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

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

x uchun yechish

$$x=5$$

Guruhlash orqali faktorlash qadamlari

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

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

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

$$\left(x^{2}+3x+2\right)\times 2+\left(x-2\right)\left(x-1\right)=\left(x^{2}-1\right)\times 4$$

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

$$2x^{2}+6x+4+\left(x-2\right)\left(x-1\right)=\left(x^{2}-1\right)\times 4$$

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

$$2x^{2}+6x+4+x^{2}-3x+2=\left(x^{2}-1\right)\times 4$$

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

$$3x^{2}+6x+4-3x+2=\left(x^{2}-1\right)\times 4$$

3x ni olish uchun 6x va -3x ni birlashtirish.

$$3x^{2}+3x+4+2=\left(x^{2}-1\right)\times 4$$

6 olish uchun 4 va 2'ni qo'shing.

$$3x^{2}+3x+6=\left(x^{2}-1\right)\times 4$$

x^{2}-1 ga 4 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$3x^{2}+3x+6=4x^{2}-4$$

Ikkala tarafdan 4x^{2} ni ayirish.

$$3x^{2}+3x+6-4x^{2}=-4$$

-x^{2} ni olish uchun 3x^{2} va -4x^{2} ni birlashtirish.

$$-x^{2}+3x+6=-4$$

4 ni ikki tarafga qo’shing.

$$-x^{2}+3x+6+4=0$$

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

$$-x^{2}+3x+10=0$$

Tenglamani yechish uchun guruhlash orqali chap qoʻl tomonni faktorlang. Avvalo, chap qoʻl tomon -x^{2}+ax+bx+10 sifatida qayta yozilishi kerak. a va b ni topish uchun yechiladigan tizimni sozlang.

$$a+b=3 ab=-10=-10$$

ab manfiy boʻlganda, a va b da qarama-qarshi belgilar bor. a+b musbat boʻlganda, musbat sonda manfiyga nisbatdan kattaroq mutlaq qiymat bor. -10-mahsulotni beruvchi bunday butun juftliklarni roʻyxat qiling.

$$-1,10 -2,5$$

Har bir juftlik yigʻindisini hisoblang.

$$-1+10=9 -2+5=3$$

Yechim – 3 yigʻindisini beruvchi juftlik.

$$a=5 b=-2$$

-x^{2}+3x+10 ni \left(-x^{2}+5x\right)+\left(-2x+10\right) sifatida qaytadan yozish.

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

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

$$-x\left(x-5\right)-2\left(x-5\right)$$

Distributiv funktsiyasidan foydalangan holda x-5 umumiy terminini chiqaring.

$$\left(x-5\right)\left(-x-2\right)$$

Tenglamani yechish uchun x-5=0 va -x-2=0 ni yeching.

$$x=5 x=-2$$

x qiymati -2 teng bo‘lmaydi.

$$x=5$$

Kvadrat tenglamalaridan foydalanish qoidalari

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

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

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

$$\left(x^{2}+3x+2\right)\times 2+\left(x-2\right)\left(x-1\right)=\left(x^{2}-1\right)\times 4$$

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

$$2x^{2}+6x+4+\left(x-2\right)\left(x-1\right)=\left(x^{2}-1\right)\times 4$$

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

$$2x^{2}+6x+4+x^{2}-3x+2=\left(x^{2}-1\right)\times 4$$

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

$$3x^{2}+6x+4-3x+2=\left(x^{2}-1\right)\times 4$$

3x ni olish uchun 6x va -3x ni birlashtirish.

$$3x^{2}+3x+4+2=\left(x^{2}-1\right)\times 4$$

6 olish uchun 4 va 2'ni qo'shing.

$$3x^{2}+3x+6=\left(x^{2}-1\right)\times 4$$

x^{2}-1 ga 4 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$3x^{2}+3x+6=4x^{2}-4$$

Ikkala tarafdan 4x^{2} ni ayirish.

$$3x^{2}+3x+6-4x^{2}=-4$$

-x^{2} ni olish uchun 3x^{2} va -4x^{2} ni birlashtirish.

$$-x^{2}+3x+6=-4$$

4 ni ikki tarafga qo’shing.

$$-x^{2}+3x+6+4=0$$

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

$$-x^{2}+3x+10=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} -1 ni a, 3 ni b va 10 ni c bilan almashtiring.

