2x-5/x^2+3x+2+4/x^2-4=1/x-2 eching

$$\frac { 2 x - 5 } { x ^ { 2 } + 3 x + 2 } + \frac { 4 } { x ^ { 2 } - 4 } = \frac { 1 } { x - 2 }$$

x uchun yechish

$$x=6$$

Faktorlash qadamlari

x qiymati -2,-1,2 qiymatlaridan birortasiga teng bo‘lmaydi, chunki nolga bo‘lish mumkin emas. Tenglamaning ikkala tarafini \left(x-2\right)\left(x+1\right)\left(x+2\right) ga, x^{2}+3x+2,x^{2}-4,x-2 ning eng kichik karralisiga ko‘paytiring.

$$\left(x-2\right)\left(2x-5\right)+\left(x+1\right)\times 4=\left(x+1\right)\left(x+2\right)$$

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

$$2x^{2}-9x+10+\left(x+1\right)\times 4=\left(x+1\right)\left(x+2\right)$$

x+1 ga 4 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$2x^{2}-9x+10+4x+4=\left(x+1\right)\left(x+2\right)$$

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

$$2x^{2}-5x+10+4=\left(x+1\right)\left(x+2\right)$$

14 olish uchun 10 va 4'ni qo'shing.

$$2x^{2}-5x+14=\left(x+1\right)\left(x+2\right)$$

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

$$2x^{2}-5x+14=x^{2}+3x+2$$

Ikkala tarafdan x^{2} ni ayirish.

$$2x^{2}-5x+14-x^{2}=3x+2$$

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

$$x^{2}-5x+14=3x+2$$

Ikkala tarafdan 3x ni ayirish.

$$x^{2}-5x+14-3x=2$$

-8x ni olish uchun -5x va -3x ni birlashtirish.

$$x^{2}-8x+14=2$$

Ikkala tarafdan 2 ni ayirish.

$$x^{2}-8x+14-2=0$$

12 olish uchun 14 dan 2 ni ayirish.

$$x^{2}-8x+12=0$$

Bu tenglamani yechish uchun x^{2}+\left(a+b\right)x+ab=\left(x+a\right)\left(x+b\right) formulasi yordamida x^{2}-8x+12 ni faktorlang. a va b ni topish uchun yechiladigan tizimni sozlang.

$$a+b=-8 ab=12$$

ab musbat boʻlganda, a va b da bir xil belgi bor. a+b manfiy boʻlganda, a va b ikkisi ham manfiy. 12-mahsulotni beruvchi bunday butun juftliklarni roʻyxat qiling.

$$-1,-12 -2,-6 -3,-4$$

Har bir juftlik yigʻindisini hisoblang.

$$-1-12=-13 -2-6=-8 -3-4=-7$$

Yechim – -8 yigʻindisini beruvchi juftlik.

$$a=-6 b=-2$$

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

$$\left(x-6\right)\left(x-2\right)$$

Tenglamani yechish uchun x-6=0 va x-2=0 ni yeching.

$$x=6 x=2$$

x qiymati 2 teng bo‘lmaydi.

$$x=6$$

Guruhlash orqali faktorlash qadamlari

x qiymati -2,-1,2 qiymatlaridan birortasiga teng bo‘lmaydi, chunki nolga bo‘lish mumkin emas. Tenglamaning ikkala tarafini \left(x-2\right)\left(x+1\right)\left(x+2\right) ga, x^{2}+3x+2,x^{2}-4,x-2 ning eng kichik karralisiga ko‘paytiring.

$$\left(x-2\right)\left(2x-5\right)+\left(x+1\right)\times 4=\left(x+1\right)\left(x+2\right)$$

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

$$2x^{2}-9x+10+\left(x+1\right)\times 4=\left(x+1\right)\left(x+2\right)$$

x+1 ga 4 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$2x^{2}-9x+10+4x+4=\left(x+1\right)\left(x+2\right)$$

