4/x-1+2/x+1=35 eching

$$\frac { 4 } { x - 1 } + \frac { 2 } { x + 1 } = 35$$

x uchun yechish

$$x = \frac{2 \sqrt{326} + 3}{35} \approx 1,117455433$$

$$x=\frac{3-2\sqrt{326}}{35}\approx -0,946026862$$

Kvadrat tenglamalaridan foydalanish qoidalari

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

$$\left(x+1\right)\times 4+\left(x-1\right)\times 2=35\left(x-1\right)\left(x+1\right)$$

x+1 ga 4 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$4x+4+\left(x-1\right)\times 2=35\left(x-1\right)\left(x+1\right)$$

x-1 ga 2 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$4x+4+2x-2=35\left(x-1\right)\left(x+1\right)$$

6x ni olish uchun 4x va 2x ni birlashtirish.

$$6x+4-2=35\left(x-1\right)\left(x+1\right)$$

2 olish uchun 4 dan 2 ni ayirish.

$$6x+2=35\left(x-1\right)\left(x+1\right)$$

35 ga x-1 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$6x+2=\left(35x-35\right)\left(x+1\right)$$

35x-35 ga x+1 ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

$$6x+2=35x^{2}-35$$

Ikkala tarafdan 35x^{2} ni ayirish.

$$6x+2-35x^{2}=-35$$

35 ni ikki tarafga qo’shing.

$$6x+2-35x^{2}+35=0$$

37 olish uchun 2 va 35'ni qo'shing.

$$6x+37-35x^{2}=0$$

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.

$$-35x^{2}+6x+37=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} -35 ni a, 6 ni b va 37 ni c bilan almashtiring.

$$x=\frac{-6±\sqrt{6^{2}-4\left(-35\right)\times 37}}{2\left(-35\right)}$$

6 kvadratini chiqarish.

$$x=\frac{-6±\sqrt{36-4\left(-35\right)\times 37}}{2\left(-35\right)}$$

-4 ni -35 marotabaga ko'paytirish.

$$x=\frac{-6±\sqrt{36+140\times 37}}{2\left(-35\right)}$$

140 ni 37 marotabaga ko'paytirish.

$$x=\frac{-6±\sqrt{36+5180}}{2\left(-35\right)}$$

36 ni 5180 ga qo'shish.

$$x=\frac{-6±\sqrt{5216}}{2\left(-35\right)}$$

5216 ning kvadrat ildizini chiqarish.

$$x=\frac{-6±4\sqrt{326}}{2\left(-35\right)}$$

2 ni -35 marotabaga ko'paytirish.

$$x=\frac{-6±4\sqrt{326}}{-70}$$

x=\frac{-6±4\sqrt{326}}{-70} tenglamasini yeching, bunda ± musbat. -6 ni 4\sqrt{326} ga qo'shish.

$$x=\frac{4\sqrt{326}-6}{-70}$$

-6+4\sqrt{326} ni -70 ga bo'lish.

$$x=\frac{3-2\sqrt{326}}{35}$$

x=\frac{-6±4\sqrt{326}}{-70} tenglamasini yeching, bunda ± manfiy. -6 dan 4\sqrt{326} ni ayirish.

$$x=\frac{-4\sqrt{326}-6}{-70}$$

-6-4\sqrt{326} ni -70 ga bo'lish.

$$x=\frac{2\sqrt{326}+3}{35}$$

Tenglama yechildi.

$$x=\frac{3-2\sqrt{326}}{35} x=\frac{2\sqrt{326}+3}{35}$$

Kvadratni yakunlash bo'yicha qadamlar

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

$$\left(x+1\right)\times 4+\left(x-1\right)\times 2=35\left(x-1\right)\left(x+1\right)$$

x+1 ga 4 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$4x+4+\left(x-1\right)\times 2=35\left(x-1\right)\left(x+1\right)$$

x-1 ga 2 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$4x+4+2x-2=35\left(x-1\right)\left(x+1\right)$$

6x ni olish uchun 4x va 2x ni birlashtirish.

$$6x+4-2=35\left(x-1\right)\left(x+1\right)$$

2 olish uchun 4 dan 2 ni ayirish.

$$6x+2=35\left(x-1\right)\left(x+1\right)$$

35 ga x-1 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$6x+2=\left(35x-35\right)\left(x+1\right)$$

35x-35 ga x+1 ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

$$6x+2=35x^{2}-35$$

Ikkala tarafdan 35x^{2} ni ayirish.

$$6x+2-35x^{2}=-35$$

Ikkala tarafdan 2 ni ayirish.

$$6x-35x^{2}=-35-2$$

-37 olish uchun -35 dan 2 ni ayirish.

$$6x-35x^{2}=-37$$

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

$$-35x^{2}+6x=-37$$

Ikki tarafini -35 ga bo‘ling.

$$\frac{-35x^{2}+6x}{-35}=-\frac{37}{-35}$$

-35 ga bo'lish -35 ga ko'paytirishni bekor qiladi.

$$x^{2}+\frac{6}{-35}x=-\frac{37}{-35}$$

6 ni -35 ga bo'lish.

$$x^{2}-\frac{6}{35}x=-\frac{37}{-35}$$

-37 ni -35 ga bo'lish.

$$x^{2}-\frac{6}{35}x=\frac{37}{35}$$

-\frac{6}{35} ni bo‘lish, x shartining koeffitsienti, 2 ga -\frac{3}{35} olish uchun. Keyin, -\frac{3}{35} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}-\frac{6}{35}x+\left(-\frac{3}{35}\right)^{2}=\frac{37}{35}+\left(-\frac{3}{35}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib -\frac{3}{35} kvadratini chiqarish.

$$x^{2}-\frac{6}{35}x+\frac{9}{1225}=\frac{37}{35}+\frac{9}{1225}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali \frac{37}{35} ni \frac{9}{1225} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$x^{2}-\frac{6}{35}x+\frac{9}{1225}=\frac{1304}{1225}$$

x^{2}-\frac{6}{35}x+\frac{9}{1225} omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(x-\frac{3}{35}\right)^{2}=\frac{1304}{1225}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x-\frac{3}{35}\right)^{2}}=\sqrt{\frac{1304}{1225}}$$

Qisqartirish.

$$x-\frac{3}{35}=\frac{2\sqrt{326}}{35} x-\frac{3}{35}=-\frac{2\sqrt{326}}{35}$$

\frac{3}{35} ni tenglamaning ikkala tarafiga qo'shish.

$$x=\frac{2\sqrt{326}+3}{35} x=\frac{3-2\sqrt{326}}{35}$$