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

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

x uchun yechish

$$x=\frac{2}{3}\approx 0,666666667$$

$$x=0$$

Faktorlash qadamlari

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, 2+x,2-x ning eng kichik karralisiga ko‘paytiring.

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

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

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

\left(2+x\right)\times \frac{x}{2} ni yagona kasrga aylantiring.

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

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

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

x+\frac{1}{2}x^{2} natijani olish uchun 2x+x^{2} ning har bir ifodasini 2 ga bo‘ling.

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

x+\frac{1}{2}x^{2} teskarisini topish uchun har birining teskarisini toping.

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

x ni ikki tarafga qo’shing.

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

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

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

\frac{1}{2}x^{2} ni ikki tarafga qo’shing.

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

\frac{3}{2}x^{2} ni olish uchun x^{2} va \frac{1}{2}x^{2} ni birlashtirish.

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

x omili.

$$x\left(\frac{3}{2}x-1\right)=0$$

Tenglamani yechish uchun x=0 va \frac{3x}{2}-1=0 ni yeching.

$$x=0 x=\frac{2}{3}$$

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, 2+x,2-x ning eng kichik karralisiga ko‘paytiring.

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

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

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

\left(2+x\right)\times \frac{x}{2} ni yagona kasrga aylantiring.

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

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

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

x+\frac{1}{2}x^{2} natijani olish uchun 2x+x^{2} ning har bir ifodasini 2 ga bo‘ling.

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

x+\frac{1}{2}x^{2} teskarisini topish uchun har birining teskarisini toping.

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

x ni ikki tarafga qo’shing.

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

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

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

\frac{1}{2}x^{2} ni ikki tarafga qo’shing.

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

\frac{3}{2}x^{2} ni olish uchun x^{2} va \frac{1}{2}x^{2} ni birlashtirish.

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

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

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

1 ning kvadrat ildizini chiqarish.

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

-1 ning teskarisi 1 ga teng.

$$x=\frac{1±1}{2\times \frac{3}{2}}$$

2 ni \frac{3}{2} marotabaga ko'paytirish.

$$x=\frac{1±1}{3}$$

x=\frac{1±1}{3} tenglamasini yeching, bunda ± musbat. 1 ni 1 ga qo'shish.

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

x=\frac{1±1}{3} tenglamasini yeching, bunda ± manfiy. 1 dan 1 ni ayirish.

$$x=\frac{0}{3}$$

0 ni 3 ga bo'lish.

$$x=0$$

Tenglama yechildi.

$$x=\frac{2}{3} x=0$$

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, 2+x,2-x ning eng kichik karralisiga ko‘paytiring.

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

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

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

\left(2+x\right)\times \frac{x}{2} ni yagona kasrga aylantiring.

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

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

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

x+\frac{1}{2}x^{2} natijani olish uchun 2x+x^{2} ning har bir ifodasini 2 ga bo‘ling.

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

x+\frac{1}{2}x^{2} teskarisini topish uchun har birining teskarisini toping.

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

x ni ikki tarafga qo’shing.

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

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

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

\frac{1}{2}x^{2} ni ikki tarafga qo’shing.

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

\frac{3}{2}x^{2} ni olish uchun x^{2} va \frac{1}{2}x^{2} ni birlashtirish.

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

Tenglamaning ikki tarafini \frac{3}{2} ga bo'lish, bu kasrni qaytarish orqali ikkala tarafga ko'paytirish bilan aynidir.

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

\frac{3}{2} ga bo'lish \frac{3}{2} ga ko'paytirishni bekor qiladi.

$$x^{2}+\left(-\frac{1}{\frac{3}{2}}\right)x=\frac{0}{\frac{3}{2}}$$

-1 ni \frac{3}{2} ga bo'lish -1 ga k'paytirish \frac{3}{2} ga qaytarish.

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

0 ni \frac{3}{2} ga bo'lish 0 ga k'paytirish \frac{3}{2} ga qaytarish.

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

-\frac{2}{3} ni bo‘lish, x shartining koeffitsienti, 2 ga -\frac{1}{3} olish uchun. Keyin, -\frac{1}{3} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

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

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib -\frac{1}{3} kvadratini chiqarish.

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

x^{2}-\frac{2}{3}x+\frac{1}{9} omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(x-\frac{1}{3}\right)^{2}=\frac{1}{9}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x-\frac{1}{3}\right)^{2}}=\sqrt{\frac{1}{9}}$$

Qisqartirish.

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

\frac{1}{3} ni tenglamaning ikkala tarafiga qo'shish.

$$x=\frac{2}{3} x=0$$