x-2/x^2-x-6=3/2x+4-x/4-x^2 eching

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

x uchun yechish

$$x=1$$

$$x = \frac{10}{3} = 3\frac{1}{3} \approx 3,333333333$$

Guruhlash orqali faktorlash qadamlari

x qiymati -2,2,3 qiymatlaridan birortasiga teng bo‘lmaydi, chunki nolga bo‘lish mumkin emas. Tenglamaning ikkala tarafini 2\left(x-3\right)\left(x-2\right)\left(x+2\right) ga, x^{2}-x-6,2x+4,4-x^{2} ning eng kichik karralisiga ko‘paytiring.

$$\left(2x-4\right)\left(x-2\right)=\left(x-3\right)\left(x-2\right)\times 3-\left(6-2x\right)x$$

2x-4 ga x-2 ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

$$2x^{2}-8x+8=\left(x-3\right)\left(x-2\right)\times 3-\left(6-2x\right)x$$

x-3 ga x-2 ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

$$2x^{2}-8x+8=\left(x^{2}-5x+6\right)\times 3-\left(6-2x\right)x$$

x^{2}-5x+6 ga 3 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$2x^{2}-8x+8=3x^{2}-15x+18-\left(6-2x\right)x$$

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

$$2x^{2}-8x+8=3x^{2}-15x+18-\left(6x-2x^{2}\right)$$

6x-2x^{2} teskarisini topish uchun har birining teskarisini toping.

$$2x^{2}-8x+8=3x^{2}-15x+18-6x+2x^{2}$$

-21x ni olish uchun -15x va -6x ni birlashtirish.

$$2x^{2}-8x+8=3x^{2}-21x+18+2x^{2}$$

5x^{2} ni olish uchun 3x^{2} va 2x^{2} ni birlashtirish.

$$2x^{2}-8x+8=5x^{2}-21x+18$$

Ikkala tarafdan 5x^{2} ni ayirish.

$$2x^{2}-8x+8-5x^{2}=-21x+18$$

-3x^{2} ni olish uchun 2x^{2} va -5x^{2} ni birlashtirish.

$$-3x^{2}-8x+8=-21x+18$$

21x ni ikki tarafga qo’shing.

$$-3x^{2}-8x+8+21x=18$$

13x ni olish uchun -8x va 21x ni birlashtirish.

$$-3x^{2}+13x+8=18$$

Ikkala tarafdan 18 ni ayirish.

$$-3x^{2}+13x+8-18=0$$

-10 olish uchun 8 dan 18 ni ayirish.

$$-3x^{2}+13x-10=0$$

Tenglamani yechish uchun guruhlash orqali chap qoʻl tomonni faktorlang. Avvalo, chap qoʻl tomon -3x^{2}+ax+bx-10 sifatida qayta yozilishi kerak. a va b ni topish uchun yechiladigan tizimni sozlang.

$$a+b=13 ab=-3\left(-10\right)=30$$

ab musbat boʻlganda, a va b da bir xil belgi bor. a+b musbat boʻlganda, a va b ikkisi ham musbat. 30-mahsulotni beruvchi bunday butun juftliklarni roʻyxat qiling.

$$1,30 2,15 3,10 5,6$$

Har bir juftlik yigʻindisini hisoblang.

$$1+30=31 2+15=17 3+10=13 5+6=11$$

Yechim – 13 yigʻindisini beruvchi juftlik.

$$a=10 b=3$$

-3x^{2}+13x-10 ni \left(-3x^{2}+10x\right)+\left(3x-10\right) sifatida qaytadan yozish.

