2x/x-4+3/x-3+4=30+5x^2-36x/x^2-7x+12 eching

$$\frac { 2 x } { x - 4 } + \frac { 3 } { x - 3 } + 4 = \frac { 30 + 5 x ^ { 2 } - 36 x } { x ^ { 2 } - 7 x + 12 }$$

x uchun yechish

$$x=-3$$

$$x=-2$$

Faktorlash qadamlari

x qiymati 3,4 qiymatlaridan birortasiga teng bo‘lmaydi, chunki nolga bo‘lish mumkin emas. Tenglamaning ikkala tarafini \left(x-4\right)\left(x-3\right) ga, x-4,x-3,x^{2}-7x+12 ning eng kichik karralisiga ko‘paytiring.

$$\left(x-3\right)\times 2x+\left(x-4\right)\times 3+\left(x-4\right)\left(x-3\right)\times 4=30+5x^{2}-36x$$

x-3 ga 2 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$\left(2x-6\right)x+\left(x-4\right)\times 3+\left(x-4\right)\left(x-3\right)\times 4=30+5x^{2}-36x$$

2x-6 ga x ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$2x^{2}-6x+\left(x-4\right)\times 3+\left(x-4\right)\left(x-3\right)\times 4=30+5x^{2}-36x$$

x-4 ga 3 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$2x^{2}-6x+3x-12+\left(x-4\right)\left(x-3\right)\times 4=30+5x^{2}-36x$$

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

$$2x^{2}-3x-12+\left(x-4\right)\left(x-3\right)\times 4=30+5x^{2}-36x$$

x-4 ga x-3 ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

$$2x^{2}-3x-12+\left(x^{2}-7x+12\right)\times 4=30+5x^{2}-36x$$

x^{2}-7x+12 ga 4 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$2x^{2}-3x-12+4x^{2}-28x+48=30+5x^{2}-36x$$

6x^{2} ni olish uchun 2x^{2} va 4x^{2} ni birlashtirish.

$$6x^{2}-3x-12-28x+48=30+5x^{2}-36x$$

-31x ni olish uchun -3x va -28x ni birlashtirish.

$$6x^{2}-31x-12+48=30+5x^{2}-36x$$

36 olish uchun -12 va 48'ni qo'shing.

$$6x^{2}-31x+36=30+5x^{2}-36x$$

Ikkala tarafdan 30 ni ayirish.

$$6x^{2}-31x+36-30=5x^{2}-36x$$

6 olish uchun 36 dan 30 ni ayirish.

$$6x^{2}-31x+6=5x^{2}-36x$$

Ikkala tarafdan 5x^{2} ni ayirish.

$$6x^{2}-31x+6-5x^{2}=-36x$$

x^{2} ni olish uchun 6x^{2} va -5x^{2} ni birlashtirish.

$$x^{2}-31x+6=-36x$$

36x ni ikki tarafga qo’shing.

$$x^{2}-31x+6+36x=0$$

5x ni olish uchun -31x va 36x ni birlashtirish.

$$x^{2}+5x+6=0$$

Bu tenglamani yechish uchun x^{2}+\left(a+b\right)x+ab=\left(x+a\right)\left(x+b\right) formulasi yordamida x^{2}+5x+6 ni faktorlang. a va b ni topish uchun yechiladigan tizimni sozlang.

$$a+b=5 ab=6$$

ab musbat boʻlganda, a va b da bir xil belgi bor. a+b musbat boʻlganda, a va b ikkisi ham musbat. 6-mahsulotni beruvchi bunday butun juftliklarni roʻyxat qiling.

$$1,6 2,3$$

Har bir juftlik yigʻindisini hisoblang.

$$1+6=7 2+3=5$$

Yechim – 5 yigʻindisini beruvchi juftlik.

$$a=2 b=3$$

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

$$\left(x+2\right)\left(x+3\right)$$

Tenglamani yechish uchun x+2=0 va x+3=0 ni yeching.

$$x=-2 x=-3$$

Guruhlash orqali faktorlash qadamlari

x qiymati 3,4 qiymatlaridan birortasiga teng bo‘lmaydi, chunki nolga bo‘lish mumkin emas. Tenglamaning ikkala tarafini \left(x-4\right)\left(x-3\right) ga, x-4,x-3,x^{2}-7x+12 ning eng kichik karralisiga ko‘paytiring.

$$\left(x-3\right)\times 2x+\left(x-4\right)\times 3+\left(x-4\right)\left(x-3\right)\times 4=30+5x^{2}-36x$$

x-3 ga 2 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$\left(2x-6\right)x+\left(x-4\right)\times 3+\left(x-4\right)\left(x-3\right)\times 4=30+5x^{2}-36x$$

2x-6 ga x ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$2x^{2}-6x+\left(x-4\right)\times 3+\left(x-4\right)\left(x-3\right)\times 4=30+5x^{2}-36x$$

x-4 ga 3 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$2x^{2}-6x+3x-12+\left(x-4\right)\left(x-3\right)\times 4=30+5x^{2}-36x$$

