7/x-3-10/x-2-6/x-1=0 eching

$$\frac { 7 } { x - 3 } - \frac { 10 } { x - 2 } - \frac { 6 } { x - 1 } = 0$$

x uchun yechish

$$x=4$$

$$x = \frac{13}{9} = 1\frac{4}{9} \approx 1,444444444$$

Guruhlash orqali faktorlash qadamlari

x qiymati 1,2,3 qiymatlaridan birortasiga teng bo‘lmaydi, chunki nolga bo‘lish mumkin emas. Tenglamaning ikkala tarafini \left(x-3\right)\left(x-2\right)\left(x-1\right) ga, x-3,x-2,x-1 ning eng kichik karralisiga ko‘paytiring.

$$\left(x-2\right)\left(x-1\right)\times 7-\left(x-3\right)\left(x-1\right)\times 10-\left(x-3\right)\left(x-2\right)\times 6=0$$

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

$$\left(x^{2}-3x+2\right)\times 7-\left(x-3\right)\left(x-1\right)\times 10-\left(x-3\right)\left(x-2\right)\times 6=0$$

x^{2}-3x+2 ga 7 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$7x^{2}-21x+14-\left(x-3\right)\left(x-1\right)\times 10-\left(x-3\right)\left(x-2\right)\times 6=0$$

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

$$7x^{2}-21x+14-\left(x^{2}-4x+3\right)\times 10-\left(x-3\right)\left(x-2\right)\times 6=0$$

x^{2}-4x+3 ga 10 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$7x^{2}-21x+14-\left(10x^{2}-40x+30\right)-\left(x-3\right)\left(x-2\right)\times 6=0$$

10x^{2}-40x+30 teskarisini topish uchun har birining teskarisini toping.

$$7x^{2}-21x+14-10x^{2}+40x-30-\left(x-3\right)\left(x-2\right)\times 6=0$$

-3x^{2} ni olish uchun 7x^{2} va -10x^{2} ni birlashtirish.

$$-3x^{2}-21x+14+40x-30-\left(x-3\right)\left(x-2\right)\times 6=0$$

19x ni olish uchun -21x va 40x ni birlashtirish.

$$-3x^{2}+19x+14-30-\left(x-3\right)\left(x-2\right)\times 6=0$$

-16 olish uchun 14 dan 30 ni ayirish.

$$-3x^{2}+19x-16-\left(x-3\right)\left(x-2\right)\times 6=0$$

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

$$-3x^{2}+19x-16-\left(x^{2}-5x+6\right)\times 6=0$$

x^{2}-5x+6 ga 6 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$-3x^{2}+19x-16-\left(6x^{2}-30x+36\right)=0$$

6x^{2}-30x+36 teskarisini topish uchun har birining teskarisini toping.

$$-3x^{2}+19x-16-6x^{2}+30x-36=0$$

-9x^{2} ni olish uchun -3x^{2} va -6x^{2} ni birlashtirish.

$$-9x^{2}+19x-16+30x-36=0$$

49x ni olish uchun 19x va 30x ni birlashtirish.

$$-9x^{2}+49x-16-36=0$$

-52 olish uchun -16 dan 36 ni ayirish.

$$-9x^{2}+49x-52=0$$

Tenglamani yechish uchun guruhlash orqali chap qoʻl tomonni faktorlang. Avvalo, chap qoʻl tomon -9x^{2}+ax+bx-52 sifatida qayta yozilishi kerak. a va b ni topish uchun yechiladigan tizimni sozlang.

$$a+b=49 ab=-9\left(-52\right)=468$$

ab musbat boʻlganda, a va b da bir xil belgi bor. a+b musbat boʻlganda, a va b ikkisi ham musbat. 468-mahsulotni beruvchi bunday butun juftliklarni roʻyxat qiling.

$$1,468 2,234 3,156 4,117 6,78 9,52 12,39 13,36 18,26$$

Har bir juftlik yigʻindisini hisoblang.

$$1+468=469 2+234=236 3+156=159 4+117=121 6+78=84 9+52=61 12+39=51 13+36=49 18+26=44$$

Yechim – 49 yigʻindisini beruvchi juftlik.

$$a=36 b=13$$

-9x^{2}+49x-52 ni \left(-9x^{2}+36x\right)+\left(13x-52\right) sifatida qaytadan yozish.

$$\left(-9x^{2}+36x\right)+\left(13x-52\right)$$

Birinchi guruhda 9x ni va ikkinchi guruhda -13 ni faktordan chiqaring.

$$9x\left(-x+4\right)-13\left(-x+4\right)$$

Distributiv funktsiyasidan foydalangan holda -x+4 umumiy terminini chiqaring.

