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

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

x uchun yechish

$$x = \frac{\sqrt{3} + 3}{2} \approx 2,366025404$$

$$x=\frac{3-\sqrt{3}}{2}\approx 0,633974596$$

Kvadrat tenglamalaridan foydalanish qoidalari

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

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

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

$$2x^{2}-5x-3+3\times 2=\left(x-3\right)\left(1-2x\right)$$

6 hosil qilish uchun 3 va 2 ni ko'paytirish.

$$2x^{2}-5x-3+6=\left(x-3\right)\left(1-2x\right)$$

3 olish uchun -3 va 6'ni qo'shing.

$$2x^{2}-5x+3=\left(x-3\right)\left(1-2x\right)$$

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

$$2x^{2}-5x+3=7x-2x^{2}-3$$

Ikkala tarafdan 7x ni ayirish.

$$2x^{2}-5x+3-7x=-2x^{2}-3$$

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

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

2x^{2} ni ikki tarafga qo’shing.

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

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

$$4x^{2}-12x+3=-3$$

3 ni ikki tarafga qo’shing.

$$4x^{2}-12x+3+3=0$$

6 olish uchun 3 va 3'ni qo'shing.

$$4x^{2}-12x+6=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 4 ni a, -12 ni b va 6 ni c bilan almashtiring.

$$x=\frac{-\left(-12\right)±\sqrt{\left(-12\right)^{2}-4\times 4\times 6}}{2\times 4}$$

-12 kvadratini chiqarish.

$$x=\frac{-\left(-12\right)±\sqrt{144-4\times 4\times 6}}{2\times 4}$$

-4 ni 4 marotabaga ko'paytirish.

$$x=\frac{-\left(-12\right)±\sqrt{144-16\times 6}}{2\times 4}$$

-16 ni 6 marotabaga ko'paytirish.

$$x=\frac{-\left(-12\right)±\sqrt{144-96}}{2\times 4}$$

144 ni -96 ga qo'shish.

$$x=\frac{-\left(-12\right)±\sqrt{48}}{2\times 4}$$

48 ning kvadrat ildizini chiqarish.

$$x=\frac{-\left(-12\right)±4\sqrt{3}}{2\times 4}$$

-12 ning teskarisi 12 ga teng.

$$x=\frac{12±4\sqrt{3}}{2\times 4}$$

2 ni 4 marotabaga ko'paytirish.

$$x=\frac{12±4\sqrt{3}}{8}$$

x=\frac{12±4\sqrt{3}}{8} tenglamasini yeching, bunda ± musbat. 12 ni 4\sqrt{3} ga qo'shish.

$$x=\frac{4\sqrt{3}+12}{8}$$

12+4\sqrt{3} ni 8 ga bo'lish.

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

x=\frac{12±4\sqrt{3}}{8} tenglamasini yeching, bunda ± manfiy. 12 dan 4\sqrt{3} ni ayirish.

$$x=\frac{12-4\sqrt{3}}{8}$$

12-4\sqrt{3} ni 8 ga bo'lish.

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

Tenglama yechildi.

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

Kvadratni yakunlash bo'yicha qadamlar

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

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

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

$$2x^{2}-5x-3+3\times 2=\left(x-3\right)\left(1-2x\right)$$

6 hosil qilish uchun 3 va 2 ni ko'paytirish.

$$2x^{2}-5x-3+6=\left(x-3\right)\left(1-2x\right)$$

3 olish uchun -3 va 6'ni qo'shing.

$$2x^{2}-5x+3=\left(x-3\right)\left(1-2x\right)$$

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

$$2x^{2}-5x+3=7x-2x^{2}-3$$

Ikkala tarafdan 7x ni ayirish.

$$2x^{2}-5x+3-7x=-2x^{2}-3$$

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

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

2x^{2} ni ikki tarafga qo’shing.

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

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

$$4x^{2}-12x+3=-3$$

Ikkala tarafdan 3 ni ayirish.

$$4x^{2}-12x=-3-3$$

-6 olish uchun -3 dan 3 ni ayirish.

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

Ikki tarafini 4 ga bo‘ling.

$$\frac{4x^{2}-12x}{4}=-\frac{6}{4}$$

4 ga bo'lish 4 ga ko'paytirishni bekor qiladi.

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

-12 ni 4 ga bo'lish.

$$x^{2}-3x=-\frac{6}{4}$$

\frac{-6}{4} ulushini 2 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

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

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

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

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib -\frac{3}{2} kvadratini chiqarish.

$$x^{2}-3x+\frac{9}{4}=-\frac{3}{2}+\frac{9}{4}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali -\frac{3}{2} ni \frac{9}{4} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$x^{2}-3x+\frac{9}{4}=\frac{3}{4}$$

x^{2}-3x+\frac{9}{4} omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

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

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

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

Qisqartirish.

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

\frac{3}{2} ni tenglamaning ikkala tarafiga qo'shish.

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