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

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

x uchun yechish

$$x=3$$

$$x=0$$

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.

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

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

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

3x ni olish uchun 2x va x ni birlashtirish.

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

-1 olish uchun -2 va 1'ni qo'shing.

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

Hisoblang: \left(x-1\right)\left(x+1\right). Ko‘paytirish qoida yordamida turli kvadratlarga aylantirilishi mumkin: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}. 1 kvadratini chiqarish.

$$3x-1=x^{2}-1$$

Ikkala tarafdan x^{2} ni ayirish.

$$3x-1-x^{2}=-1$$

1 ni ikki tarafga qo’shing.

$$3x-1-x^{2}+1=0$$

0 olish uchun -1 va 1'ni qo'shing.

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

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.

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

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} -1 ni a, 3 ni b va 0 ni c bilan almashtiring.

$$x=\frac{-3±\sqrt{3^{2}}}{2\left(-1\right)}$$

3^{2} ning kvadrat ildizini chiqarish.

$$x=\frac{-3±3}{2\left(-1\right)}$$

2 ni -1 marotabaga ko'paytirish.

$$x=\frac{-3±3}{-2}$$

x=\frac{-3±3}{-2} tenglamasini yeching, bunda ± musbat. -3 ni 3 ga qo'shish.

$$x=\frac{0}{-2}$$

0 ni -2 ga bo'lish.

$$x=0$$

x=\frac{-3±3}{-2} tenglamasini yeching, bunda ± manfiy. -3 dan 3 ni ayirish.

$$x=-\frac{6}{-2}$$

-6 ni -2 ga bo'lish.

$$x=3$$

Tenglama yechildi.

$$x=0 x=3$$

Kvadratni yakunlash bo'yicha qadamlar

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

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

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

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

3x ni olish uchun 2x va x ni birlashtirish.

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

-1 olish uchun -2 va 1'ni qo'shing.

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

Hisoblang: \left(x-1\right)\left(x+1\right). Ko‘paytirish qoida yordamida turli kvadratlarga aylantirilishi mumkin: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}. 1 kvadratini chiqarish.

$$3x-1=x^{2}-1$$

Ikkala tarafdan x^{2} ni ayirish.

$$3x-1-x^{2}=-1$$

1 ni ikki tarafga qo’shing.

$$3x-x^{2}=-1+1$$

0 olish uchun -1 va 1'ni qo'shing.

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

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

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

Ikki tarafini -1 ga bo‘ling.

$$\frac{-x^{2}+3x}{-1}=\frac{0}{-1}$$

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

$$x^{2}+\frac{3}{-1}x=\frac{0}{-1}$$

3 ni -1 ga bo'lish.

$$x^{2}-3x=\frac{0}{-1}$$

0 ni -1 ga bo'lish.

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

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

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

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib -\frac{3}{2} kvadratini chiqarish.

$$x^{2}-3x+\frac{9}{4}=\frac{9}{4}$$

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

$$\left(x-\frac{3}{2}\right)^{2}=\frac{9}{4}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x-\frac{3}{2}\right)^{2}}=\sqrt{\frac{9}{4}}$$

Qisqartirish.

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

\frac{3}{2} ni tenglamaning ikkala tarafiga qo'shish.

$$x=3 x=0$$