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

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

x uchun yechish

$$x=\frac{2\sqrt{6}}{3}+1\approx 2,632993162$$

$$x=-\frac{2\sqrt{6}}{3}+1\approx -0,632993162$$

Kvadrat tenglamalaridan foydalanish qoidalari

x qiymati -1,1 qiymatlaridan birortasiga teng bo‘lmaydi, chunki nolga bo‘lish mumkin emas. Tenglamaning ikkala tarafini \left(x-1\right)\left(x+1\right) ga, x-1,x+1 ning eng kichik karralisiga ko‘paytiring.

$$\left(x+1\right)\times 4+\left(x-1\right)\times 2=3\left(x-1\right)\left(x+1\right)$$

x+1 ga 4 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$4x+4+\left(x-1\right)\times 2=3\left(x-1\right)\left(x+1\right)$$

x-1 ga 2 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$4x+4+2x-2=3\left(x-1\right)\left(x+1\right)$$

6x ni olish uchun 4x va 2x ni birlashtirish.

$$6x+4-2=3\left(x-1\right)\left(x+1\right)$$

2 olish uchun 4 dan 2 ni ayirish.

$$6x+2=3\left(x-1\right)\left(x+1\right)$$

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

$$6x+2=\left(3x-3\right)\left(x+1\right)$$

3x-3 ga x+1 ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

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

Ikkala tarafdan 3x^{2} ni ayirish.

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

3 ni ikki tarafga qo’shing.

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

5 olish uchun 2 va 3'ni qo'shing.

$$6x+5-3x^{2}=0$$

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.

$$-3x^{2}+6x+5=0$$

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 5 ni c bilan almashtiring.

$$x=\frac{-6±\sqrt{6^{2}-4\left(-3\right)\times 5}}{2\left(-3\right)}$$

6 kvadratini chiqarish.

$$x=\frac{-6±\sqrt{36-4\left(-3\right)\times 5}}{2\left(-3\right)}$$

-4 ni -3 marotabaga ko'paytirish.

$$x=\frac{-6±\sqrt{36+12\times 5}}{2\left(-3\right)}$$

12 ni 5 marotabaga ko'paytirish.

$$x=\frac{-6±\sqrt{36+60}}{2\left(-3\right)}$$

36 ni 60 ga qo'shish.

$$x=\frac{-6±\sqrt{96}}{2\left(-3\right)}$$

96 ning kvadrat ildizini chiqarish.

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

2 ni -3 marotabaga ko'paytirish.

$$x=\frac{-6±4\sqrt{6}}{-6}$$

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

$$x=\frac{4\sqrt{6}-6}{-6}$$

-6+4\sqrt{6} ni -6 ga bo'lish.

$$x=-\frac{2\sqrt{6}}{3}+1$$

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

$$x=\frac{-4\sqrt{6}-6}{-6}$$

-6-4\sqrt{6} ni -6 ga bo'lish.

$$x=\frac{2\sqrt{6}}{3}+1$$

Tenglama yechildi.

$$x=-\frac{2\sqrt{6}}{3}+1 x=\frac{2\sqrt{6}}{3}+1$$

Kvadratni yakunlash bo'yicha qadamlar

x qiymati -1,1 qiymatlaridan birortasiga teng bo‘lmaydi, chunki nolga bo‘lish mumkin emas. Tenglamaning ikkala tarafini \left(x-1\right)\left(x+1\right) ga, x-1,x+1 ning eng kichik karralisiga ko‘paytiring.

$$\left(x+1\right)\times 4+\left(x-1\right)\times 2=3\left(x-1\right)\left(x+1\right)$$

x+1 ga 4 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$4x+4+\left(x-1\right)\times 2=3\left(x-1\right)\left(x+1\right)$$

x-1 ga 2 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$4x+4+2x-2=3\left(x-1\right)\left(x+1\right)$$

6x ni olish uchun 4x va 2x ni birlashtirish.

$$6x+4-2=3\left(x-1\right)\left(x+1\right)$$

2 olish uchun 4 dan 2 ni ayirish.

$$6x+2=3\left(x-1\right)\left(x+1\right)$$

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

$$6x+2=\left(3x-3\right)\left(x+1\right)$$

3x-3 ga x+1 ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

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

Ikkala tarafdan 3x^{2} ni ayirish.

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

Ikkala tarafdan 2 ni ayirish.

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

-5 olish uchun -3 dan 2 ni ayirish.

$$6x-3x^{2}=-5$$

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

$$-3x^{2}+6x=-5$$

Ikki tarafini -3 ga bo‘ling.

$$\frac{-3x^{2}+6x}{-3}=-\frac{5}{-3}$$

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

$$x^{2}+\frac{6}{-3}x=-\frac{5}{-3}$$

6 ni -3 ga bo'lish.

$$x^{2}-2x=-\frac{5}{-3}$$

-5 ni -3 ga bo'lish.

$$x^{2}-2x=\frac{5}{3}$$

-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.

$$x^{2}-2x+1=\frac{5}{3}+1$$

\frac{5}{3} ni 1 ga qo'shish.

$$x^{2}-2x+1=\frac{8}{3}$$

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.

$$\left(x-1\right)^{2}=\frac{8}{3}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x-1\right)^{2}}=\sqrt{\frac{8}{3}}$$

Qisqartirish.

$$x-1=\frac{2\sqrt{6}}{3} x-1=-\frac{2\sqrt{6}}{3}$$

1 ni tenglamaning ikkala tarafiga qo'shish.

$$x=\frac{2\sqrt{6}}{3}+1 x=-\frac{2\sqrt{6}}{3}+1$$