$$\left(-x+4\right)\left(9x-13\right)$$

Tenglamani yechish uchun -x+4=0 va 9x-13=0 ni yeching.

$$x=4 x=\frac{13}{9}$$

Kvadrat tenglamalaridan foydalanish qoidalari

x qiymati 1,2,3 qiymatlaridan birortasiga teng bo‘lmaydi, chunki nolga bo‘lish mumkin emas. Tenglamaning ikkala tarafini \left(x-3\right)\left(x-2\right)\left(x-1\right) ga, x-3,x-2,x-1 ning eng kichik karralisiga ko‘paytiring.

$$\left(x-2\right)\left(x-1\right)\times 7-\left(x-3\right)\left(x-1\right)\times 10-\left(x-3\right)\left(x-2\right)\times 6=0$$

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

$$\left(x^{2}-3x+2\right)\times 7-\left(x-3\right)\left(x-1\right)\times 10-\left(x-3\right)\left(x-2\right)\times 6=0$$

x^{2}-3x+2 ga 7 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$7x^{2}-21x+14-\left(x-3\right)\left(x-1\right)\times 10-\left(x-3\right)\left(x-2\right)\times 6=0$$

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

$$7x^{2}-21x+14-\left(x^{2}-4x+3\right)\times 10-\left(x-3\right)\left(x-2\right)\times 6=0$$

x^{2}-4x+3 ga 10 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$7x^{2}-21x+14-\left(10x^{2}-40x+30\right)-\left(x-3\right)\left(x-2\right)\times 6=0$$

10x^{2}-40x+30 teskarisini topish uchun har birining teskarisini toping.

$$7x^{2}-21x+14-10x^{2}+40x-30-\left(x-3\right)\left(x-2\right)\times 6=0$$

-3x^{2} ni olish uchun 7x^{2} va -10x^{2} ni birlashtirish.

$$-3x^{2}-21x+14+40x-30-\left(x-3\right)\left(x-2\right)\times 6=0$$

19x ni olish uchun -21x va 40x ni birlashtirish.

$$-3x^{2}+19x+14-30-\left(x-3\right)\left(x-2\right)\times 6=0$$

-16 olish uchun 14 dan 30 ni ayirish.

$$-3x^{2}+19x-16-\left(x-3\right)\left(x-2\right)\times 6=0$$

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

$$-3x^{2}+19x-16-\left(x^{2}-5x+6\right)\times 6=0$$

x^{2}-5x+6 ga 6 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$-3x^{2}+19x-16-\left(6x^{2}-30x+36\right)=0$$

6x^{2}-30x+36 teskarisini topish uchun har birining teskarisini toping.

$$-3x^{2}+19x-16-6x^{2}+30x-36=0$$

-9x^{2} ni olish uchun -3x^{2} va -6x^{2} ni birlashtirish.

$$-9x^{2}+19x-16+30x-36=0$$

49x ni olish uchun 19x va 30x ni birlashtirish.

$$-9x^{2}+49x-16-36=0$$

-52 olish uchun -16 dan 36 ni ayirish.

$$-9x^{2}+49x-52=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} -9 ni a, 49 ni b va -52 ni c bilan almashtiring.

$$x=\frac{-49±\sqrt{49^{2}-4\left(-9\right)\left(-52\right)}}{2\left(-9\right)}$$

49 kvadratini chiqarish.

$$x=\frac{-49±\sqrt{2401-4\left(-9\right)\left(-52\right)}}{2\left(-9\right)}$$

-4 ni -9 marotabaga ko'paytirish.

$$x=\frac{-49±\sqrt{2401+36\left(-52\right)}}{2\left(-9\right)}$$

36 ni -52 marotabaga ko'paytirish.

$$x=\frac{-49±\sqrt{2401-1872}}{2\left(-9\right)}$$

2401 ni -1872 ga qo'shish.

$$x=\frac{-49±\sqrt{529}}{2\left(-9\right)}$$

529 ning kvadrat ildizini chiqarish.

$$x=\frac{-49±23}{2\left(-9\right)}$$

2 ni -9 marotabaga ko'paytirish.

$$x=\frac{-49±23}{-18}$$

x=\frac{-49±23}{-18} tenglamasini yeching, bunda ± musbat. -49 ni 23 ga qo'shish.

$$x=-\frac{26}{-18}$$

\frac{-26}{-18} ulushini 2 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

$$x=\frac{13}{9}$$

x=\frac{-49±23}{-18} tenglamasini yeching, bunda ± manfiy. -49 dan 23 ni ayirish.

$$x=-\frac{72}{-18}$$

-72 ni -18 ga bo'lish.

$$x=4$$

Tenglama yechildi.