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

-3x^{2}+10x ichida -x ni ajrating.

$$-x\left(3x-10\right)+3x-10$$

Distributiv funktsiyasidan foydalangan holda 3x-10 umumiy terminini chiqaring.

$$\left(3x-10\right)\left(-x+1\right)$$

Tenglamani yechish uchun 3x-10=0 va -x+1=0 ni yeching.

$$x=\frac{10}{3} x=1$$

Kvadrat tenglamalaridan foydalanish qoidalari

x qiymati -2,2,3 qiymatlaridan birortasiga teng bo‘lmaydi, chunki nolga bo‘lish mumkin emas. Tenglamaning ikkala tarafini 2\left(x-3\right)\left(x-2\right)\left(x+2\right) ga, x^{2}-x-6,2x+4,4-x^{2} ning eng kichik karralisiga ko‘paytiring.

$$\left(2x-4\right)\left(x-2\right)=\left(x-3\right)\left(x-2\right)\times 3-\left(6-2x\right)x$$

2x-4 ga x-2 ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

$$2x^{2}-8x+8=\left(x-3\right)\left(x-2\right)\times 3-\left(6-2x\right)x$$

x-3 ga x-2 ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

$$2x^{2}-8x+8=\left(x^{2}-5x+6\right)\times 3-\left(6-2x\right)x$$

x^{2}-5x+6 ga 3 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$2x^{2}-8x+8=3x^{2}-15x+18-\left(6-2x\right)x$$

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

$$2x^{2}-8x+8=3x^{2}-15x+18-\left(6x-2x^{2}\right)$$

6x-2x^{2} teskarisini topish uchun har birining teskarisini toping.

$$2x^{2}-8x+8=3x^{2}-15x+18-6x+2x^{2}$$

-21x ni olish uchun -15x va -6x ni birlashtirish.

$$2x^{2}-8x+8=3x^{2}-21x+18+2x^{2}$$

5x^{2} ni olish uchun 3x^{2} va 2x^{2} ni birlashtirish.

$$2x^{2}-8x+8=5x^{2}-21x+18$$

Ikkala tarafdan 5x^{2} ni ayirish.

$$2x^{2}-8x+8-5x^{2}=-21x+18$$

-3x^{2} ni olish uchun 2x^{2} va -5x^{2} ni birlashtirish.

$$-3x^{2}-8x+8=-21x+18$$

21x ni ikki tarafga qo’shing.

$$-3x^{2}-8x+8+21x=18$$

13x ni olish uchun -8x va 21x ni birlashtirish.

$$-3x^{2}+13x+8=18$$

Ikkala tarafdan 18 ni ayirish.

$$-3x^{2}+13x+8-18=0$$

-10 olish uchun 8 dan 18 ni ayirish.

$$-3x^{2}+13x-10=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} -3 ni a, 13 ni b va -10 ni c bilan almashtiring.

$$x=\frac{-13±\sqrt{13^{2}-4\left(-3\right)\left(-10\right)}}{2\left(-3\right)}$$

13 kvadratini chiqarish.

$$x=\frac{-13±\sqrt{169-4\left(-3\right)\left(-10\right)}}{2\left(-3\right)}$$

-4 ni -3 marotabaga ko'paytirish.

$$x=\frac{-13±\sqrt{169+12\left(-10\right)}}{2\left(-3\right)}$$

12 ni -10 marotabaga ko'paytirish.

$$x=\frac{-13±\sqrt{169-120}}{2\left(-3\right)}$$

169 ni -120 ga qo'shish.

$$x=\frac{-13±\sqrt{49}}{2\left(-3\right)}$$

49 ning kvadrat ildizini chiqarish.

$$x=\frac{-13±7}{2\left(-3\right)}$$

2 ni -3 marotabaga ko'paytirish.

$$x=\frac{-13±7}{-6}$$

x=\frac{-13±7}{-6} tenglamasini yeching, bunda ± musbat. -13 ni 7 ga qo'shish.