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

$$2x^{2}-3x-12+\left(x-4\right)\left(x-3\right)\times 4=30+5x^{2}-36x$$

x-4 ga x-3 ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

$$2x^{2}-3x-12+\left(x^{2}-7x+12\right)\times 4=30+5x^{2}-36x$$

x^{2}-7x+12 ga 4 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$2x^{2}-3x-12+4x^{2}-28x+48=30+5x^{2}-36x$$

6x^{2} ni olish uchun 2x^{2} va 4x^{2} ni birlashtirish.

$$6x^{2}-3x-12-28x+48=30+5x^{2}-36x$$

-31x ni olish uchun -3x va -28x ni birlashtirish.

$$6x^{2}-31x-12+48=30+5x^{2}-36x$$

36 olish uchun -12 va 48'ni qo'shing.

$$6x^{2}-31x+36=30+5x^{2}-36x$$

Ikkala tarafdan 30 ni ayirish.

$$6x^{2}-31x+36-30=5x^{2}-36x$$

6 olish uchun 36 dan 30 ni ayirish.

$$6x^{2}-31x+6=5x^{2}-36x$$

Ikkala tarafdan 5x^{2} ni ayirish.

$$6x^{2}-31x+6-5x^{2}=-36x$$

x^{2} ni olish uchun 6x^{2} va -5x^{2} ni birlashtirish.

$$x^{2}-31x+6=-36x$$

36x ni ikki tarafga qo’shing.

$$x^{2}-31x+6+36x=0$$

5x ni olish uchun -31x va 36x ni birlashtirish.

$$x^{2}+5x+6=0$$

Tenglamani yechish uchun guruhlash orqali chap qoʻl tomonni faktorlang. Avvalo, chap qoʻl tomon x^{2}+ax+bx+6 sifatida qayta yozilishi kerak. a va b ni topish uchun yechiladigan tizimni sozlang.

$$a+b=5 ab=1\times 6=6$$

ab musbat boʻlganda, a va b da bir xil belgi bor. a+b musbat boʻlganda, a va b ikkisi ham musbat. 6-mahsulotni beruvchi bunday butun juftliklarni roʻyxat qiling.

$$1,6 2,3$$

Har bir juftlik yigʻindisini hisoblang.

$$1+6=7 2+3=5$$

Yechim – 5 yigʻindisini beruvchi juftlik.

$$a=2 b=3$$

x^{2}+5x+6 ni \left(x^{2}+2x\right)+\left(3x+6\right) sifatida qaytadan yozish.

$$\left(x^{2}+2x\right)+\left(3x+6\right)$$

Birinchi guruhda x ni va ikkinchi guruhda 3 ni faktordan chiqaring.

$$x\left(x+2\right)+3\left(x+2\right)$$

Distributiv funktsiyasidan foydalangan holda x+2 umumiy terminini chiqaring.

$$\left(x+2\right)\left(x+3\right)$$

Tenglamani yechish uchun x+2=0 va x+3=0 ni yeching.

$$x=-2 x=-3$$

Kvadrat tenglamalaridan foydalanish qoidalari

x qiymati 3,4 qiymatlaridan birortasiga teng bo‘lmaydi, chunki nolga bo‘lish mumkin emas. Tenglamaning ikkala tarafini \left(x-4\right)\left(x-3\right) ga, x-4,x-3,x^{2}-7x+12 ning eng kichik karralisiga ko‘paytiring.

$$\left(x-3\right)\times 2x+\left(x-4\right)\times 3+\left(x-4\right)\left(x-3\right)\times 4=30+5x^{2}-36x$$

x-3 ga 2 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$\left(2x-6\right)x+\left(x-4\right)\times 3+\left(x-4\right)\left(x-3\right)\times 4=30+5x^{2}-36x$$

2x-6 ga x ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$2x^{2}-6x+\left(x-4\right)\times 3+\left(x-4\right)\left(x-3\right)\times 4=30+5x^{2}-36x$$

x-4 ga 3 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$2x^{2}-6x+3x-12+\left(x-4\right)\left(x-3\right)\times 4=30+5x^{2}-36x$$

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

$$2x^{2}-3x-12+\left(x-4\right)\left(x-3\right)\times 4=30+5x^{2}-36x$$

x-4 ga x-3 ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

$$2x^{2}-3x-12+\left(x^{2}-7x+12\right)\times 4=30+5x^{2}-36x$$

x^{2}-7x+12 ga 4 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$2x^{2}-3x-12+4x^{2}-28x+48=30+5x^{2}-36x$$

6x^{2} ni olish uchun 2x^{2} va 4x^{2} ni birlashtirish.

$$6x^{2}-3x-12-28x+48=30+5x^{2}-36x$$

-31x ni olish uchun -3x va -28x ni birlashtirish.

$$6x^{2}-31x-12+48=30+5x^{2}-36x$$

36 olish uchun -12 va 48'ni qo'shing.

$$6x^{2}-31x+36=30+5x^{2}-36x$$

Ikkala tarafdan 30 ni ayirish.

$$6x^{2}-31x+36-30=5x^{2}-36x$$

6 olish uchun 36 dan 30 ni ayirish.

