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

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

x uchun yechish

$$x=\frac{\sqrt{13}-1}{6}\approx 0,434258546$$

$$x=\frac{-\sqrt{13}-1}{6}\approx -0,767591879$$

Kvadrat tenglamalaridan foydalanish qoidalari

x qiymati -\frac{1}{2},1 qiymatlaridan birortasiga teng bo‘lmaydi, chunki nolga bo‘lish mumkin emas. Tenglamaning ikkala tarafini \left(x-1\right)\left(2x+1\right) ga, 2x+1,x-1 ning eng kichik karralisiga ko‘paytiring.

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

\left(x-1\right)^{2} hosil qilish uchun x-1 va x-1 ni ko'paytirish.

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

\left(2x+1\right)^{2} hosil qilish uchun 2x+1 va 2x+1 ni ko'paytirish.

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

\left(a-b\right)^{2}=a^{2}-2ab+b^{2} binom teoremasini \left(x-1\right)^{2} kengaytirilishi uchun ishlating.

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

\left(a+b\right)^{2}=a^{2}+2ab+b^{2} binom teoremasini \left(2x+1\right)^{2} kengaytirilishi uchun ishlating.

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

x-1 ga 2x+1 ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

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

2x^{2}-x-1 ga 3 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$x^{2}-2x+1=4x^{2}+4x+1+6x^{2}-3x-3$$

10x^{2} ni olish uchun 4x^{2} va 6x^{2} ni birlashtirish.

$$x^{2}-2x+1=10x^{2}+4x+1-3x-3$$

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

$$x^{2}-2x+1=10x^{2}+x+1-3$$

-2 olish uchun 1 dan 3 ni ayirish.

$$x^{2}-2x+1=10x^{2}+x-2$$

Ikkala tarafdan 10x^{2} ni ayirish.

$$x^{2}-2x+1-10x^{2}=x-2$$

-9x^{2} ni olish uchun x^{2} va -10x^{2} ni birlashtirish.

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

Ikkala tarafdan x ni ayirish.

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

-3x ni olish uchun -2x va -x ni birlashtirish.

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

2 ni ikki tarafga qo’shing.

$$-9x^{2}-3x+1+2=0$$

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

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

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} -9 ni a, -3 ni b va 3 ni c bilan almashtiring.

$$x=\frac{-\left(-3\right)±\sqrt{\left(-3\right)^{2}-4\left(-9\right)\times 3}}{2\left(-9\right)}$$

-3 kvadratini chiqarish.

$$x=\frac{-\left(-3\right)±\sqrt{9-4\left(-9\right)\times 3}}{2\left(-9\right)}$$

-4 ni -9 marotabaga ko'paytirish.

$$x=\frac{-\left(-3\right)±\sqrt{9+36\times 3}}{2\left(-9\right)}$$

36 ni 3 marotabaga ko'paytirish.

$$x=\frac{-\left(-3\right)±\sqrt{9+108}}{2\left(-9\right)}$$

9 ni 108 ga qo'shish.

$$x=\frac{-\left(-3\right)±\sqrt{117}}{2\left(-9\right)}$$

117 ning kvadrat ildizini chiqarish.

$$x=\frac{-\left(-3\right)±3\sqrt{13}}{2\left(-9\right)}$$

-3 ning teskarisi 3 ga teng.

$$x=\frac{3±3\sqrt{13}}{2\left(-9\right)}$$

2 ni -9 marotabaga ko'paytirish.

$$x=\frac{3±3\sqrt{13}}{-18}$$

x=\frac{3±3\sqrt{13}}{-18} tenglamasini yeching, bunda ± musbat. 3 ni 3\sqrt{13} ga qo'shish.

$$x=\frac{3\sqrt{13}+3}{-18}$$

3+3\sqrt{13} ni -18 ga bo'lish.

$$x=\frac{-\sqrt{13}-1}{6}$$

x=\frac{3±3\sqrt{13}}{-18} tenglamasini yeching, bunda ± manfiy. 3 dan 3\sqrt{13} ni ayirish.