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

$$2x^{2}-5x+10+4=\left(x+1\right)\left(x+2\right)$$

14 olish uchun 10 va 4'ni qo'shing.

$$2x^{2}-5x+14=\left(x+1\right)\left(x+2\right)$$

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

$$2x^{2}-5x+14=x^{2}+3x+2$$

Ikkala tarafdan x^{2} ni ayirish.

$$2x^{2}-5x+14-x^{2}=3x+2$$

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

$$x^{2}-5x+14=3x+2$$

Ikkala tarafdan 3x ni ayirish.

$$x^{2}-5x+14-3x=2$$

-8x ni olish uchun -5x va -3x ni birlashtirish.

$$x^{2}-8x+14=2$$

Ikkala tarafdan 2 ni ayirish.

$$x^{2}-8x+14-2=0$$

12 olish uchun 14 dan 2 ni ayirish.

$$x^{2}-8x+12=0$$

Tenglamani yechish uchun guruhlash orqali chap qoʻl tomonni faktorlang. Avvalo, chap qoʻl tomon x^{2}+ax+bx+12 sifatida qayta yozilishi kerak. a va b ni topish uchun yechiladigan tizimni sozlang.

$$a+b=-8 ab=1\times 12=12$$

ab musbat boʻlganda, a va b da bir xil belgi bor. a+b manfiy boʻlganda, a va b ikkisi ham manfiy. 12-mahsulotni beruvchi bunday butun juftliklarni roʻyxat qiling.

$$-1,-12 -2,-6 -3,-4$$

Har bir juftlik yigʻindisini hisoblang.

$$-1-12=-13 -2-6=-8 -3-4=-7$$

Yechim – -8 yigʻindisini beruvchi juftlik.

$$a=-6 b=-2$$

x^{2}-8x+12 ni \left(x^{2}-6x\right)+\left(-2x+12\right) sifatida qaytadan yozish.

$$\left(x^{2}-6x\right)+\left(-2x+12\right)$$

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

$$x\left(x-6\right)-2\left(x-6\right)$$

Distributiv funktsiyasidan foydalangan holda x-6 umumiy terminini chiqaring.

$$\left(x-6\right)\left(x-2\right)$$

Tenglamani yechish uchun x-6=0 va x-2=0 ni yeching.

$$x=6 x=2$$

x qiymati 2 teng bo‘lmaydi.

$$x=6$$

Kvadrat tenglamalaridan foydalanish qoidalari

x qiymati -2,-1,2 qiymatlaridan birortasiga teng bo‘lmaydi, chunki nolga bo‘lish mumkin emas. Tenglamaning ikkala tarafini \left(x-2\right)\left(x+1\right)\left(x+2\right) ga, x^{2}+3x+2,x^{2}-4,x-2 ning eng kichik karralisiga ko‘paytiring.

$$\left(x-2\right)\left(2x-5\right)+\left(x+1\right)\times 4=\left(x+1\right)\left(x+2\right)$$

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

$$2x^{2}-9x+10+\left(x+1\right)\times 4=\left(x+1\right)\left(x+2\right)$$

x+1 ga 4 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$2x^{2}-9x+10+4x+4=\left(x+1\right)\left(x+2\right)$$

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

$$2x^{2}-5x+10+4=\left(x+1\right)\left(x+2\right)$$

14 olish uchun 10 va 4'ni qo'shing.

$$2x^{2}-5x+14=\left(x+1\right)\left(x+2\right)$$

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

$$2x^{2}-5x+14=x^{2}+3x+2$$

Ikkala tarafdan x^{2} ni ayirish.

$$2x^{2}-5x+14-x^{2}=3x+2$$

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

$$x^{2}-5x+14=3x+2$$

Ikkala tarafdan 3x ni ayirish.

$$x^{2}-5x+14-3x=2$$

-8x ni olish uchun -5x va -3x ni birlashtirish.

$$x^{2}-8x+14=2$$

Ikkala tarafdan 2 ni ayirish.