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

-6 ni -6 ga bo'lish.

$$x=1$$

x=\frac{-13±7}{-6} tenglamasini yeching, bunda ± manfiy. -13 dan 7 ni ayirish.

$$x=-\frac{20}{-6}$$

\frac{-20}{-6} ulushini 2 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

$$x=\frac{10}{3}$$

Tenglama yechildi.

$$x=1 x=\frac{10}{3}$$

Kvadratni yakunlash bo'yicha qadamlar

x qiymati -2,2,3 qiymatlaridan birortasiga teng bo‘lmaydi, chunki nolga bo‘lish mumkin emas. Tenglamaning ikkala tarafini 2\left(x-3\right)\left(x-2\right)\left(x+2\right) ga, x^{2}-x-6,2x+4,4-x^{2} ning eng kichik karralisiga ko‘paytiring.

$$\left(2x-4\right)\left(x-2\right)=\left(x-3\right)\left(x-2\right)\times 3-\left(6-2x\right)x$$

2x-4 ga x-2 ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

$$2x^{2}-8x+8=\left(x-3\right)\left(x-2\right)\times 3-\left(6-2x\right)x$$

x-3 ga x-2 ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

$$2x^{2}-8x+8=\left(x^{2}-5x+6\right)\times 3-\left(6-2x\right)x$$

x^{2}-5x+6 ga 3 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$2x^{2}-8x+8=3x^{2}-15x+18-\left(6-2x\right)x$$

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

$$2x^{2}-8x+8=3x^{2}-15x+18-\left(6x-2x^{2}\right)$$

6x-2x^{2} teskarisini topish uchun har birining teskarisini toping.

$$2x^{2}-8x+8=3x^{2}-15x+18-6x+2x^{2}$$

-21x ni olish uchun -15x va -6x ni birlashtirish.

$$2x^{2}-8x+8=3x^{2}-21x+18+2x^{2}$$

5x^{2} ni olish uchun 3x^{2} va 2x^{2} ni birlashtirish.

$$2x^{2}-8x+8=5x^{2}-21x+18$$

Ikkala tarafdan 5x^{2} ni ayirish.

$$2x^{2}-8x+8-5x^{2}=-21x+18$$

-3x^{2} ni olish uchun 2x^{2} va -5x^{2} ni birlashtirish.

$$-3x^{2}-8x+8=-21x+18$$

21x ni ikki tarafga qo’shing.

$$-3x^{2}-8x+8+21x=18$$

13x ni olish uchun -8x va 21x ni birlashtirish.

$$-3x^{2}+13x+8=18$$

Ikkala tarafdan 8 ni ayirish.

$$-3x^{2}+13x=18-8$$

10 olish uchun 18 dan 8 ni ayirish.

$$-3x^{2}+13x=10$$

Ikki tarafini -3 ga bo‘ling.

$$\frac{-3x^{2}+13x}{-3}=\frac{10}{-3}$$

-3 ga bo'lish -3 ga ko'paytirishni bekor qiladi.

$$x^{2}+\frac{13}{-3}x=\frac{10}{-3}$$

13 ni -3 ga bo'lish.

$$x^{2}-\frac{13}{3}x=\frac{10}{-3}$$

10 ni -3 ga bo'lish.

$$x^{2}-\frac{13}{3}x=-\frac{10}{3}$$

-\frac{13}{3} ni bo‘lish, x shartining koeffitsienti, 2 ga -\frac{13}{6} olish uchun. Keyin, -\frac{13}{6} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}-\frac{13}{3}x+\left(-\frac{13}{6}\right)^{2}=-\frac{10}{3}+\left(-\frac{13}{6}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib -\frac{13}{6} kvadratini chiqarish.

$$x^{2}-\frac{13}{3}x+\frac{169}{36}=-\frac{10}{3}+\frac{169}{36}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali -\frac{10}{3} ni \frac{169}{36} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$x^{2}-\frac{13}{3}x+\frac{169}{36}=\frac{49}{36}$$

x^{2}-\frac{13}{3}x+\frac{169}{36} omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(x-\frac{13}{6}\right)^{2}=\frac{49}{36}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x-\frac{13}{6}\right)^{2}}=\sqrt{\frac{49}{36}}$$

Qisqartirish.

$$x-\frac{13}{6}=\frac{7}{6} x-\frac{13}{6}=-\frac{7}{6}$$

\frac{13}{6} ni tenglamaning ikkala tarafiga qo'shish.

$$x=\frac{10}{3} x=1$$