5/x-3-x-1/x-2=7 eching

$$\frac { 5 } { x - 3 } - \frac { x - 1 } { x - 2 } = 7$$

x uchun yechish

$$x = \frac{\sqrt{11} + 11}{4} \approx 3,579156198$$

$$x = \frac{11 - \sqrt{11}}{4} \approx 1,920843802$$

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.

$$\left(x-2\right)\times 5-\left(x-3\right)\left(x-1\right)=7\left(x-3\right)\left(x-2\right)$$

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

$$5x-10-\left(x-3\right)\left(x-1\right)=7\left(x-3\right)\left(x-2\right)$$

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

$$5x-10-\left(x^{2}-4x+3\right)=7\left(x-3\right)\left(x-2\right)$$

x^{2}-4x+3 teskarisini topish uchun har birining teskarisini toping.

$$5x-10-x^{2}+4x-3=7\left(x-3\right)\left(x-2\right)$$

9x ni olish uchun 5x va 4x ni birlashtirish.

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

-13 olish uchun -10 dan 3 ni ayirish.

$$9x-13-x^{2}=7\left(x-3\right)\left(x-2\right)$$

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

$$9x-13-x^{2}=\left(7x-21\right)\left(x-2\right)$$

7x-21 ga x-2 ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

$$9x-13-x^{2}=7x^{2}-35x+42$$

Ikkala tarafdan 7x^{2} ni ayirish.

$$9x-13-x^{2}-7x^{2}=-35x+42$$

-8x^{2} ni olish uchun -x^{2} va -7x^{2} ni birlashtirish.

$$9x-13-8x^{2}=-35x+42$$

35x ni ikki tarafga qo’shing.

$$9x-13-8x^{2}+35x=42$$

44x ni olish uchun 9x va 35x ni birlashtirish.

$$44x-13-8x^{2}=42$$

Ikkala tarafdan 42 ni ayirish.

$$44x-13-8x^{2}-42=0$$

-55 olish uchun -13 dan 42 ni ayirish.

$$44x-55-8x^{2}=0$$

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.

$$-8x^{2}+44x-55=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} -8 ni a, 44 ni b va -55 ni c bilan almashtiring.

$$x=\frac{-44±\sqrt{44^{2}-4\left(-8\right)\left(-55\right)}}{2\left(-8\right)}$$

44 kvadratini chiqarish.

$$x=\frac{-44±\sqrt{1936-4\left(-8\right)\left(-55\right)}}{2\left(-8\right)}$$

-4 ni -8 marotabaga ko'paytirish.

$$x=\frac{-44±\sqrt{1936+32\left(-55\right)}}{2\left(-8\right)}$$

32 ni -55 marotabaga ko'paytirish.

$$x=\frac{-44±\sqrt{1936-1760}}{2\left(-8\right)}$$

1936 ni -1760 ga qo'shish.

$$x=\frac{-44±\sqrt{176}}{2\left(-8\right)}$$

176 ning kvadrat ildizini chiqarish.

$$x=\frac{-44±4\sqrt{11}}{2\left(-8\right)}$$

2 ni -8 marotabaga ko'paytirish.

$$x=\frac{-44±4\sqrt{11}}{-16}$$

x=\frac{-44±4\sqrt{11}}{-16} tenglamasini yeching, bunda ± musbat. -44 ni 4\sqrt{11} ga qo'shish.

$$x=\frac{4\sqrt{11}-44}{-16}$$

-44+4\sqrt{11} ni -16 ga bo'lish.

$$x=\frac{11-\sqrt{11}}{4}$$

x=\frac{-44±4\sqrt{11}}{-16} tenglamasini yeching, bunda ± manfiy. -44 dan 4\sqrt{11} ni ayirish.

$$x=\frac{-4\sqrt{11}-44}{-16}$$

-44-4\sqrt{11} ni -16 ga bo'lish.

$$x=\frac{\sqrt{11}+11}{4}$$

Tenglama yechildi.

$$x=\frac{11-\sqrt{11}}{4} x=\frac{\sqrt{11}+11}{4}$$

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.

$$\left(x-2\right)\times 5-\left(x-3\right)\left(x-1\right)=7\left(x-3\right)\left(x-2\right)$$

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

$$5x-10-\left(x-3\right)\left(x-1\right)=7\left(x-3\right)\left(x-2\right)$$

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

$$5x-10-\left(x^{2}-4x+3\right)=7\left(x-3\right)\left(x-2\right)$$

x^{2}-4x+3 teskarisini topish uchun har birining teskarisini toping.

$$5x-10-x^{2}+4x-3=7\left(x-3\right)\left(x-2\right)$$

9x ni olish uchun 5x va 4x ni birlashtirish.

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

-13 olish uchun -10 dan 3 ni ayirish.

$$9x-13-x^{2}=7\left(x-3\right)\left(x-2\right)$$

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

$$9x-13-x^{2}=\left(7x-21\right)\left(x-2\right)$$

7x-21 ga x-2 ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

$$9x-13-x^{2}=7x^{2}-35x+42$$

Ikkala tarafdan 7x^{2} ni ayirish.

$$9x-13-x^{2}-7x^{2}=-35x+42$$

-8x^{2} ni olish uchun -x^{2} va -7x^{2} ni birlashtirish.

$$9x-13-8x^{2}=-35x+42$$

35x ni ikki tarafga qo’shing.

$$9x-13-8x^{2}+35x=42$$

44x ni olish uchun 9x va 35x ni birlashtirish.

$$44x-13-8x^{2}=42$$

13 ni ikki tarafga qo’shing.

$$44x-8x^{2}=42+13$$

55 olish uchun 42 va 13'ni qo'shing.

$$44x-8x^{2}=55$$

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

$$-8x^{2}+44x=55$$

Ikki tarafini -8 ga bo‘ling.

$$\frac{-8x^{2}+44x}{-8}=\frac{55}{-8}$$

-8 ga bo'lish -8 ga ko'paytirishni bekor qiladi.

$$x^{2}+\frac{44}{-8}x=\frac{55}{-8}$$

\frac{44}{-8} ulushini 4 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

$$x^{2}-\frac{11}{2}x=\frac{55}{-8}$$

55 ni -8 ga bo'lish.

$$x^{2}-\frac{11}{2}x=-\frac{55}{8}$$

-\frac{11}{2} ni bo‘lish, x shartining koeffitsienti, 2 ga -\frac{11}{4} olish uchun. Keyin, -\frac{11}{4} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}-\frac{11}{2}x+\left(-\frac{11}{4}\right)^{2}=-\frac{55}{8}+\left(-\frac{11}{4}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib -\frac{11}{4} kvadratini chiqarish.

$$x^{2}-\frac{11}{2}x+\frac{121}{16}=-\frac{55}{8}+\frac{121}{16}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali -\frac{55}{8} ni \frac{121}{16} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$x^{2}-\frac{11}{2}x+\frac{121}{16}=\frac{11}{16}$$

x^{2}-\frac{11}{2}x+\frac{121}{16} omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(x-\frac{11}{4}\right)^{2}=\frac{11}{16}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x-\frac{11}{4}\right)^{2}}=\sqrt{\frac{11}{16}}$$

Qisqartirish.

$$x-\frac{11}{4}=\frac{\sqrt{11}}{4} x-\frac{11}{4}=-\frac{\sqrt{11}}{4}$$

\frac{11}{4} ni tenglamaning ikkala tarafiga qo'shish.

$$x=\frac{\sqrt{11}+11}{4} x=\frac{11-\sqrt{11}}{4}$$