1/x-1/x-2=3 eching

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

x uchun yechish

$$x=\frac{\sqrt{3}}{3}+1\approx 1,577350269$$

$$x=-\frac{\sqrt{3}}{3}+1\approx 0,422649731$$

Kvadrat tenglamalaridan foydalanish qoidalari

x qiymati 0,2 qiymatlaridan birortasiga teng bo‘lmaydi, chunki nolga bo‘lish mumkin emas. Tenglamaning ikkala tarafini x\left(x-2\right) ga, x,x-2 ning eng kichik karralisiga ko‘paytiring.

$$x-2-x=3x\left(x-2\right)$$

3x ga x-2 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$x-2-x=3x^{2}-6x$$

Ikkala tarafdan 3x^{2} ni ayirish.

$$x-2-x-3x^{2}=-6x$$

6x ni ikki tarafga qo’shing.

$$x-2-x-3x^{2}+6x=0$$

7x ni olish uchun x va 6x ni birlashtirish.

$$7x-2-x-3x^{2}=0$$

6x ni olish uchun 7x va -x ni birlashtirish.

$$6x-2-3x^{2}=0$$

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.

$$-3x^{2}+6x-2=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} -3 ni a, 6 ni b va -2 ni c bilan almashtiring.

$$x=\frac{-6±\sqrt{6^{2}-4\left(-3\right)\left(-2\right)}}{2\left(-3\right)}$$

6 kvadratini chiqarish.

$$x=\frac{-6±\sqrt{36-4\left(-3\right)\left(-2\right)}}{2\left(-3\right)}$$

-4 ni -3 marotabaga ko'paytirish.

$$x=\frac{-6±\sqrt{36+12\left(-2\right)}}{2\left(-3\right)}$$

12 ni -2 marotabaga ko'paytirish.

$$x=\frac{-6±\sqrt{36-24}}{2\left(-3\right)}$$

36 ni -24 ga qo'shish.

$$x=\frac{-6±\sqrt{12}}{2\left(-3\right)}$$

12 ning kvadrat ildizini chiqarish.

$$x=\frac{-6±2\sqrt{3}}{2\left(-3\right)}$$

2 ni -3 marotabaga ko'paytirish.

$$x=\frac{-6±2\sqrt{3}}{-6}$$

x=\frac{-6±2\sqrt{3}}{-6} tenglamasini yeching, bunda ± musbat. -6 ni 2\sqrt{3} ga qo'shish.

$$x=\frac{2\sqrt{3}-6}{-6}$$

-6+2\sqrt{3} ni -6 ga bo'lish.

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

x=\frac{-6±2\sqrt{3}}{-6} tenglamasini yeching, bunda ± manfiy. -6 dan 2\sqrt{3} ni ayirish.

$$x=\frac{-2\sqrt{3}-6}{-6}$$

-6-2\sqrt{3} ni -6 ga bo'lish.

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

Tenglama yechildi.

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

Kvadratni yakunlash bo'yicha qadamlar

x qiymati 0,2 qiymatlaridan birortasiga teng bo‘lmaydi, chunki nolga bo‘lish mumkin emas. Tenglamaning ikkala tarafini x\left(x-2\right) ga, x,x-2 ning eng kichik karralisiga ko‘paytiring.

$$x-2-x=3x\left(x-2\right)$$

3x ga x-2 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$x-2-x=3x^{2}-6x$$

Ikkala tarafdan 3x^{2} ni ayirish.

$$x-2-x-3x^{2}=-6x$$

6x ni ikki tarafga qo’shing.

$$x-2-x-3x^{2}+6x=0$$

7x ni olish uchun x va 6x ni birlashtirish.

$$7x-2-x-3x^{2}=0$$

2 ni ikki tarafga qo’shing. Har qanday songa nolni qo‘shsangiz, o‘zi chiqadi.

$$7x-x-3x^{2}=2$$

6x ni olish uchun 7x va -x ni birlashtirish.

$$6x-3x^{2}=2$$

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

$$-3x^{2}+6x=2$$

Ikki tarafini -3 ga bo‘ling.

$$\frac{-3x^{2}+6x}{-3}=\frac{2}{-3}$$

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

$$x^{2}+\frac{6}{-3}x=\frac{2}{-3}$$

6 ni -3 ga bo'lish.

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

2 ni -3 ga bo'lish.

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

-2 ni bo‘lish, x shartining koeffitsienti, 2 ga -1 olish uchun. Keyin, -1 ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

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

-\frac{2}{3} ni 1 ga qo'shish.

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

x^{2}-2x+1 omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

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

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

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

Qisqartirish.

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

1 ni tenglamaning ikkala tarafiga qo'shish.

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