$$x=\frac{-3±\sqrt{3^{2}-4\left(-1\right)\times 10}}{2\left(-1\right)}$$

3 kvadratini chiqarish.

$$x=\frac{-3±\sqrt{9-4\left(-1\right)\times 10}}{2\left(-1\right)}$$

-4 ni -1 marotabaga ko'paytirish.

$$x=\frac{-3±\sqrt{9+4\times 10}}{2\left(-1\right)}$$

4 ni 10 marotabaga ko'paytirish.

$$x=\frac{-3±\sqrt{9+40}}{2\left(-1\right)}$$

9 ni 40 ga qo'shish.

$$x=\frac{-3±\sqrt{49}}{2\left(-1\right)}$$

49 ning kvadrat ildizini chiqarish.

$$x=\frac{-3±7}{2\left(-1\right)}$$

2 ni -1 marotabaga ko'paytirish.

$$x=\frac{-3±7}{-2}$$

x=\frac{-3±7}{-2} tenglamasini yeching, bunda ± musbat. -3 ni 7 ga qo'shish.

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

4 ni -2 ga bo'lish.

$$x=-2$$

x=\frac{-3±7}{-2} tenglamasini yeching, bunda ± manfiy. -3 dan 7 ni ayirish.

$$x=-\frac{10}{-2}$$

-10 ni -2 ga bo'lish.

$$x=5$$

Tenglama yechildi.

$$x=-2 x=5$$

x qiymati -2 teng bo‘lmaydi.

$$x=5$$

Kvadratni yakunlash bo'yicha qadamlar

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

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

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

$$\left(x^{2}+3x+2\right)\times 2+\left(x-2\right)\left(x-1\right)=\left(x^{2}-1\right)\times 4$$

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

$$2x^{2}+6x+4+\left(x-2\right)\left(x-1\right)=\left(x^{2}-1\right)\times 4$$

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

$$2x^{2}+6x+4+x^{2}-3x+2=\left(x^{2}-1\right)\times 4$$

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

$$3x^{2}+6x+4-3x+2=\left(x^{2}-1\right)\times 4$$

3x ni olish uchun 6x va -3x ni birlashtirish.

$$3x^{2}+3x+4+2=\left(x^{2}-1\right)\times 4$$

6 olish uchun 4 va 2'ni qo'shing.

$$3x^{2}+3x+6=\left(x^{2}-1\right)\times 4$$

x^{2}-1 ga 4 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$3x^{2}+3x+6=4x^{2}-4$$

Ikkala tarafdan 4x^{2} ni ayirish.

$$3x^{2}+3x+6-4x^{2}=-4$$

-x^{2} ni olish uchun 3x^{2} va -4x^{2} ni birlashtirish.

$$-x^{2}+3x+6=-4$$

Ikkala tarafdan 6 ni ayirish.

$$-x^{2}+3x=-4-6$$

-10 olish uchun -4 dan 6 ni ayirish.

$$-x^{2}+3x=-10$$

Ikki tarafini -1 ga bo‘ling.

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

-1 ga bo'lish -1 ga ko'paytirishni bekor qiladi.

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

3 ni -1 ga bo'lish.

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

-10 ni -1 ga bo'lish.

$$x^{2}-3x=10$$

-3 ni bo‘lish, x shartining koeffitsienti, 2 ga -\frac{3}{2} olish uchun. Keyin, -\frac{3}{2} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

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

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib -\frac{3}{2} kvadratini chiqarish.

$$x^{2}-3x+\frac{9}{4}=10+\frac{9}{4}$$

10 ni \frac{9}{4} ga qo'shish.

$$x^{2}-3x+\frac{9}{4}=\frac{49}{4}$$

x^{2}-3x+\frac{9}{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.

$$\left(x-\frac{3}{2}\right)^{2}=\frac{49}{4}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x-\frac{3}{2}\right)^{2}}=\sqrt{\frac{49}{4}}$$

Qisqartirish.

$$x-\frac{3}{2}=\frac{7}{2} x-\frac{3}{2}=-\frac{7}{2}$$

\frac{3}{2} ni tenglamaning ikkala tarafiga qo'shish.

$$x=5 x=-2$$

x qiymati -2 teng bo‘lmaydi.

$$x=5$$