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

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

x uchun yechish (complex solution)

$$x=\frac{7+\sqrt{119}i}{12}\approx 0,583333333+0,909059343i$$

$$x=\frac{-\sqrt{119}i+7}{12}\approx 0,583333333-0,909059343i$$

Kvadrat tenglamalaridan foydalanish qoidalari

x qiymati \frac{1}{3} teng bo‘lmaydi, chunki nolga bo‘lish mumkin emas. Tenglamaning ikkala tarafini 4\left(3x-1\right) ga, 3x-1,4 ning eng kichik karralisiga ko‘paytiring.

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

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

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

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

$$4x+4=12x-4-\left(3x-1\right)\left(2x+1\right)$$

3x-1 ga 2x+1 ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

$$4x+4=12x-4-\left(6x^{2}+x-1\right)$$

6x^{2}+x-1 teskarisini topish uchun har birining teskarisini toping.

$$4x+4=12x-4-6x^{2}-x+1$$

11x ni olish uchun 12x va -x ni birlashtirish.

$$4x+4=11x-4-6x^{2}+1$$

-3 olish uchun -4 va 1'ni qo'shing.

$$4x+4=11x-3-6x^{2}$$

Ikkala tarafdan 11x ni ayirish.

$$4x+4-11x=-3-6x^{2}$$

-7x ni olish uchun 4x va -11x ni birlashtirish.

$$-7x+4=-3-6x^{2}$$

Ikkala tarafdan -3 ni ayirish.

$$-7x+4-\left(-3\right)=-6x^{2}$$

-3 ning teskarisi 3 ga teng.

$$-7x+4+3=-6x^{2}$$

6x^{2} ni ikki tarafga qo’shing.

$$-7x+4+3+6x^{2}=0$$

7 olish uchun 4 va 3'ni qo'shing.

$$-7x+7+6x^{2}=0$$

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.

$$6x^{2}-7x+7=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 6 ni a, -7 ni b va 7 ni c bilan almashtiring.

$$x=\frac{-\left(-7\right)±\sqrt{\left(-7\right)^{2}-4\times 6\times 7}}{2\times 6}$$

-7 kvadratini chiqarish.

$$x=\frac{-\left(-7\right)±\sqrt{49-4\times 6\times 7}}{2\times 6}$$

-4 ni 6 marotabaga ko'paytirish.

$$x=\frac{-\left(-7\right)±\sqrt{49-24\times 7}}{2\times 6}$$

-24 ni 7 marotabaga ko'paytirish.

$$x=\frac{-\left(-7\right)±\sqrt{49-168}}{2\times 6}$$

49 ni -168 ga qo'shish.

$$x=\frac{-\left(-7\right)±\sqrt{-119}}{2\times 6}$$

-119 ning kvadrat ildizini chiqarish.

$$x=\frac{-\left(-7\right)±\sqrt{119}i}{2\times 6}$$

-7 ning teskarisi 7 ga teng.

$$x=\frac{7±\sqrt{119}i}{2\times 6}$$

2 ni 6 marotabaga ko'paytirish.

$$x=\frac{7±\sqrt{119}i}{12}$$

x=\frac{7±\sqrt{119}i}{12} tenglamasini yeching, bunda ± musbat. 7 ni i\sqrt{119} ga qo'shish.

$$x=\frac{7+\sqrt{119}i}{12}$$

x=\frac{7±\sqrt{119}i}{12} tenglamasini yeching, bunda ± manfiy. 7 dan i\sqrt{119} ni ayirish.

$$x=\frac{-\sqrt{119}i+7}{12}$$

Tenglama yechildi.

$$x=\frac{7+\sqrt{119}i}{12} x=\frac{-\sqrt{119}i+7}{12}$$

Kvadratni yakunlash bo'yicha qadamlar

x qiymati \frac{1}{3} teng bo‘lmaydi, chunki nolga bo‘lish mumkin emas. Tenglamaning ikkala tarafini 4\left(3x-1\right) ga, 3x-1,4 ning eng kichik karralisiga ko‘paytiring.

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

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

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

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

$$4x+4=12x-4-\left(3x-1\right)\left(2x+1\right)$$

3x-1 ga 2x+1 ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

$$4x+4=12x-4-\left(6x^{2}+x-1\right)$$

6x^{2}+x-1 teskarisini topish uchun har birining teskarisini toping.

$$4x+4=12x-4-6x^{2}-x+1$$

11x ni olish uchun 12x va -x ni birlashtirish.

$$4x+4=11x-4-6x^{2}+1$$

-3 olish uchun -4 va 1'ni qo'shing.

$$4x+4=11x-3-6x^{2}$$

Ikkala tarafdan 11x ni ayirish.

$$4x+4-11x=-3-6x^{2}$$

-7x ni olish uchun 4x va -11x ni birlashtirish.

$$-7x+4=-3-6x^{2}$$

6x^{2} ni ikki tarafga qo’shing.

$$-7x+4+6x^{2}=-3$$

Ikkala tarafdan 4 ni ayirish.

$$-7x+6x^{2}=-3-4$$

-7 olish uchun -3 dan 4 ni ayirish.

$$-7x+6x^{2}=-7$$

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

$$6x^{2}-7x=-7$$

Ikki tarafini 6 ga bo‘ling.

$$\frac{6x^{2}-7x}{6}=-\frac{7}{6}$$

6 ga bo'lish 6 ga ko'paytirishni bekor qiladi.

$$x^{2}-\frac{7}{6}x=-\frac{7}{6}$$

-\frac{7}{6} ni bo‘lish, x shartining koeffitsienti, 2 ga -\frac{7}{12} olish uchun. Keyin, -\frac{7}{12} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}-\frac{7}{6}x+\left(-\frac{7}{12}\right)^{2}=-\frac{7}{6}+\left(-\frac{7}{12}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib -\frac{7}{12} kvadratini chiqarish.

$$x^{2}-\frac{7}{6}x+\frac{49}{144}=-\frac{7}{6}+\frac{49}{144}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali -\frac{7}{6} ni \frac{49}{144} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$x^{2}-\frac{7}{6}x+\frac{49}{144}=-\frac{119}{144}$$

x^{2}-\frac{7}{6}x+\frac{49}{144} omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(x-\frac{7}{12}\right)^{2}=-\frac{119}{144}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x-\frac{7}{12}\right)^{2}}=\sqrt{-\frac{119}{144}}$$

Qisqartirish.

$$x-\frac{7}{12}=\frac{\sqrt{119}i}{12} x-\frac{7}{12}=-\frac{\sqrt{119}i}{12}$$

\frac{7}{12} ni tenglamaning ikkala tarafiga qo'shish.

$$x=\frac{7+\sqrt{119}i}{12} x=\frac{-\sqrt{119}i+7}{12}$$