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

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

x uchun yechish

$$x = \frac{\sqrt{7} + 10}{3} \approx 4,215250437$$

$$x = \frac{10 - \sqrt{7}}{3} \approx 2,45141623$$

Kvadrat tenglamalaridan foydalanish qoidalari

x qiymati 2,3 qiymatlaridan birortasiga teng bo‘lmaydi, chunki nolga bo‘lish mumkin emas. Tenglamaning ikkala tarafini \left(x-3\right)\left(x-2\right) ga, x-3,x-2 ning eng kichik karralisiga ko‘paytiring.

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

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

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

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

$$2x-4+3x-9=3\left(x-3\right)\left(x-2\right)$$

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

$$5x-4-9=3\left(x-3\right)\left(x-2\right)$$

-13 olish uchun -4 dan 9 ni ayirish.

$$5x-13=3\left(x-3\right)\left(x-2\right)$$

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

$$5x-13=\left(3x-9\right)\left(x-2\right)$$

3x-9 ga x-2 ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

$$5x-13=3x^{2}-15x+18$$

Ikkala tarafdan 3x^{2} ni ayirish.

$$5x-13-3x^{2}=-15x+18$$

15x ni ikki tarafga qo’shing.

$$5x-13-3x^{2}+15x=18$$

20x ni olish uchun 5x va 15x ni birlashtirish.

$$20x-13-3x^{2}=18$$

Ikkala tarafdan 18 ni ayirish.

$$20x-13-3x^{2}-18=0$$

-31 olish uchun -13 dan 18 ni ayirish.

$$20x-31-3x^{2}=0$$

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.

$$-3x^{2}+20x-31=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} -3 ni a, 20 ni b va -31 ni c bilan almashtiring.

$$x=\frac{-20±\sqrt{20^{2}-4\left(-3\right)\left(-31\right)}}{2\left(-3\right)}$$

20 kvadratini chiqarish.

$$x=\frac{-20±\sqrt{400-4\left(-3\right)\left(-31\right)}}{2\left(-3\right)}$$

-4 ni -3 marotabaga ko'paytirish.

$$x=\frac{-20±\sqrt{400+12\left(-31\right)}}{2\left(-3\right)}$$

12 ni -31 marotabaga ko'paytirish.

$$x=\frac{-20±\sqrt{400-372}}{2\left(-3\right)}$$

400 ni -372 ga qo'shish.

$$x=\frac{-20±\sqrt{28}}{2\left(-3\right)}$$

28 ning kvadrat ildizini chiqarish.

$$x=\frac{-20±2\sqrt{7}}{2\left(-3\right)}$$

2 ni -3 marotabaga ko'paytirish.

$$x=\frac{-20±2\sqrt{7}}{-6}$$

x=\frac{-20±2\sqrt{7}}{-6} tenglamasini yeching, bunda ± musbat. -20 ni 2\sqrt{7} ga qo'shish.

$$x=\frac{2\sqrt{7}-20}{-6}$$

-20+2\sqrt{7} ni -6 ga bo'lish.

$$x=\frac{10-\sqrt{7}}{3}$$

x=\frac{-20±2\sqrt{7}}{-6} tenglamasini yeching, bunda ± manfiy. -20 dan 2\sqrt{7} ni ayirish.

$$x=\frac{-2\sqrt{7}-20}{-6}$$

-20-2\sqrt{7} ni -6 ga bo'lish.

$$x=\frac{\sqrt{7}+10}{3}$$

Tenglama yechildi.

$$x=\frac{10-\sqrt{7}}{3} x=\frac{\sqrt{7}+10}{3}$$

Kvadratni yakunlash bo'yicha qadamlar

x qiymati 2,3 qiymatlaridan birortasiga teng bo‘lmaydi, chunki nolga bo‘lish mumkin emas. Tenglamaning ikkala tarafini \left(x-3\right)\left(x-2\right) ga, x-3,x-2 ning eng kichik karralisiga ko‘paytiring.

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

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

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

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

$$2x-4+3x-9=3\left(x-3\right)\left(x-2\right)$$

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

$$5x-4-9=3\left(x-3\right)\left(x-2\right)$$

-13 olish uchun -4 dan 9 ni ayirish.

$$5x-13=3\left(x-3\right)\left(x-2\right)$$

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

$$5x-13=\left(3x-9\right)\left(x-2\right)$$

3x-9 ga x-2 ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

$$5x-13=3x^{2}-15x+18$$

Ikkala tarafdan 3x^{2} ni ayirish.

$$5x-13-3x^{2}=-15x+18$$

15x ni ikki tarafga qo’shing.

$$5x-13-3x^{2}+15x=18$$

20x ni olish uchun 5x va 15x ni birlashtirish.

$$20x-13-3x^{2}=18$$

13 ni ikki tarafga qo’shing.

$$20x-3x^{2}=18+13$$

31 olish uchun 18 va 13'ni qo'shing.

$$20x-3x^{2}=31$$

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

$$-3x^{2}+20x=31$$

Ikki tarafini -3 ga bo‘ling.

$$\frac{-3x^{2}+20x}{-3}=\frac{31}{-3}$$

-3 ga bo'lish -3 ga ko'paytirishni bekor qiladi.

$$x^{2}+\frac{20}{-3}x=\frac{31}{-3}$$

20 ni -3 ga bo'lish.

$$x^{2}-\frac{20}{3}x=\frac{31}{-3}$$

31 ni -3 ga bo'lish.

$$x^{2}-\frac{20}{3}x=-\frac{31}{3}$$

-\frac{20}{3} ni bo‘lish, x shartining koeffitsienti, 2 ga -\frac{10}{3} olish uchun. Keyin, -\frac{10}{3} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}-\frac{20}{3}x+\left(-\frac{10}{3}\right)^{2}=-\frac{31}{3}+\left(-\frac{10}{3}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib -\frac{10}{3} kvadratini chiqarish.

$$x^{2}-\frac{20}{3}x+\frac{100}{9}=-\frac{31}{3}+\frac{100}{9}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali -\frac{31}{3} ni \frac{100}{9} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$x^{2}-\frac{20}{3}x+\frac{100}{9}=\frac{7}{9}$$

x^{2}-\frac{20}{3}x+\frac{100}{9} omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(x-\frac{10}{3}\right)^{2}=\frac{7}{9}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x-\frac{10}{3}\right)^{2}}=\sqrt{\frac{7}{9}}$$

Qisqartirish.

$$x-\frac{10}{3}=\frac{\sqrt{7}}{3} x-\frac{10}{3}=-\frac{\sqrt{7}}{3}$$

\frac{10}{3} ni tenglamaning ikkala tarafiga qo'shish.

$$x=\frac{\sqrt{7}+10}{3} x=\frac{10-\sqrt{7}}{3}$$