$$x^{2}-8x+14-2=0$$

12 olish uchun 14 dan 2 ni ayirish.

$$x^{2}-8x+12=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 1 ni a, -8 ni b va 12 ni c bilan almashtiring.

$$x=\frac{-\left(-8\right)±\sqrt{\left(-8\right)^{2}-4\times 12}}{2}$$

-8 kvadratini chiqarish.

$$x=\frac{-\left(-8\right)±\sqrt{64-4\times 12}}{2}$$

-4 ni 12 marotabaga ko'paytirish.

$$x=\frac{-\left(-8\right)±\sqrt{64-48}}{2}$$

64 ni -48 ga qo'shish.

$$x=\frac{-\left(-8\right)±\sqrt{16}}{2}$$

16 ning kvadrat ildizini chiqarish.

$$x=\frac{-\left(-8\right)±4}{2}$$

-8 ning teskarisi 8 ga teng.

$$x=\frac{8±4}{2}$$

x=\frac{8±4}{2} tenglamasini yeching, bunda ± musbat. 8 ni 4 ga qo'shish.

$$x=\frac{12}{2}$$

12 ni 2 ga bo'lish.

$$x=6$$

x=\frac{8±4}{2} tenglamasini yeching, bunda ± manfiy. 8 dan 4 ni ayirish.

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

4 ni 2 ga bo'lish.

$$x=2$$

Tenglama yechildi.

$$x=6 x=2$$

x qiymati 2 teng bo‘lmaydi.

$$x=6$$

Kvadratni yakunlash bo'yicha qadamlar

x qiymati -2,-1,2 qiymatlaridan birortasiga teng bo‘lmaydi, chunki nolga bo‘lish mumkin emas. Tenglamaning ikkala tarafini \left(x-2\right)\left(x+1\right)\left(x+2\right) ga, x^{2}+3x+2,x^{2}-4,x-2 ning eng kichik karralisiga ko‘paytiring.

$$\left(x-2\right)\left(2x-5\right)+\left(x+1\right)\times 4=\left(x+1\right)\left(x+2\right)$$

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

$$2x^{2}-9x+10+\left(x+1\right)\times 4=\left(x+1\right)\left(x+2\right)$$

x+1 ga 4 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$2x^{2}-9x+10+4x+4=\left(x+1\right)\left(x+2\right)$$

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

$$2x^{2}-5x+10+4=\left(x+1\right)\left(x+2\right)$$

14 olish uchun 10 va 4'ni qo'shing.

$$2x^{2}-5x+14=\left(x+1\right)\left(x+2\right)$$

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

$$2x^{2}-5x+14=x^{2}+3x+2$$

Ikkala tarafdan x^{2} ni ayirish.

$$2x^{2}-5x+14-x^{2}=3x+2$$

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

$$x^{2}-5x+14=3x+2$$

Ikkala tarafdan 3x ni ayirish.

$$x^{2}-5x+14-3x=2$$

-8x ni olish uchun -5x va -3x ni birlashtirish.

$$x^{2}-8x+14=2$$

Ikkala tarafdan 14 ni ayirish.

$$x^{2}-8x=2-14$$

-12 olish uchun 2 dan 14 ni ayirish.

$$x^{2}-8x=-12$$

-8 ni bo‘lish, x shartining koeffitsienti, 2 ga -4 olish uchun. Keyin, -4 ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}-8x+\left(-4\right)^{2}=-12+\left(-4\right)^{2}$$

-4 kvadratini chiqarish.

$$x^{2}-8x+16=-12+16$$

-12 ni 16 ga qo'shish.

$$x^{2}-8x+16=4$$

x^{2}-8x+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.

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

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

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

Qisqartirish.

$$x-4=2 x-4=-2$$

4 ni tenglamaning ikkala tarafiga qo'shish.

$$x=6 x=2$$

x qiymati 2 teng bo‘lmaydi.

$$x=6$$