{-4±\sqrt{16+12\left(-1\right)}}{2\left(-3\right)}$$

12 ni -1 marotabaga ko'paytirish.

$$x=\frac{-4±\sqrt{16-12}}{2\left(-3\right)}$$

16 ni -12 ga qo'shish.

$$x=\frac{-4±\sqrt{4}}{2\left(-3\right)}$$

4 ning kvadrat ildizini chiqarish.

$$x=\frac{-4±2}{2\left(-3\right)}$$

2 ni -3 marotabaga ko'paytirish.

$$x=\frac{-4±2}{-6}$$

x=\frac{-4±2}{-6} tenglamasini yeching, bunda ± musbat. -4 ni 2 ga qo'shish.

$$x=-\frac{2}{-6}$$

\frac{-2}{-6} ulushini 2 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

$$x=\frac{1}{3}$$

x=\frac{-4±2}{-6} tenglamasini yeching, bunda ± manfiy. -4 dan 2 ni ayirish.

$$x=-\frac{6}{-6}$$

-6 ni -6 ga bo'lish.

$$x=1$$

Tenglama yechildi.

$$x=\frac{1}{3} x=1$$

x qiymati 1 teng bo‘lmaydi.

$$x=\frac{1}{3}$$

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.

$$x=3x\left(x-1\right)+1$$

3x ga x-1 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$x=3x^{2}-3x+1$$

Ikkala tarafdan 3x^{2} ni ayirish.

$$x-3x^{2}=-3x+1$$

3x ni ikki tarafga qo’shing.

$$x-3x^{2}+3x=1$$

4x ni olish uchun x va 3x ni birlashtirish.

$$4x-3x^{2}=1$$

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

$$-3x^{2}+4x=1$$

Ikki tarafini -3 ga bo‘ling.

$$\frac{-3x^{2}+4x}{-3}=\frac{1}{-3}$$

-3 ga bo'lish -3 ga ko'paytirishni bekor qiladi.

$$x^{2}+\frac{4}{-3}x=\frac{1}{-3}$$

4 ni -3 ga bo'lish.

$$x^{2}-\frac{4}{3}x=\frac{1}{-3}$$

1 ni -3 ga bo'lish.

$$x^{2}-\frac{4}{3}x=-\frac{1}{3}$$

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

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

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib -\frac{2}{3} kvadratini chiqarish.

$$x^{2}-\frac{4}{3}x+\frac{4}{9}=-\frac{1}{3}+\frac{4}{9}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali -\frac{1}{3} ni \frac{4}{9} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$x^{2}-\frac{4}{3}x+\frac{4}{9}=\frac{1}{9}$$

x^{2}-\frac{4}{3}x+\frac{4}{9} omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(x-\frac{2}{3}\right)^{2}=\frac{1}{9}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x-\frac{2}{3}\right)^{2}}=\sqrt{\frac{1}{9}}$$

Qisqartirish.

$$x-\frac{2}{3}=\frac{1}{3} x-\frac{2}{3}=-\frac{1}{3}$$

\frac{2}{3} ni tenglamaning ikkala tarafiga qo'shish.

$$x=1 x=\frac{1}{3}$$

x qiymati 1 teng bo‘lmaydi.

$$x=\frac{1}{3}$$