3/x^2+2x-8-4/x^2+4x-12=x-4/x^2+10x+24 eching

$$\frac { 3 } { x ^ { 2 } + 2 x - 8 } - \frac { 4 } { x ^ { 2 } + 4 x - 12 } = \frac { x - 4 } { x ^ { 2 } + 10 x + 24 }$$

x uchun yechish

$$x=3$$

Guruhlash orqali faktorlash qadamlari

x qiymati -6,-4,2 qiymatlaridan birortasiga teng bo‘lmaydi, chunki nolga bo‘lish mumkin emas. Tenglamaning ikkala tarafini \left(x-2\right)\left(x+4\right)\left(x+6\right) ga, x^{2}+2x-8,x^{2}+4x-12,x^{2}+10x+24 ning eng kichik karralisiga ko‘paytiring.

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

x+6 ga 3 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$3x+18-\left(x+4\right)\times 4=\left(x-2\right)\left(x-4\right)$$

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

$$3x+18-\left(4x+16\right)=\left(x-2\right)\left(x-4\right)$$

4x+16 teskarisini topish uchun har birining teskarisini toping.

$$3x+18-4x-16=\left(x-2\right)\left(x-4\right)$$

-x ni olish uchun 3x va -4x ni birlashtirish.

$$-x+18-16=\left(x-2\right)\left(x-4\right)$$

2 olish uchun 18 dan 16 ni ayirish.

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

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

$$-x+2=x^{2}-6x+8$$

Ikkala tarafdan x^{2} ni ayirish.

$$-x+2-x^{2}=-6x+8$$

6x ni ikki tarafga qo’shing.

$$-x+2-x^{2}+6x=8$$

5x ni olish uchun -x va 6x ni birlashtirish.

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

Ikkala tarafdan 8 ni ayirish.

$$5x+2-x^{2}-8=0$$

-6 olish uchun 2 dan 8 ni ayirish.

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

Polinomni standart shaklga keltirish uchun uni qayta tartiblang. Shartlarni eng yuqoridan eng pastki qiymat ko'rsatgichiga joylashtirish.

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

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.

$$a+b=5 ab=-\left(-6\right)=6$$

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.

$$a=3 b=2$$

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

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

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

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

Distributiv funktsiyasidan foydalangan holda x-3 umumiy terminini chiqaring.

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

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

$$x=3 x=2$$

x qiymati 2 teng bo‘lmaydi.

$$x=3$$

Kvadrat tenglamalaridan foydalanish qoidalari

x qiymati -6,-4,2 qiymatlaridan birortasiga teng bo‘lmaydi, chunki nolga bo‘lish mumkin emas. Tenglamaning ikkala tarafini \left(x-2\right)\left(x+4\right)\left(x+6\right) ga, x^{2}+2x-8,x^{2}+4x-12,x^{2}+10x+24 ning eng kichik karralisiga ko‘paytiring.

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

x+6 ga 3 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$3x+18-\left(x+4\right)\times 4=\left(x-2\right)\left(x-4\right)$$

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

$$3x+18-\left(4x+16\right)=\left(x-2\right)\left(x-4\right)$$

4x+16 teskarisini topish uchun har birining teskarisini toping.

$$3x+18-4x-16=\left(x-2\right)\left(x-4\right)$$

-x ni olish uchun 3x va -4x ni birlashtirish.

$$-x+18-16=\left(x-2\right)\left(x-4\right)$$

2 olish uchun 18 dan 16 ni ayirish.

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

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

$$-x+2=x^{2}-6x+8$$

Ikkala tarafdan x^{2} ni ayirish.

$$-x+2-x^{2}=-6x+8$$

6x ni ikki tarafga qo’shing.

$$-x+2-x^{2}+6x=8$$

5x ni olish uchun -x va 6x ni birlashtirish.

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

Ikkala tarafdan 8 ni ayirish.

$$5x+2-x^{2}-8=0$$

-6 olish uchun 2 dan 8 ni ayirish.

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

ax^{2}+bx+c=0 shaklidagi barcha tenglamalarni kvadrat formulasi bilan yechish mumkin: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. Kvadrat formula ikki yechmni taqdim qiladi, biri ± qo'shish bo'lganda, va ikkinchisi ayiruv bo'lganda.

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

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.

$$x=\frac{-5±\sqrt{5^{2}-4\left(-1\right)\left(-6\right)}}{2\left(-1\right)}$$

5 kvadratini chiqarish.

$$x=\frac{-5±\sqrt{25-4\left(-1\right)\left(-6\right)}}{2\left(-1\right)}$$

-4 ni -1 marotabaga ko'paytirish.

$$x=\frac{-5±\sqrt{25+4\left(-6\right)}}{2\left(-1\right)}$$

4 ni -6 marotabaga ko'paytirish.

$$x=\frac{-5±\sqrt{25-24}}{2\left(-1\right)}$$

25 ni -24 ga qo'shish.

$$x=\frac{-5±\sqrt{1}}{2\left(-1\right)}$$

1 ning kvadrat ildizini chiqarish.

$$x=\frac{-5±1}{2\left(-1\right)}$$

2 ni -1 marotabaga ko'paytirish.

$$x=\frac{-5±1}{-2}$$

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.

$$x=2 x=3$$

x qiymati 2 teng bo‘lmaydi.

$$x=3$$

Kvadratni yakunlash bo'yicha qadamlar

x qiymati -6,-4,2 qiymatlaridan birortasiga teng bo‘lmaydi, chunki nolga bo‘lish mumkin emas. Tenglamaning ikkala tarafini \left(x-2\right)\left(x+4\right)\left(x+6\right) ga, x^{2}+2x-8,x^{2}+4x-12,x^{2}+10x+24 ning eng kichik karralisiga ko‘paytiring.

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

x+6 ga 3 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$3x+18-\left(x+4\right)\times 4=\left(x-2\right)\left(x-4\right)$$

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

$$3x+18-\left(4x+16\right)=\left(x-2\right)\left(x-4\right)$$

4x+16 teskarisini topish uchun har birining teskarisini toping.

$$3x+18-4x-16=\left(x-2\right)\left(x-4\right)$$

-x ni olish uchun 3x va -4x ni birlashtirish.

$$-x+18-16=\left(x-2\right)\left(x-4\right)$$

2 olish uchun 18 dan 16 ni ayirish.

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

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

$$-x+2=x^{2}-6x+8$$

Ikkala tarafdan x^{2} ni ayirish.

$$-x+2-x^{2}=-6x+8$$

6x ni ikki tarafga qo’shing.

$$-x+2-x^{2}+6x=8$$

5x ni olish uchun -x va 6x ni birlashtirish.

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

Ikkala tarafdan 2 ni ayirish.

$$5x-x^{2}=8-2$$

6 olish uchun 8 dan 2 ni ayirish.

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

Bu kabi kvadrat tenglamalarni kvadratni yakunlab yechish mumkin. Kvadratni yechish uchun tenglama avval ushbu shaklda bo'lishi shart: x^{2}+bx=c.

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

Ikki tarafini -1 ga bo‘ling.

$$\frac{-x^{2}+5x}{-1}=\frac{6}{-1}$$

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

$$x^{2}+\frac{5}{-1}x=\frac{6}{-1}$$

5 ni -1 ga bo'lish.

$$x^{2}-5x=\frac{6}{-1}$$

6 ni -1 ga bo'lish.

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

-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.

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

\frac{5}{2} ni tenglamaning ikkala tarafiga qo'shish.

$$x=3 x=2$$

x qiymati 2 teng bo‘lmaydi.

$$x=3$$