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

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

x uchun yechish

$$x=-\frac{5}{9}\approx -0,555555556$$

$$x=0$$

Kvadrat tenglamalaridan foydalanish qoidalari

x qiymati -1,-\frac{1}{3} qiymatlaridan birortasiga teng bo‘lmaydi, chunki nolga bo‘lish mumkin emas. Tenglamaning ikkala tarafini \left(x+1\right)\left(3x+1\right) ga, 3x+1,x+1 ning eng kichik karralisiga ko‘paytiring.

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

3x+1 ga 2 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

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

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

$$7x+1+2=3\left(x+1\right)\left(3x+1\right)$$

3 olish uchun 1 va 2'ni qo'shing.

$$7x+3=3\left(x+1\right)\left(3x+1\right)$$

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

$$7x+3=\left(3x+3\right)\left(3x+1\right)$$

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

$$7x+3=9x^{2}+12x+3$$

Ikkala tarafdan 9x^{2} ni ayirish.

$$7x+3-9x^{2}=12x+3$$

Ikkala tarafdan 12x ni ayirish.

$$7x+3-9x^{2}-12x=3$$

-5x ni olish uchun 7x va -12x ni birlashtirish.

$$-5x+3-9x^{2}=3$$

Ikkala tarafdan 3 ni ayirish.

$$-5x+3-9x^{2}-3=0$$

0 olish uchun 3 dan 3 ni ayirish.

$$-5x-9x^{2}=0$$

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.

$$-9x^{2}-5x=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} -9 ni a, -5 ni b va 0 ni c bilan almashtiring.

$$x=\frac{-\left(-5\right)±\sqrt{\left(-5\right)^{2}}}{2\left(-9\right)}$$

\left(-5\right)^{2} ning kvadrat ildizini chiqarish.

$$x=\frac{-\left(-5\right)±5}{2\left(-9\right)}$$

-5 ning teskarisi 5 ga teng.

$$x=\frac{5±5}{2\left(-9\right)}$$

2 ni -9 marotabaga ko'paytirish.

$$x=\frac{5±5}{-18}$$

x=\frac{5±5}{-18} tenglamasini yeching, bunda ± musbat. 5 ni 5 ga qo'shish.

$$x=\frac{10}{-18}$$

\frac{10}{-18} ulushini 2 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

$$x=-\frac{5}{9}$$

x=\frac{5±5}{-18} tenglamasini yeching, bunda ± manfiy. 5 dan 5 ni ayirish.

$$x=\frac{0}{-18}$$

0 ni -18 ga bo'lish.

$$x=0$$

Tenglama yechildi.

$$x=-\frac{5}{9} x=0$$

Kvadratni yakunlash bo'yicha qadamlar

x qiymati -1,-\frac{1}{3} qiymatlaridan birortasiga teng bo‘lmaydi, chunki nolga bo‘lish mumkin emas. Tenglamaning ikkala tarafini \left(x+1\right)\left(3x+1\right) ga, 3x+1,x+1 ning eng kichik karralisiga ko‘paytiring.

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

3x+1 ga 2 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

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

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

$$7x+1+2=3\left(x+1\right)\left(3x+1\right)$$

3 olish uchun 1 va 2'ni qo'shing.

$$7x+3=3\left(x+1\right)\left(3x+1\right)$$

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

$$7x+3=\left(3x+3\right)\left(3x+1\right)$$

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

$$7x+3=9x^{2}+12x+3$$

Ikkala tarafdan 9x^{2} ni ayirish.

$$7x+3-9x^{2}=12x+3$$

Ikkala tarafdan 12x ni ayirish.

$$7x+3-9x^{2}-12x=3$$

-5x ni olish uchun 7x va -12x ni birlashtirish.

$$-5x+3-9x^{2}=3$$

Ikkala tarafdan 3 ni ayirish.

$$-5x-9x^{2}=3-3$$

0 olish uchun 3 dan 3 ni ayirish.

$$-5x-9x^{2}=0$$

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

$$-9x^{2}-5x=0$$

Ikki tarafini -9 ga bo‘ling.

$$\frac{-9x^{2}-5x}{-9}=\frac{0}{-9}$$

-9 ga bo'lish -9 ga ko'paytirishni bekor qiladi.

$$x^{2}+\left(-\frac{5}{-9}\right)x=\frac{0}{-9}$$

-5 ni -9 ga bo'lish.

$$x^{2}+\frac{5}{9}x=\frac{0}{-9}$$

0 ni -9 ga bo'lish.

$$x^{2}+\frac{5}{9}x=0$$

\frac{5}{9} ni bo‘lish, x shartining koeffitsienti, 2 ga \frac{5}{18} olish uchun. Keyin, \frac{5}{18} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}+\frac{5}{9}x+\left(\frac{5}{18}\right)^{2}=\left(\frac{5}{18}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib \frac{5}{18} kvadratini chiqarish.

$$x^{2}+\frac{5}{9}x+\frac{25}{324}=\frac{25}{324}$$

x^{2}+\frac{5}{9}x+\frac{25}{324} omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(x+\frac{5}{18}\right)^{2}=\frac{25}{324}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x+\frac{5}{18}\right)^{2}}=\sqrt{\frac{25}{324}}$$

Qisqartirish.

$$x+\frac{5}{18}=\frac{5}{18} x+\frac{5}{18}=-\frac{5}{18}$$

Tenglamaning ikkala tarafidan \frac{5}{18} ni ayirish.

$$x=0 x=-\frac{5}{9}$$