3/x-1+3/x+1=-4 eching

$$\frac{ 3 }{ x-1 } + \frac{ 3 }{ x+1 } = -4$$

x uchun yechish

$$x=-2$$

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

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.

$$\left(x+1\right)\times 3+\left(x-1\right)\times 3=-4\left(x-1\right)\left(x+1\right)$$

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

$$3x+3+\left(x-1\right)\times 3=-4\left(x-1\right)\left(x+1\right)$$

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

$$3x+3+3x-3=-4\left(x-1\right)\left(x+1\right)$$

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

$$6x+3-3=-4\left(x-1\right)\left(x+1\right)$$

0 olish uchun 3 dan 3 ni ayirish.

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

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

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

-4x+4 ga x+1 ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

$$6x=-4x^{2}+4$$

4x^{2} ni ikki tarafga qo’shing.

$$6x+4x^{2}=4$$

Ikkala tarafdan 4 ni ayirish.

$$6x+4x^{2}-4=0$$

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.

$$4x^{2}+6x-4=0$$

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

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

6 kvadratini chiqarish.

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

-4 ni 4 marotabaga ko'paytirish.

$$x=\frac{-6±\sqrt{36-16\left(-4\right)}}{2\times 4}$$

-16 ni -4 marotabaga ko'paytirish.

$$x=\frac{-6±\sqrt{36+64}}{2\times 4}$$

36 ni 64 ga qo'shish.

$$x=\frac{-6±\sqrt{100}}{2\times 4}$$

100 ning kvadrat ildizini chiqarish.

$$x=\frac{-6±10}{2\times 4}$$

2 ni 4 marotabaga ko'paytirish.

$$x=\frac{-6±10}{8}$$

x=\frac{-6±10}{8} tenglamasini yeching, bunda ± musbat. -6 ni 10 ga qo'shish.

$$x=\frac{4}{8}$$

\frac{4}{8} ulushini 4 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

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

x=\frac{-6±10}{8} tenglamasini yeching, bunda ± manfiy. -6 dan 10 ni ayirish.

$$x=-\frac{16}{8}$$

-16 ni 8 ga bo'lish.

$$x=-2$$

Tenglama yechildi.

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

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.

$$\left(x+1\right)\times 3+\left(x-1\right)\times 3=-4\left(x-1\right)\left(x+1\right)$$

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

$$3x+3+\left(x-1\right)\times 3=-4\left(x-1\right)\left(x+1\right)$$

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

$$3x+3+3x-3=-4\left(x-1\right)\left(x+1\right)$$

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

$$6x+3-3=-4\left(x-1\right)\left(x+1\right)$$

0 olish uchun 3 dan 3 ni ayirish.

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

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

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

-4x+4 ga x+1 ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

$$6x=-4x^{2}+4$$

4x^{2} ni ikki tarafga qo’shing.

$$6x+4x^{2}=4$$

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

$$4x^{2}+6x=4$$

Ikki tarafini 4 ga bo‘ling.

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

4 ga bo'lish 4 ga ko'paytirishni bekor qiladi.

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

\frac{6}{4} ulushini 2 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

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

4 ni 4 ga bo'lish.

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

\frac{3}{2} ni bo‘lish, x shartining koeffitsienti, 2 ga \frac{3}{4} olish uchun. Keyin, \frac{3}{4} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}+\frac{3}{2}x+\left(\frac{3}{4}\right)^{2}=1+\left(\frac{3}{4}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib \frac{3}{4} kvadratini chiqarish.

$$x^{2}+\frac{3}{2}x+\frac{9}{16}=1+\frac{9}{16}$$

1 ni \frac{9}{16} ga qo'shish.

$$x^{2}+\frac{3}{2}x+\frac{9}{16}=\frac{25}{16}$$

x^{2}+\frac{3}{2}x+\frac{9}{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.

$$\left(x+\frac{3}{4}\right)^{2}=\frac{25}{16}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x+\frac{3}{4}\right)^{2}}=\sqrt{\frac{25}{16}}$$

Qisqartirish.

$$x+\frac{3}{4}=\frac{5}{4} x+\frac{3}{4}=-\frac{5}{4}$$

Tenglamaning ikkala tarafidan \frac{3}{4} ni ayirish.

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