$$6x^{2}-31x+6=5x^{2}-36x$$

Ikkala tarafdan 5x^{2} ni ayirish.

$$6x^{2}-31x+6-5x^{2}=-36x$$

x^{2} ni olish uchun 6x^{2} va -5x^{2} ni birlashtirish.

$$x^{2}-31x+6=-36x$$

36x ni ikki tarafga qo’shing.

$$x^{2}-31x+6+36x=0$$

5x ni olish uchun -31x va 36x ni birlashtirish.

$$x^{2}+5x+6=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 1 ni a, 5 ni b va 6 ni c bilan almashtiring.

$$x=\frac{-5±\sqrt{5^{2}-4\times 6}}{2}$$

5 kvadratini chiqarish.

$$x=\frac{-5±\sqrt{25-4\times 6}}{2}$$

-4 ni 6 marotabaga ko'paytirish.

$$x=\frac{-5±\sqrt{25-24}}{2}$$

25 ni -24 ga qo'shish.

$$x=\frac{-5±\sqrt{1}}{2}$$

1 ning kvadrat ildizini chiqarish.

$$x=\frac{-5±1}{2}$$

x=\frac{-5±1}{2} tenglamasini yeching, bunda ± musbat. -5 ni 1 ga qo'shish.

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

-4 ni 2 ga bo'lish.

$$x=-2$$

x=\frac{-5±1}{2} tenglamasini yeching, bunda ± manfiy. -5 dan 1 ni ayirish.

$$x=-\frac{6}{2}$$

-6 ni 2 ga bo'lish.

$$x=-3$$

Tenglama yechildi.

$$x=-2 x=-3$$

Kvadratni yakunlash bo'yicha qadamlar

x qiymati 3,4 qiymatlaridan birortasiga teng bo‘lmaydi, chunki nolga bo‘lish mumkin emas. Tenglamaning ikkala tarafini \left(x-4\right)\left(x-3\right) ga, x-4,x-3,x^{2}-7x+12 ning eng kichik karralisiga ko‘paytiring.

$$\left(x-3\right)\times 2x+\left(x-4\right)\times 3+\left(x-4\right)\left(x-3\right)\times 4=30+5x^{2}-36x$$

x-3 ga 2 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$\left(2x-6\right)x+\left(x-4\right)\times 3+\left(x-4\right)\left(x-3\right)\times 4=30+5x^{2}-36x$$

2x-6 ga x ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$2x^{2}-6x+\left(x-4\right)\times 3+\left(x-4\right)\left(x-3\right)\times 4=30+5x^{2}-36x$$

x-4 ga 3 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$2x^{2}-6x+3x-12+\left(x-4\right)\left(x-3\right)\times 4=30+5x^{2}-36x$$

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

$$2x^{2}-3x-12+\left(x-4\right)\left(x-3\right)\times 4=30+5x^{2}-36x$$

x-4 ga x-3 ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

$$2x^{2}-3x-12+\left(x^{2}-7x+12\right)\times 4=30+5x^{2}-36x$$

x^{2}-7x+12 ga 4 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$2x^{2}-3x-12+4x^{2}-28x+48=30+5x^{2}-36x$$

6x^{2} ni olish uchun 2x^{2} va 4x^{2} ni birlashtirish.

$$6x^{2}-3x-12-28x+48=30+5x^{2}-36x$$

-31x ni olish uchun -3x va -28x ni birlashtirish.

$$6x^{2}-31x-12+48=30+5x^{2}-36x$$

36 olish uchun -12 va 48'ni qo'shing.

$$6x^{2}-31x+36=30+5x^{2}-36x$$

Ikkala tarafdan 5x^{2} ni ayirish.

$$6x^{2}-31x+36-5x^{2}=30-36x$$

x^{2} ni olish uchun 6x^{2} va -5x^{2} ni birlashtirish.

$$x^{2}-31x+36=30-36x$$

36x ni ikki tarafga qo’shing.

$$x^{2}-31x+36+36x=30$$

5x ni olish uchun -31x va 36x ni birlashtirish.

$$x^{2}+5x+36=30$$

Ikkala tarafdan 36 ni ayirish.

$$x^{2}+5x=30-36$$

-6 olish uchun 30 dan 36 ni ayirish.

$$x^{2}+5x=-6$$

5 ni bo‘lish, x shartining koeffitsienti, 2 ga \frac{5}{2} olish uchun. Keyin, \frac{5}{2} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}+5x+\left(\frac{5}{2}\right)^{2}=-6+\left(\frac{5}{2}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib \frac{5}{2} kvadratini chiqarish.

$$x^{2}+5x+\frac{25}{4}=-6+\frac{25}{4}$$

-6 ni \frac{25}{4} ga qo'shish.

$$x^{2}+5x+\frac{25}{4}=\frac{1}{4}$$

x^{2}+5x+\frac{25}{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.

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

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

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

Qisqartirish.

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

Tenglamaning ikkala tarafidan \frac{5}{2} ni ayirish.

$$x=-2 x=-3$$