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

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

x uchun yechish

$$x = \frac{28}{9} = 3\frac{1}{9} \approx 3,111111111$$

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.

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

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

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

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

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

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

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

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

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

-9x^{2} ni olish uchun x^{2} va -10x^{2} ni birlashtirish.

$$-9x^{2}-3x+2+40x-30-\left(x-3\right)\left(x-2\right)\times 0=0$$

37x ni olish uchun -3x va 40x ni birlashtirish.

$$-9x^{2}+37x+2-30-\left(x-3\right)\left(x-2\right)\times 0=0$$

-28 olish uchun 2 dan 30 ni ayirish.

$$-9x^{2}+37x-28-\left(x-3\right)\left(x-2\right)\times 0=0$$

Har qanday sonni nolga ko‘paytirsangiz, nol chiqadi.

$$-9x^{2}+37x-28+0=0$$

-28 olish uchun -28 va 0'ni qo'shing.

$$-9x^{2}+37x-28=0$$

Tenglamani yechish uchun guruhlash orqali chap qoʻl tomonni faktorlang. Avvalo, chap qoʻl tomon -9x^{2}+ax+bx-28 sifatida qayta yozilishi kerak. a va b ni topish uchun yechiladigan tizimni sozlang.

$$a+b=37 ab=-9\left(-28\right)=252$$

ab musbat boʻlganda, a va b da bir xil belgi bor. a+b musbat boʻlganda, a va b ikkisi ham musbat. 252-mahsulotni beruvchi bunday butun juftliklarni roʻyxat qiling.

$$1,252 2,126 3,84 4,63 6,42 7,36 9,28 12,21 14,18$$

Har bir juftlik yigʻindisini hisoblang.

$$1+252=253 2+126=128 3+84=87 4+63=67 6+42=48 7+36=43 9+28=37 12+21=33 14+18=32$$

Yechim – 37 yigʻindisini beruvchi juftlik.

$$a=28 b=9$$

-9x^{2}+37x-28 ni \left(-9x^{2}+28x\right)+\left(9x-28\right) sifatida qaytadan yozish.

$$\left(-9x^{2}+28x\right)+\left(9x-28\right)$$

-9x^{2}+28x ichida -x ni ajrating.

$$-x\left(9x-28\right)+9x-28$$

Distributiv funktsiyasidan foydalangan holda 9x-28 umumiy terminini chiqaring.

$$\left(9x-28\right)\left(-x+1\right)$$

Tenglamani yechish uchun 9x-28=0 va -x+1=0 ni yeching.

$$x=\frac{28}{9} x=1$$

x qiymati 1 teng bo‘lmaydi.

$$x=\frac{28}{9}$$

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.

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

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

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

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

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

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

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

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

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

-9x^{2} ni olish uchun x^{2} va -10x^{2} ni birlashtirish.

$$-9x^{2}-3x+2+40x-30-\left(x-3\right)\left(x-2\right)\times 0=0$$

37x ni olish uchun -3x va 40x ni birlashtirish.

$$-9x^{2}+37x+2-30-\left(x-3\right)\left(x-2\right)\times 0=0$$

-28 olish uchun 2 dan 30 ni ayirish.

$$-9x^{2}+37x-28-\left(x-3\right)\left(x-2\right)\times 0=0$$

Har qanday sonni nolga ko‘paytirsangiz, nol chiqadi.

$$-9x^{2}+37x-28+0=0$$

-28 olish uchun -28 va 0'ni qo'shing.

$$-9x^{2}+37x-28=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} -9 ni a, 37 ni b va -28 ni c bilan almashtiring.

$$x=\frac{-37±\sqrt{37^{2}-4\left(-9\right)\left(-28\right)}}{2\left(-9\right)}$$

37 kvadratini chiqarish.

$$x=\frac{-37±\sqrt{1369-4\left(-9\right)\left(-28\right)}}{2\left(-9\right)}$$

-4 ni -9 marotabaga ko'paytirish.

