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

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

x uchun yechish

$$x = \frac{\sqrt{41} + 7}{2} \approx 6,701562119$$

$$x=\frac{7-\sqrt{41}}{2}\approx 0,298437881$$

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.

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

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

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

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

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

3x^{2} ni olish uchun 2x^{2} va x^{2} ni birlashtirish.

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

-7x ni olish uchun -5x va -2x ni birlashtirish.

$$3x^{2}-7x+3-3=2\left(x-1\right)\left(x+1\right)$$

0 olish uchun 3 dan 3 ni ayirish.

$$3x^{2}-7x=2\left(x-1\right)\left(x+1\right)$$

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

$$3x^{2}-7x=\left(2x-2\right)\left(x+1\right)$$

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

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

Ikkala tarafdan 2x^{2} ni ayirish.

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

x^{2} ni olish uchun 3x^{2} va -2x^{2} ni birlashtirish.

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

2 ni ikki tarafga qo’shing.

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

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 1 ni a, -7 ni b va 2 ni c bilan almashtiring.

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

-7 kvadratini chiqarish.

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

-4 ni 2 marotabaga ko'paytirish.

$$x=\frac{-\left(-7\right)±\sqrt{49-8}}{2}$$

49 ni -8 ga qo'shish.

$$x=\frac{-\left(-7\right)±\sqrt{41}}{2}$$

-7 ning teskarisi 7 ga teng.

$$x=\frac{7±\sqrt{41}}{2}$$

x=\frac{7±\sqrt{41}}{2} tenglamasini yeching, bunda ± musbat. 7 ni \sqrt{41} ga qo'shish.

$$x=\frac{\sqrt{41}+7}{2}$$

x=\frac{7±\sqrt{41}}{2} tenglamasini yeching, bunda ± manfiy. 7 dan \sqrt{41} ni ayirish.

$$x=\frac{7-\sqrt{41}}{2}$$

Tenglama yechildi.

$$x=\frac{\sqrt{41}+7}{2} x=\frac{7-\sqrt{41}}{2}$$

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.

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

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

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

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

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

3x^{2} ni olish uchun 2x^{2} va x^{2} ni birlashtirish.

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

-7x ni olish uchun -5x va -2x ni birlashtirish.

$$3x^{2}-7x+3-3=2\left(x-1\right)\left(x+1\right)$$

0 olish uchun 3 dan 3 ni ayirish.

$$3x^{2}-7x=2\left(x-1\right)\left(x+1\right)$$

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

$$3x^{2}-7x=\left(2x-2\right)\left(x+1\right)$$

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

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

Ikkala tarafdan 2x^{2} ni ayirish.

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

x^{2} ni olish uchun 3x^{2} va -2x^{2} ni birlashtirish.

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

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

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

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib -\frac{7}{2} kvadratini chiqarish.

$$x^{2}-7x+\frac{49}{4}=-2+\frac{49}{4}$$

-2 ni \frac{49}{4} ga qo'shish.

$$x^{2}-7x+\frac{49}{4}=\frac{41}{4}$$

x^{2}-7x+\frac{49}{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.

$$\left(x-\frac{7}{2}\right)^{2}=\frac{41}{4}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x-\frac{7}{2}\right)^{2}}=\sqrt{\frac{41}{4}}$$

Qisqartirish.

$$x-\frac{7}{2}=\frac{\sqrt{41}}{2} x-\frac{7}{2}=-\frac{\sqrt{41}}{2}$$

\frac{7}{2} ni tenglamaning ikkala tarafiga qo'shish.

$$x=\frac{\sqrt{41}+7}{2} x=\frac{7-\sqrt{41}}{2}$$