3x/2x+1=x-19/2x^2+3x+1+x+5/x+1 eching

$$\frac { 3 x } { 2 x + 1 } = \frac { x - 19 } { 2 x ^ { 2 } + 3 x + 1 } + \frac { x + 5 } { x + 1 }$$

x uchun yechish

$$x=2$$

$$x=7$$

Faktorlash qadamlari

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

$$\left(x+1\right)\times 3x=x-19+\left(2x+1\right)\left(x+5\right)$$

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

$$\left(3x+3\right)x=x-19+\left(2x+1\right)\left(x+5\right)$$

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

$$3x^{2}+3x=x-19+\left(2x+1\right)\left(x+5\right)$$

2x+1 ga x+5 ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

$$3x^{2}+3x=x-19+2x^{2}+11x+5$$

12x ni olish uchun x va 11x ni birlashtirish.

$$3x^{2}+3x=12x-19+2x^{2}+5$$

-14 olish uchun -19 va 5'ni qo'shing.

$$3x^{2}+3x=12x-14+2x^{2}$$

Ikkala tarafdan 12x ni ayirish.

$$3x^{2}+3x-12x=-14+2x^{2}$$

-9x ni olish uchun 3x va -12x ni birlashtirish.

$$3x^{2}-9x=-14+2x^{2}$$

Ikkala tarafdan -14 ni ayirish.

$$3x^{2}-9x-\left(-14\right)=2x^{2}$$

-14 ning teskarisi 14 ga teng.

$$3x^{2}-9x+14=2x^{2}$$

Ikkala tarafdan 2x^{2} ni ayirish.

$$3x^{2}-9x+14-2x^{2}=0$$

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

$$x^{2}-9x+14=0$$

Bu tenglamani yechish uchun x^{2}+\left(a+b\right)x+ab=\left(x+a\right)\left(x+b\right) formulasi yordamida x^{2}-9x+14 ni faktorlang. a va b ni topish uchun yechiladigan tizimni sozlang.

$$a+b=-9 ab=14$$

ab musbat boʻlganda, a va b da bir xil belgi bor. a+b manfiy boʻlganda, a va b ikkisi ham manfiy. 14-mahsulotni beruvchi bunday butun juftliklarni roʻyxat qiling.

$$-1,-14 -2,-7$$

Har bir juftlik yigʻindisini hisoblang.

$$-1-14=-15 -2-7=-9$$

Yechim – -9 yigʻindisini beruvchi juftlik.

$$a=-7 b=-2$$

Faktorlangan \left(x+a\right)\left(x+b\right) ifodani olingan qiymatlar bilan qaytadan yozing.

$$\left(x-7\right)\left(x-2\right)$$

Tenglamani yechish uchun x-7=0 va x-2=0 ni yeching.

$$x=7 x=2$$

Guruhlash orqali faktorlash qadamlari

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

$$\left(x+1\right)\times 3x=x-19+\left(2x+1\right)\left(x+5\right)$$

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

$$\left(3x+3\right)x=x-19+\left(2x+1\right)\left(x+5\right)$$

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

$$3x^{2}+3x=x-19+\left(2x+1\right)\left(x+5\right)$$

2x+1 ga x+5 ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

$$3x^{2}+3x=x-19+2x^{2}+11x+5$$

12x ni olish uchun x va 11x ni birlashtirish.

$$3x^{2}+3x=12x-19+2x^{2}+5$$

-14 olish uchun -19 va 5'ni qo'shing.

$$3x^{2}+3x=12x-14+2x^{2}$$

Ikkala tarafdan 12x ni ayirish.

$$3x^{2}+3x-12x=-14+2x^{2}$$

-9x ni olish uchun 3x va -12x ni birlashtirish.

$$3x^{2}-9x=-14+2x^{2}$$

Ikkala tarafdan -14 ni ayirish.

$$3x^{2}-9x-\left(-14\right)=2x^{2}$$

-14 ning teskarisi 14 ga teng.

$$3x^{2}-9x+14=2x^{2}$$

Ikkala tarafdan 2x^{2} ni ayirish.

$$3x^{2}-9x+14-2x^{2}=0$$

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

$$x^{2}-9x+14=0$$

Tenglamani yechish uchun guruhlash orqali chap qoʻl tomonni faktorlang. Avvalo, chap qoʻl tomon x^{2}+ax+bx+14 sifatida qayta yozilishi kerak. a va b ni topish uchun yechiladigan tizimni sozlang.

$$a+b=-9 ab=1\times 14=14$$

ab musbat boʻlganda, a va b da bir xil belgi bor. a+b manfiy boʻlganda, a va b ikkisi ham manfiy. 14-mahsulotni beruvchi bunday butun juftliklarni roʻyxat qiling.

$$-1,-14 -2,-7$$

Har bir juftlik yigʻindisini hisoblang.

$$-1-14=-15 -2-7=-9$$

Yechim – -9 yigʻindisini beruvchi juftlik.

$$a=-7 b=-2$$

x^{2}-9x+14 ni \left(x^{2}-7x\right)+\left(-2x+14\right) sifatida qaytadan yozish.