$$x=\frac{-37±\sqrt{1369+36\left(-28\right)}}{2\left(-9\right)}$$

36 ni -28 marotabaga ko'paytirish.

$$x=\frac{-37±\sqrt{1369-1008}}{2\left(-9\right)}$$

1369 ni -1008 ga qo'shish.

$$x=\frac{-37±\sqrt{361}}{2\left(-9\right)}$$

361 ning kvadrat ildizini chiqarish.

$$x=\frac{-37±19}{2\left(-9\right)}$$

2 ni -9 marotabaga ko'paytirish.

$$x=\frac{-37±19}{-18}$$

x=\frac{-37±19}{-18} tenglamasini yeching, bunda ± musbat. -37 ni 19 ga qo'shish.

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

-18 ni -18 ga bo'lish.

$$x=1$$

x=\frac{-37±19}{-18} tenglamasini yeching, bunda ± manfiy. -37 dan 19 ni ayirish.

$$x=-\frac{56}{-18}$$

\frac{-56}{-18} ulushini 2 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

$$x=\frac{28}{9}$$

Tenglama yechildi.

$$x=1 x=\frac{28}{9}$$

x qiymati 1 teng bo‘lmaydi.

$$x=\frac{28}{9}$$

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.

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

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

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

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

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

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

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

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

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

-9x^{2} ni olish uchun x^{2} va -10x^{2} ni birlashtirish.

$$-9x^{2}-3x+2+40x-30-\left(x-3\right)\left(x-2\right)\times 0=0$$

37x ni olish uchun -3x va 40x ni birlashtirish.

$$-9x^{2}+37x+2-30-\left(x-3\right)\left(x-2\right)\times 0=0$$

-28 olish uchun 2 dan 30 ni ayirish.

$$-9x^{2}+37x-28-\left(x-3\right)\left(x-2\right)\times 0=0$$

Har qanday sonni nolga ko‘paytirsangiz, nol chiqadi.

$$-9x^{2}+37x-28+0=0$$

-28 olish uchun -28 va 0'ni qo'shing.

$$-9x^{2}+37x-28=0$$

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

$$-9x^{2}+37x=28$$

Ikki tarafini -9 ga bo‘ling.

$$\frac{-9x^{2}+37x}{-9}=\frac{28}{-9}$$

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

$$x^{2}+\frac{37}{-9}x=\frac{28}{-9}$$

37 ni -9 ga bo'lish.

$$x^{2}-\frac{37}{9}x=\frac{28}{-9}$$

28 ni -9 ga bo'lish.

$$x^{2}-\frac{37}{9}x=-\frac{28}{9}$$

-\frac{37}{9} ni bo‘lish, x shartining koeffitsienti, 2 ga -\frac{37}{18} olish uchun. Keyin, -\frac{37}{18} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}-\frac{37}{9}x+\left(-\frac{37}{18}\right)^{2}=-\frac{28}{9}+\left(-\frac{37}{18}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib -\frac{37}{18} kvadratini chiqarish.

$$x^{2}-\frac{37}{9}x+\frac{1369}{324}=-\frac{28}{9}+\frac{1369}{324}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali -\frac{28}{9} ni \frac{1369}{324} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$x^{2}-\frac{37}{9}x+\frac{1369}{324}=\frac{361}{324}$$

x^{2}-\frac{37}{9}x+\frac{1369}{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.

$$\left(x-\frac{37}{18}\right)^{2}=\frac{361}{324}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x-\frac{37}{18}\right)^{2}}=\sqrt{\frac{361}{324}}$$

Qisqartirish.

$$x-\frac{37}{18}=\frac{19}{18} x-\frac{37}{18}=-\frac{19}{18}$$

\frac{37}{18} ni tenglamaning ikkala tarafiga qo'shish.

$$x=\frac{28}{9} x=1$$

x qiymati 1 teng bo‘lmaydi.

$$x=\frac{28}{9}$$