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

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

x uchun yechish

$$x=3$$

$$x=\frac{1}{2}=0,5$$

Guruhlash orqali faktorlash qadamlari

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

$$\left(x+1\right)\times 3+\left(2x-2\right)\times 3=\left(2x+2\right)x$$

x+1 ga 3 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

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

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

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

9x ni olish uchun 3x va 6x ni birlashtirish.

$$9x+3-6=\left(2x+2\right)x$$

-3 olish uchun 3 dan 6 ni ayirish.

$$9x-3=\left(2x+2\right)x$$

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

$$9x-3=2x^{2}+2x$$

Ikkala tarafdan 2x^{2} ni ayirish.

$$9x-3-2x^{2}=2x$$

Ikkala tarafdan 2x ni ayirish.

$$9x-3-2x^{2}-2x=0$$

7x ni olish uchun 9x va -2x ni birlashtirish.

$$7x-3-2x^{2}=0$$

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

$$-2x^{2}+7x-3=0$$

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

$$a+b=7 ab=-2\left(-3\right)=6$$

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 – 7 yigʻindisini beruvchi juftlik.

$$a=6 b=1$$

-2x^{2}+7x-3 ni \left(-2x^{2}+6x\right)+\left(x-3\right) sifatida qaytadan yozish.

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

Birinchi guruhda 2x ni va ikkinchi guruhda -1 ni faktordan chiqaring.

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

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

$$\left(-x+3\right)\left(2x-1\right)$$

Tenglamani yechish uchun -x+3=0 va 2x-1=0 ni yeching.

$$x=3 x=\frac{1}{2}$$

Kvadrat tenglamalaridan foydalanish qoidalari

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

$$\left(x+1\right)\times 3+\left(2x-2\right)\times 3=\left(2x+2\right)x$$

x+1 ga 3 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

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

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

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

9x ni olish uchun 3x va 6x ni birlashtirish.

$$9x+3-6=\left(2x+2\right)x$$

-3 olish uchun 3 dan 6 ni ayirish.

$$9x-3=\left(2x+2\right)x$$

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

$$9x-3=2x^{2}+2x$$

Ikkala tarafdan 2x^{2} ni ayirish.

$$9x-3-2x^{2}=2x$$

Ikkala tarafdan 2x ni ayirish.

$$9x-3-2x^{2}-2x=0$$

7x ni olish uchun 9x va -2x ni birlashtirish.

$$7x-3-2x^{2}=0$$

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.

$$-2x^{2}+7x-3=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} -2 ni a, 7 ni b va -3 ni c bilan almashtiring.

$$x=\frac{-7±\sqrt{7^{2}-4\left(-2\right)\left(-3\right)}}{2\left(-2\right)}$$

7 kvadratini chiqarish.

$$x=\frac{-7±\sqrt{49-4\left(-2\right)\left(-3\right)}}{2\left(-2\right)}$$

-4 ni -2 marotabaga ko'paytirish.

$$x=\frac{-7±\sqrt{49+8\left(-3\right)}}{2\left(-2\right)}$$

8 ni -3 marotabaga ko'paytirish.

$$x=\frac{-7±\sqrt{49-24}}{2\left(-2\right)}$$

49 ni -24 ga qo'shish.

$$x=\frac{-7±\sqrt{25}}{2\left(-2\right)}$$

25 ning kvadrat ildizini chiqarish.

$$x=\frac{-7±5}{2\left(-2\right)}$$

2 ni -2 marotabaga ko'paytirish.

$$x=\frac{-7±5}{-4}$$

x=\frac{-7±5}{-4} tenglamasini yeching, bunda ± musbat. -7 ni 5 ga qo'shish.

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

\frac{-2}{-4} ulushini 2 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

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

x=\frac{-7±5}{-4} tenglamasini yeching, bunda ± manfiy. -7 dan 5 ni ayirish.

$$x=-\frac{12}{-4}$$

-12 ni -4 ga bo'lish.

$$x=3$$

Tenglama yechildi.

$$x=\frac{1}{2} x=3$$

Kvadratni yakunlash bo'yicha qadamlar

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

$$\left(x+1\right)\times 3+\left(2x-2\right)\times 3=\left(2x+2\right)x$$

x+1 ga 3 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

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

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

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

9x ni olish uchun 3x va 6x ni birlashtirish.

$$9x+3-6=\left(2x+2\right)x$$

-3 olish uchun 3 dan 6 ni ayirish.

$$9x-3=\left(2x+2\right)x$$

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

$$9x-3=2x^{2}+2x$$

Ikkala tarafdan 2x^{2} ni ayirish.

$$9x-3-2x^{2}=2x$$

Ikkala tarafdan 2x ni ayirish.

$$9x-3-2x^{2}-2x=0$$

7x ni olish uchun 9x va -2x ni birlashtirish.

$$7x-3-2x^{2}=0$$

3 ni ikki tarafga qo’shing. Har qanday songa nolni qo‘shsangiz, o‘zi chiqadi.

$$7x-2x^{2}=3$$

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

$$-2x^{2}+7x=3$$

Ikki tarafini -2 ga bo‘ling.

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

-2 ga bo'lish -2 ga ko'paytirishni bekor qiladi.

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

7 ni -2 ga bo'lish.

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

3 ni -2 ga bo'lish.

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

-\frac{7}{2} ni bo‘lish, x shartining koeffitsienti, 2 ga -\frac{7}{4} olish uchun. Keyin, -\frac{7}{4} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}-\frac{7}{2}x+\left(-\frac{7}{4}\right)^{2}=-\frac{3}{2}+\left(-\frac{7}{4}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib -\frac{7}{4} kvadratini chiqarish.

$$x^{2}-\frac{7}{2}x+\frac{49}{16}=-\frac{3}{2}+\frac{49}{16}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali -\frac{3}{2} ni \frac{49}{16} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$x^{2}-\frac{7}{2}x+\frac{49}{16}=\frac{25}{16}$$

x^{2}-\frac{7}{2}x+\frac{49}{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.

$$\left(x-\frac{7}{4}\right)^{2}=\frac{25}{16}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x-\frac{7}{4}\right)^{2}}=\sqrt{\frac{25}{16}}$$

Qisqartirish.

$$x-\frac{7}{4}=\frac{5}{4} x-\frac{7}{4}=-\frac{5}{4}$$

\frac{7}{4} ni tenglamaning ikkala tarafiga qo'shish.

$$x=3 x=\frac{1}{2}$$