$$x=\frac{13}{9} x=4$$

Kvadratni yakunlash bo'yicha qadamlar

x qiymati 1,2,3 qiymatlaridan birortasiga teng bo‘lmaydi, chunki nolga bo‘lish mumkin emas. Tenglamaning ikkala tarafini \left(x-3\right)\left(x-2\right)\left(x-1\right) ga, x-3,x-2,x-1 ning eng kichik karralisiga ko‘paytiring.

$$\left(x-2\right)\left(x-1\right)\times 7-\left(x-3\right)\left(x-1\right)\times 10-\left(x-3\right)\left(x-2\right)\times 6=0$$

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

$$\left(x^{2}-3x+2\right)\times 7-\left(x-3\right)\left(x-1\right)\times 10-\left(x-3\right)\left(x-2\right)\times 6=0$$

x^{2}-3x+2 ga 7 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$7x^{2}-21x+14-\left(x-3\right)\left(x-1\right)\times 10-\left(x-3\right)\left(x-2\right)\times 6=0$$

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

$$7x^{2}-21x+14-\left(x^{2}-4x+3\right)\times 10-\left(x-3\right)\left(x-2\right)\times 6=0$$

x^{2}-4x+3 ga 10 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$7x^{2}-21x+14-\left(10x^{2}-40x+30\right)-\left(x-3\right)\left(x-2\right)\times 6=0$$

10x^{2}-40x+30 teskarisini topish uchun har birining teskarisini toping.

$$7x^{2}-21x+14-10x^{2}+40x-30-\left(x-3\right)\left(x-2\right)\times 6=0$$

-3x^{2} ni olish uchun 7x^{2} va -10x^{2} ni birlashtirish.

$$-3x^{2}-21x+14+40x-30-\left(x-3\right)\left(x-2\right)\times 6=0$$

19x ni olish uchun -21x va 40x ni birlashtirish.

$$-3x^{2}+19x+14-30-\left(x-3\right)\left(x-2\right)\times 6=0$$

-16 olish uchun 14 dan 30 ni ayirish.

$$-3x^{2}+19x-16-\left(x-3\right)\left(x-2\right)\times 6=0$$

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

$$-3x^{2}+19x-16-\left(x^{2}-5x+6\right)\times 6=0$$

x^{2}-5x+6 ga 6 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$-3x^{2}+19x-16-\left(6x^{2}-30x+36\right)=0$$

6x^{2}-30x+36 teskarisini topish uchun har birining teskarisini toping.

$$-3x^{2}+19x-16-6x^{2}+30x-36=0$$

-9x^{2} ni olish uchun -3x^{2} va -6x^{2} ni birlashtirish.

$$-9x^{2}+19x-16+30x-36=0$$

49x ni olish uchun 19x va 30x ni birlashtirish.

$$-9x^{2}+49x-16-36=0$$

-52 olish uchun -16 dan 36 ni ayirish.

$$-9x^{2}+49x-52=0$$

52 ni ikki tarafga qo’shing. Har qanday songa nolni qo‘shsangiz, o‘zi chiqadi.

$$-9x^{2}+49x=52$$

Ikki tarafini -9 ga bo‘ling.

$$\frac{-9x^{2}+49x}{-9}=\frac{52}{-9}$$

-9 ga bo'lish -9 ga ko'paytirishni bekor qiladi.

$$x^{2}+\frac{49}{-9}x=\frac{52}{-9}$$

49 ni -9 ga bo'lish.

$$x^{2}-\frac{49}{9}x=\frac{52}{-9}$$

52 ni -9 ga bo'lish.

$$x^{2}-\frac{49}{9}x=-\frac{52}{9}$$

-\frac{49}{9} ni bo‘lish, x shartining koeffitsienti, 2 ga -\frac{49}{18} olish uchun. Keyin, -\frac{49}{18} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}-\frac{49}{9}x+\left(-\frac{49}{18}\right)^{2}=-\frac{52}{9}+\left(-\frac{49}{18}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib -\frac{49}{18} kvadratini chiqarish.

$$x^{2}-\frac{49}{9}x+\frac{2401}{324}=-\frac{52}{9}+\frac{2401}{324}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali -\frac{52}{9} ni \frac{2401}{324} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$x^{2}-\frac{49}{9}x+\frac{2401}{324}=\frac{529}{324}$$

x^{2}-\frac{49}{9}x+\frac{2401}{324} omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(x-\frac{49}{18}\right)^{2}=\frac{529}{324}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x-\frac{49}{18}\right)^{2}}=\sqrt{\frac{529}{324}}$$

Qisqartirish.

$$x-\frac{49}{18}=\frac{23}{18} x-\frac{49}{18}=-\frac{23}{18}$$

\frac{49}{18} ni tenglamaning ikkala tarafiga qo'shish.

$$x=4 x=\frac{13}{9}$$