$$x=\frac{3-3\sqrt{13}}{-18}$$

3-3\sqrt{13} ni -18 ga bo'lish.

$$x=\frac{\sqrt{13}-1}{6}$$

Tenglama yechildi.

$$x=\frac{-\sqrt{13}-1}{6} x=\frac{\sqrt{13}-1}{6}$$

Kvadratni yakunlash bo'yicha qadamlar

x qiymati -\frac{1}{2},1 qiymatlaridan birortasiga teng bo‘lmaydi, chunki nolga bo‘lish mumkin emas. Tenglamaning ikkala tarafini \left(x-1\right)\left(2x+1\right) ga, 2x+1,x-1 ning eng kichik karralisiga ko‘paytiring.

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

\left(x-1\right)^{2} hosil qilish uchun x-1 va x-1 ni ko'paytirish.

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

\left(2x+1\right)^{2} hosil qilish uchun 2x+1 va 2x+1 ni ko'paytirish.

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

\left(a-b\right)^{2}=a^{2}-2ab+b^{2} binom teoremasini \left(x-1\right)^{2} kengaytirilishi uchun ishlating.

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

\left(a+b\right)^{2}=a^{2}+2ab+b^{2} binom teoremasini \left(2x+1\right)^{2} kengaytirilishi uchun ishlating.

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

x-1 ga 2x+1 ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

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

2x^{2}-x-1 ga 3 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$x^{2}-2x+1=4x^{2}+4x+1+6x^{2}-3x-3$$

10x^{2} ni olish uchun 4x^{2} va 6x^{2} ni birlashtirish.

$$x^{2}-2x+1=10x^{2}+4x+1-3x-3$$

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

$$x^{2}-2x+1=10x^{2}+x+1-3$$

-2 olish uchun 1 dan 3 ni ayirish.

$$x^{2}-2x+1=10x^{2}+x-2$$

Ikkala tarafdan 10x^{2} ni ayirish.

$$x^{2}-2x+1-10x^{2}=x-2$$

-9x^{2} ni olish uchun x^{2} va -10x^{2} ni birlashtirish.

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

Ikkala tarafdan x ni ayirish.

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

-3x ni olish uchun -2x va -x ni birlashtirish.

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

Ikkala tarafdan 1 ni ayirish.

$$-9x^{2}-3x=-2-1$$

-3 olish uchun -2 dan 1 ni ayirish.

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

Ikki tarafini -9 ga bo‘ling.

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

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

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

\frac{-3}{-9} ulushini 3 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

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

\frac{-3}{-9} ulushini 3 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

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

\frac{1}{3} ni bo‘lish, x shartining koeffitsienti, 2 ga \frac{1}{6} olish uchun. Keyin, \frac{1}{6} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}+\frac{1}{3}x+\left(\frac{1}{6}\right)^{2}=\frac{1}{3}+\left(\frac{1}{6}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib \frac{1}{6} kvadratini chiqarish.

$$x^{2}+\frac{1}{3}x+\frac{1}{36}=\frac{1}{3}+\frac{1}{36}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali \frac{1}{3} ni \frac{1}{36} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$x^{2}+\frac{1}{3}x+\frac{1}{36}=\frac{13}{36}$$

x^{2}+\frac{1}{3}x+\frac{1}{36} omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(x+\frac{1}{6}\right)^{2}=\frac{13}{36}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x+\frac{1}{6}\right)^{2}}=\sqrt{\frac{13}{36}}$$

Qisqartirish.

$$x+\frac{1}{6}=\frac{\sqrt{13}}{6} x+\frac{1}{6}=-\frac{\sqrt{13}}{6}$$

Tenglamaning ikkala tarafidan \frac{1}{6} ni ayirish.

$$x=\frac{\sqrt{13}-1}{6} x=\frac{-\sqrt{13}-1}{6}$$