1/9x^2-6x+1+16/15x^2+25x-10=1/3x-1 eching

$$\frac { 1 } { 9 x ^ { 2 } - 6 x + 1 } + \frac { 16 } { 15 x ^ { 2 } + 25 x - 10 } = \frac { 1 } { 3 x - 1 }$$

x uchun yechish

$$x=-\frac{2}{15}\approx -0,133333333$$

$$x=2$$

Guruhlash orqali faktorlash qadamlari

x qiymati -2,\frac{1}{3} qiymatlaridan birortasiga teng bo‘lmaydi, chunki nolga bo‘lish mumkin emas. Tenglamaning ikkala tarafini 5\left(x+2\right)\left(3x-1\right)^{2} ga, 9x^{2}-6x+1,15x^{2}+25x-10,3x-1 ning eng kichik karralisiga ko‘paytiring.

$$5x+10+\left(3x-1\right)\times 16=5\left(x+2\right)\left(3x-1\right)$$

3x-1 ga 16 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$5x+10+48x-16=5\left(x+2\right)\left(3x-1\right)$$

53x ni olish uchun 5x va 48x ni birlashtirish.

$$53x+10-16=5\left(x+2\right)\left(3x-1\right)$$

-6 olish uchun 10 dan 16 ni ayirish.

$$53x-6=5\left(x+2\right)\left(3x-1\right)$$

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

$$53x-6=\left(5x+10\right)\left(3x-1\right)$$

5x+10 ga 3x-1 ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

$$53x-6=15x^{2}+25x-10$$

Ikkala tarafdan 15x^{2} ni ayirish.

$$53x-6-15x^{2}=25x-10$$

Ikkala tarafdan 25x ni ayirish.

$$53x-6-15x^{2}-25x=-10$$

28x ni olish uchun 53x va -25x ni birlashtirish.

$$28x-6-15x^{2}=-10$$

10 ni ikki tarafga qo’shing.

$$28x-6-15x^{2}+10=0$$

4 olish uchun -6 va 10'ni qo'shing.

$$28x+4-15x^{2}=0$$

Polinomni standart shaklga keltirish uchun uni qayta tartiblang. Shartlarni eng yuqoridan eng pastki qiymat ko'rsatgichiga joylashtirish.

$$-15x^{2}+28x+4=0$$

Tenglamani yechish uchun guruhlash orqali chap qoʻl tomonni faktorlang. Avvalo, chap qoʻl tomon -15x^{2}+ax+bx+4 sifatida qayta yozilishi kerak. a va b ni topish uchun yechiladigan tizimni sozlang.

$$a+b=28 ab=-15\times 4=-60$$

ab manfiy boʻlganda, a va b da qarama-qarshi belgilar bor. a+b musbat boʻlganda, musbat sonda manfiyga nisbatdan kattaroq mutlaq qiymat bor. -60-mahsulotni beruvchi bunday butun juftliklarni roʻyxat qiling.

$$-1,60 -2,30 -3,20 -4,15 -5,12 -6,10$$

Har bir juftlik yigʻindisini hisoblang.

$$-1+60=59 -2+30=28 -3+20=17 -4+15=11 -5+12=7 -6+10=4$$

Yechim – 28 yigʻindisini beruvchi juftlik.

$$a=30 b=-2$$

-15x^{2}+28x+4 ni \left(-15x^{2}+30x\right)+\left(-2x+4\right) sifatida qaytadan yozish.

$$\left(-15x^{2}+30x\right)+\left(-2x+4\right)$$

Birinchi guruhda 15x ni va ikkinchi guruhda 2 ni faktordan chiqaring.

$$15x\left(-x+2\right)+2\left(-x+2\right)$$

Distributiv funktsiyasidan foydalangan holda -x+2 umumiy terminini chiqaring.

$$\left(-x+2\right)\left(15x+2\right)$$

Tenglamani yechish uchun -x+2=0 va 15x+2=0 ni yeching.

$$x=2 x=-\frac{2}{15}$$

Kvadrat tenglamalaridan foydalanish qoidalari

x qiymati -2,\frac{1}{3} qiymatlaridan birortasiga teng bo‘lmaydi, chunki nolga bo‘lish mumkin emas. Tenglamaning ikkala tarafini 5\left(x+2\right)\left(3x-1\right)^{2} ga, 9x^{2}-6x+1,15x^{2}+25x-10,3x-1 ning eng kichik karralisiga ko‘paytiring.

$$5x+10+\left(3x-1\right)\times 16=5\left(x+2\right)\left(3x-1\right)$$

3x-1 ga 16 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$5x+10+48x-16=5\left(x+2\right)\left(3x-1\right)$$

53x ni olish uchun 5x va 48x ni birlashtirish.

$$53x+10-16=5\left(x+2\right)\left(3x-1\right)$$

-6 olish uchun 10 dan 16 ni ayirish.

$$53x-6=5\left(x+2\right)\left(3x-1\right)$$

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

$$53x-6=\left(5x+10\right)\left(3x-1\right)$$

5x+10 ga 3x-1 ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

$$53x-6=15x^{2}+25x-10$$

Ikkala tarafdan 15x^{2} ni ayirish.

$$53x-6-15x^{2}=25x-10$$

Ikkala tarafdan 25x ni ayirish.

$$53x-6-15x^{2}-25x=-10$$

28x ni olish uchun 53x va -25x ni birlashtirish.