$$\left(x^{2}-7x\right)+\left(-2x+14\right)$$

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

$$x\left(x-7\right)-2\left(x-7\right)$$

Distributiv funktsiyasidan foydalangan holda x-7 umumiy terminini chiqaring.

$$\left(x-7\right)\left(x-2\right)$$

Tenglamani yechish uchun x-7=0 va x-2=0 ni yeching.

$$x=7 x=2$$

Kvadrat tenglamalaridan foydalanish qoidalari

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

$$\left(x+1\right)\times 3x=x-19+\left(2x+1\right)\left(x+5\right)$$

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

$$\left(3x+3\right)x=x-19+\left(2x+1\right)\left(x+5\right)$$

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

$$3x^{2}+3x=x-19+\left(2x+1\right)\left(x+5\right)$$

2x+1 ga x+5 ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

$$3x^{2}+3x=x-19+2x^{2}+11x+5$$

12x ni olish uchun x va 11x ni birlashtirish.

$$3x^{2}+3x=12x-19+2x^{2}+5$$

-14 olish uchun -19 va 5'ni qo'shing.

$$3x^{2}+3x=12x-14+2x^{2}$$

Ikkala tarafdan 12x ni ayirish.

$$3x^{2}+3x-12x=-14+2x^{2}$$

-9x ni olish uchun 3x va -12x ni birlashtirish.

$$3x^{2}-9x=-14+2x^{2}$$

Ikkala tarafdan -14 ni ayirish.

$$3x^{2}-9x-\left(-14\right)=2x^{2}$$

-14 ning teskarisi 14 ga teng.

$$3x^{2}-9x+14=2x^{2}$$

Ikkala tarafdan 2x^{2} ni ayirish.

$$3x^{2}-9x+14-2x^{2}=0$$

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

$$x^{2}-9x+14=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 1 ni a, -9 ni b va 14 ni c bilan almashtiring.

$$x=\frac{-\left(-9\right)±\sqrt{\left(-9\right)^{2}-4\times 14}}{2}$$

-9 kvadratini chiqarish.

$$x=\frac{-\left(-9\right)±\sqrt{81-4\times 14}}{2}$$

-4 ni 14 marotabaga ko'paytirish.

$$x=\frac{-\left(-9\right)±\sqrt{81-56}}{2}$$

81 ni -56 ga qo'shish.

$$x=\frac{-\left(-9\right)±\sqrt{25}}{2}$$

25 ning kvadrat ildizini chiqarish.

$$x=\frac{-\left(-9\right)±5}{2}$$

-9 ning teskarisi 9 ga teng.

$$x=\frac{9±5}{2}$$

x=\frac{9±5}{2} tenglamasini yeching, bunda ± musbat. 9 ni 5 ga qo'shish.

$$x=\frac{14}{2}$$

14 ni 2 ga bo'lish.

$$x=7$$

x=\frac{9±5}{2} tenglamasini yeching, bunda ± manfiy. 9 dan 5 ni ayirish.

$$x=\frac{4}{2}$$

4 ni 2 ga bo'lish.

$$x=2$$

Tenglama yechildi.

$$x=7 x=2$$

Kvadratni yakunlash bo'yicha qadamlar

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

$$\left(x+1\right)\times 3x=x-19+\left(2x+1\right)\left(x+5\right)$$

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

$$\left(3x+3\right)x=x-19+\left(2x+1\right)\left(x+5\right)$$

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

$$3x^{2}+3x=x-19+\left(2x+1\right)\left(x+5\right)$$

2x+1 ga x+5 ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

$$3x^{2}+3x=x-19+2x^{2}+11x+5$$

12x ni olish uchun x va 11x ni birlashtirish.

$$3x^{2}+3x=12x-19+2x^{2}+5$$

-14 olish uchun -19 va 5'ni qo'shing.

$$3x^{2}+3x=12x-14+2x^{2}$$

Ikkala tarafdan 12x ni ayirish.

$$3x^{2}+3x-12x=-14+2x^{2}$$

-9x ni olish uchun 3x va -12x ni birlashtirish.

$$3x^{2}-9x=-14+2x^{2}$$

Ikkala tarafdan 2x^{2} ni ayirish.

$$3x^{2}-9x-2x^{2}=-14$$

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

$$x^{2}-9x=-14$$

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

$$x^{2}-9x+\left(-\frac{9}{2}\right)^{2}=-14+\left(-\frac{9}{2}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib -\frac{9}{2} kvadratini chiqarish.

$$x^{2}-9x+\frac{81}{4}=-14+\frac{81}{4}$$

-14 ni \frac{81}{4} ga qo'shish.

$$x^{2}-9x+\frac{81}{4}=\frac{25}{4}$$

x^{2}-9x+\frac{81}{4} omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(x-\frac{9}{2}\right)^{2}=\frac{25}{4}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x-\frac{9}{2}\right)^{2}}=\sqrt{\frac{25}{4}}$$

Qisqartirish.

$$x-\frac{9}{2}=\frac{5}{2} x-\frac{9}{2}=-\frac{5}{2}$$

\frac{9}{2} ni tenglamaning ikkala tarafiga qo'shish.

$$x=7 x=2$$