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

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

x uchun yechish

$$x = \frac{\sqrt{5} + 1}{2} \approx 1,618033989$$

$$x=\frac{1-\sqrt{5}}{2}\approx -0,618033989$$

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,x^{2}-1 ning eng kichik karralisiga ko‘paytiring.

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

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

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

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

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

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

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

Ikkala tarafdan x^{2} ni ayirish.

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

Ikkala tarafdan 2x ni ayirish.

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

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

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

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.

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

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

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

1 kvadratini chiqarish.

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

-4 ni -1 marotabaga ko'paytirish.

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

1 ni 4 ga qo'shish.

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

2 ni -1 marotabaga ko'paytirish.

$$x=\frac{-1±\sqrt{5}}{-2}$$

x=\frac{-1±\sqrt{5}}{-2} tenglamasini yeching, bunda ± musbat. -1 ni \sqrt{5} ga qo'shish.

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

-1+\sqrt{5} ni -2 ga bo'lish.

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

x=\frac{-1±\sqrt{5}}{-2} tenglamasini yeching, bunda ± manfiy. -1 dan \sqrt{5} ni ayirish.

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

-1-\sqrt{5} ni -2 ga bo'lish.

$$x=\frac{\sqrt{5}+1}{2}$$

Tenglama yechildi.

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

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,x^{2}-1 ning eng kichik karralisiga ko‘paytiring.

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

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

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

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

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

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

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

Ikkala tarafdan x^{2} ni ayirish.

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

Ikkala tarafdan 2x ni ayirish.

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

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

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

Ikkala tarafdan 1 ni ayirish. Har qanday sonni noldan ayirsangiz, o‘zining manfiyi chiqadi.

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

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

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

Ikki tarafini -1 ga bo‘ling.

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

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

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

1 ni -1 ga bo'lish.

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

-1 ni -1 ga bo'lish.

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

-1 ni bo‘lish, x shartining koeffitsienti, 2 ga -\frac{1}{2} olish uchun. Keyin, -\frac{1}{2} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

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

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib -\frac{1}{2} kvadratini chiqarish.

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

1 ni \frac{1}{4} ga qo'shish.

$$x^{2}-x+\frac{1}{4}=\frac{5}{4}$$

x^{2}-x+\frac{1}{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.

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

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

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

Qisqartirish.

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

\frac{1}{2} ni tenglamaning ikkala tarafiga qo'shish.

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