$$28x-6-15x^{2}=-10$$

10 ni ikki tarafga qo’shing.

$$28x-6-15x^{2}+10=0$$

4 olish uchun -6 va 10'ni qo'shing.

$$28x+4-15x^{2}=0$$

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.

$$-15x^{2}+28x+4=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} -15 ni a, 28 ni b va 4 ni c bilan almashtiring.

$$x=\frac{-28±\sqrt{28^{2}-4\left(-15\right)\times 4}}{2\left(-15\right)}$$

28 kvadratini chiqarish.

$$x=\frac{-28±\sqrt{784-4\left(-15\right)\times 4}}{2\left(-15\right)}$$

-4 ni -15 marotabaga ko'paytirish.

$$x=\frac{-28±\sqrt{784+60\times 4}}{2\left(-15\right)}$$

60 ni 4 marotabaga ko'paytirish.

$$x=\frac{-28±\sqrt{784+240}}{2\left(-15\right)}$$

784 ni 240 ga qo'shish.

$$x=\frac{-28±\sqrt{1024}}{2\left(-15\right)}$$

1024 ning kvadrat ildizini chiqarish.

$$x=\frac{-28±32}{2\left(-15\right)}$$

2 ni -15 marotabaga ko'paytirish.

$$x=\frac{-28±32}{-30}$$

x=\frac{-28±32}{-30} tenglamasini yeching, bunda ± musbat. -28 ni 32 ga qo'shish.

$$x=\frac{4}{-30}$$

\frac{4}{-30} ulushini 2 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

$$x=-\frac{2}{15}$$

x=\frac{-28±32}{-30} tenglamasini yeching, bunda ± manfiy. -28 dan 32 ni ayirish.

$$x=-\frac{60}{-30}$$

-60 ni -30 ga bo'lish.

$$x=2$$

Tenglama yechildi.

$$x=-\frac{2}{15} x=2$$

Kvadratni yakunlash bo'yicha qadamlar

x qiymati -2,\frac{1}{3} qiymatlaridan birortasiga teng bo‘lmaydi, chunki nolga bo‘lish mumkin emas. Tenglamaning ikkala tarafini 5\left(x+2\right)\left(3x-1\right)^{2} ga, 9x^{2}-6x+1,15x^{2}+25x-10,3x-1 ning eng kichik karralisiga ko‘paytiring.

$$5x+10+\left(3x-1\right)\times 16=5\left(x+2\right)\left(3x-1\right)$$

3x-1 ga 16 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$5x+10+48x-16=5\left(x+2\right)\left(3x-1\right)$$

53x ni olish uchun 5x va 48x ni birlashtirish.

$$53x+10-16=5\left(x+2\right)\left(3x-1\right)$$

-6 olish uchun 10 dan 16 ni ayirish.

$$53x-6=5\left(x+2\right)\left(3x-1\right)$$

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

$$53x-6=\left(5x+10\right)\left(3x-1\right)$$

5x+10 ga 3x-1 ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

$$53x-6=15x^{2}+25x-10$$

Ikkala tarafdan 15x^{2} ni ayirish.

$$53x-6-15x^{2}=25x-10$$

Ikkala tarafdan 25x ni ayirish.

$$53x-6-15x^{2}-25x=-10$$

28x ni olish uchun 53x va -25x ni birlashtirish.

$$28x-6-15x^{2}=-10$$

6 ni ikki tarafga qo’shing.

$$28x-15x^{2}=-10+6$$

-4 olish uchun -10 va 6'ni qo'shing.

$$28x-15x^{2}=-4$$

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

$$-15x^{2}+28x=-4$$

Ikki tarafini -15 ga bo‘ling.

$$\frac{-15x^{2}+28x}{-15}=-\frac{4}{-15}$$

-15 ga bo'lish -15 ga ko'paytirishni bekor qiladi.

$$x^{2}+\frac{28}{-15}x=-\frac{4}{-15}$$

28 ni -15 ga bo'lish.

$$x^{2}-\frac{28}{15}x=-\frac{4}{-15}$$

-4 ni -15 ga bo'lish.

$$x^{2}-\frac{28}{15}x=\frac{4}{15}$$

-\frac{28}{15} ni bo‘lish, x shartining koeffitsienti, 2 ga -\frac{14}{15} olish uchun. Keyin, -\frac{14}{15} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}-\frac{28}{15}x+\left(-\frac{14}{15}\right)^{2}=\frac{4}{15}+\left(-\frac{14}{15}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib -\frac{14}{15} kvadratini chiqarish.

$$x^{2}-\frac{28}{15}x+\frac{196}{225}=\frac{4}{15}+\frac{196}{225}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali \frac{4}{15} ni \frac{196}{225} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$x^{2}-\frac{28}{15}x+\frac{196}{225}=\frac{256}{225}$$

x^{2}-\frac{28}{15}x+\frac{196}{225} omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(x-\frac{14}{15}\right)^{2}=\frac{256}{225}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x-\frac{14}{15}\right)^{2}}=\sqrt{\frac{256}{225}}$$

Qisqartirish.

$$x-\frac{14}{15}=\frac{16}{15} x-\frac{14}{15}=-\frac{16}{15}$$

\frac{14}{15} ni tenglamaning ikkala tarafiga qo'shish.

$$x=2 x=-\frac{2}{15}$$