x-2/x-5=2*+1/x-1 eching

$$\frac{ x-2 }{ x-5 } = \frac{ 2 \times +1 }{ x-1 }$$

x uchun yechish (complex solution)

$$x=\frac{5+\sqrt{23}i}{2}\approx 2,5+2,397915762i$$

$$x=\frac{-\sqrt{23}i+5}{2}\approx 2,5-2,397915762i$$

Kvadrat tenglamalaridan foydalanish qoidalari

x qiymati 1,5 qiymatlaridan birortasiga teng bo‘lmaydi, chunki nolga bo‘lish mumkin emas. Tenglamaning ikkala tarafini \left(x-5\right)\left(x-1\right) ga, x-5,x-1 ning eng kichik karralisiga ko‘paytiring.

$$\left(x-1\right)\left(x-2\right)=\left(x-5\right)\times 2\times 1$$

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

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

2 hosil qilish uchun 2 va 1 ni ko'paytirish.

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

x-5 ga 2 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

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

Ikkala tarafdan 2x ni ayirish.

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

-5x ni olish uchun -3x va -2x ni birlashtirish.

$$x^{2}-5x+2=-10$$

10 ni ikki tarafga qo’shing.

$$x^{2}-5x+2+10=0$$

12 olish uchun 2 va 10'ni qo'shing.

$$x^{2}-5x+12=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 1 ni a, -5 ni b va 12 ni c bilan almashtiring.

$$x=\frac{-\left(-5\right)±\sqrt{\left(-5\right)^{2}-4\times 12}}{2}$$

-5 kvadratini chiqarish.

$$x=\frac{-\left(-5\right)±\sqrt{25-4\times 12}}{2}$$

-4 ni 12 marotabaga ko'paytirish.

$$x=\frac{-\left(-5\right)±\sqrt{25-48}}{2}$$

25 ni -48 ga qo'shish.

$$x=\frac{-\left(-5\right)±\sqrt{-23}}{2}$$

-23 ning kvadrat ildizini chiqarish.

$$x=\frac{-\left(-5\right)±\sqrt{23}i}{2}$$

-5 ning teskarisi 5 ga teng.

$$x=\frac{5±\sqrt{23}i}{2}$$

x=\frac{5±\sqrt{23}i}{2} tenglamasini yeching, bunda ± musbat. 5 ni i\sqrt{23} ga qo'shish.

$$x=\frac{5+\sqrt{23}i}{2}$$

x=\frac{5±\sqrt{23}i}{2} tenglamasini yeching, bunda ± manfiy. 5 dan i\sqrt{23} ni ayirish.

$$x=\frac{-\sqrt{23}i+5}{2}$$

Tenglama yechildi.

$$x=\frac{5+\sqrt{23}i}{2} x=\frac{-\sqrt{23}i+5}{2}$$

Kvadratni yakunlash bo'yicha qadamlar

x qiymati 1,5 qiymatlaridan birortasiga teng bo‘lmaydi, chunki nolga bo‘lish mumkin emas. Tenglamaning ikkala tarafini \left(x-5\right)\left(x-1\right) ga, x-5,x-1 ning eng kichik karralisiga ko‘paytiring.

$$\left(x-1\right)\left(x-2\right)=\left(x-5\right)\times 2\times 1$$

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

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

2 hosil qilish uchun 2 va 1 ni ko'paytirish.

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

x-5 ga 2 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

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

Ikkala tarafdan 2x ni ayirish.

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

-5x ni olish uchun -3x va -2x ni birlashtirish.

$$x^{2}-5x+2=-10$$

Ikkala tarafdan 2 ni ayirish.

$$x^{2}-5x=-10-2$$

-12 olish uchun -10 dan 2 ni ayirish.

$$x^{2}-5x=-12$$

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

$$x^{2}-5x+\left(-\frac{5}{2}\right)^{2}=-12+\left(-\frac{5}{2}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib -\frac{5}{2} kvadratini chiqarish.

$$x^{2}-5x+\frac{25}{4}=-12+\frac{25}{4}$$

-12 ni \frac{25}{4} ga qo'shish.

$$x^{2}-5x+\frac{25}{4}=-\frac{23}{4}$$

x^{2}-5x+\frac{25}{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.

$$\left(x-\frac{5}{2}\right)^{2}=-\frac{23}{4}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x-\frac{5}{2}\right)^{2}}=\sqrt{-\frac{23}{4}}$$

Qisqartirish.

$$x-\frac{5}{2}=\frac{\sqrt{23}i}{2} x-\frac{5}{2}=-\frac{\sqrt{23}i}{2}$$

\frac{5}{2} ni tenglamaning ikkala tarafiga qo'shish.

$$x=\frac{5+\sqrt{23}i}{2} x=\frac{-\sqrt{23}i+5}{2}$$