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

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

x uchun yechish (complex solution)

$$x=\sqrt{3}-1\approx 0,732050808$$

$$x=-\left(\sqrt{3}+1\right)\approx -2,732050808$$

Kvadrat tenglamalaridan foydalanish qoidalari

x qiymati -2,2 qiymatlaridan birortasiga teng bo‘lmaydi, chunki nolga bo‘lish mumkin emas. Tenglamaning ikkala tarafini \left(x-2\right)\left(x+2\right) ga, x+2,x-2,x^{2}-4 ning eng kichik karralisiga ko‘paytiring.

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

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

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

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

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

Ikkala tarafdan x ni ayirish.

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

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

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

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.

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

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

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

2 kvadratini chiqarish.

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

-4 ni -2 marotabaga ko'paytirish.

$$x=\frac{-2±\sqrt{4+8}}{2}$$

4 ni 8 ga qo'shish.

$$x=\frac{-2±\sqrt{12}}{2}$$

12 ning kvadrat ildizini chiqarish.

$$x=\frac{-2±2\sqrt{3}}{2}$$

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

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

-2+2\sqrt{3} ni 2 ga bo'lish.

$$x=\sqrt{3}-1$$

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

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

-2-2\sqrt{3} ni 2 ga bo'lish.

$$x=-\sqrt{3}-1$$

Tenglama yechildi.

$$x=\sqrt{3}-1 x=-\sqrt{3}-1$$

Kvadratni yakunlash bo'yicha qadamlar

x qiymati -2,2 qiymatlaridan birortasiga teng bo‘lmaydi, chunki nolga bo‘lish mumkin emas. Tenglamaning ikkala tarafini \left(x-2\right)\left(x+2\right) ga, x+2,x-2,x^{2}-4 ning eng kichik karralisiga ko‘paytiring.

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

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

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

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

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

Ikkala tarafdan x ni ayirish.

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

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

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

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

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

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

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

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

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

1 kvadratini chiqarish.

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

2 ni 1 ga qo'shish.

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

x^{2}+2x+1 omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

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

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

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

Qisqartirish.

$$x+1=\sqrt{3} x+1=-\sqrt{3}$$

Tenglamaning ikkala tarafidan 1 ni ayirish.

$$x=\sqrt{3}-1 x=-\sqrt{3}-1$$

x uchun yechish

$$x=\sqrt{3}-1\approx 0,732050808$$

$$x=-\sqrt{3}-1\approx -2,732050808$$

Kvadrat tenglamalaridan foydalanish qoidalari

x qiymati -2,2 qiymatlaridan birortasiga teng bo‘lmaydi, chunki nolga bo‘lish mumkin emas. Tenglamaning ikkala tarafini \left(x-2\right)\left(x+2\right) ga, x+2,x-2,x^{2}-4 ning eng kichik karralisiga ko‘paytiring.

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

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

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

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

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

Ikkala tarafdan x ni ayirish.

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

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

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

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.

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

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

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

2 kvadratini chiqarish.

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

-4 ni -2 marotabaga ko'paytirish.

$$x=\frac{-2±\sqrt{4+8}}{2}$$

4 ni 8 ga qo'shish.

$$x=\frac{-2±\sqrt{12}}{2}$$

12 ning kvadrat ildizini chiqarish.

$$x=\frac{-2±2\sqrt{3}}{2}$$

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

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

-2+2\sqrt{3} ni 2 ga bo'lish.

$$x=\sqrt{3}-1$$

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

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

-2-2\sqrt{3} ni 2 ga bo'lish.

$$x=-\sqrt{3}-1$$

Tenglama yechildi.

$$x=\sqrt{3}-1 x=-\sqrt{3}-1$$

Kvadratni yakunlash bo'yicha qadamlar

x qiymati -2,2 qiymatlaridan birortasiga teng bo‘lmaydi, chunki nolga bo‘lish mumkin emas. Tenglamaning ikkala tarafini \left(x-2\right)\left(x+2\right) ga, x+2,x-2,x^{2}-4 ning eng kichik karralisiga ko‘paytiring.

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

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

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

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

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

Ikkala tarafdan x ni ayirish.

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

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

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

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

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

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

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

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

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

1 kvadratini chiqarish.

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

2 ni 1 ga qo'shish.

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

x^{2}+2x+1 omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

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

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

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

Qisqartirish.

$$x+1=\sqrt{3} x+1=-\sqrt{3}$$

Tenglamaning ikkala tarafidan 1 ni ayirish.

$$x=\sqrt{3}-1 x=-\sqrt{3}-1$$