(x-3/2)^2+2x(x-1/2)<3(x^2+1/4) eching

$$( x - \frac { 3 } { 2 } ) ^ { 2 } + 2 x ( x - \frac { 1 } { 2 } ) < 3 ( x ^ { 2 } + \frac { 1 } { 4 } )$$

x uchun yechish

$$x>\frac{3}{8}$$

Yechim qadamlari

\left(a-b\right)^{2}=a^{2}-2ab+b^{2} binom teoremasini \left(x-\frac{3}{2}\right)^{2} kengaytirilishi uchun ishlating.

$$x^{2}-3x+\frac{9}{4}+2x\left(x-\frac{1}{2}\right)<3\left(x^{2}+\frac{1}{4}\right)$$

2x ga x-\frac{1}{2} ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$x^{2}-3x+\frac{9}{4}+2x^{2}-x<3\left(x^{2}+\frac{1}{4}\right)$$

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

$$3x^{2}-3x+\frac{9}{4}-x<3\left(x^{2}+\frac{1}{4}\right)$$

-4x ni olish uchun -3x va -x ni birlashtirish.

$$3x^{2}-4x+\frac{9}{4}<3\left(x^{2}+\frac{1}{4}\right)$$

3 ga x^{2}+\frac{1}{4} ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

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

Ikkala tarafdan 3x^{2} ni ayirish.

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

0 ni olish uchun 3x^{2} va -3x^{2} ni birlashtirish.

$$-4x+\frac{9}{4}<\frac{3}{4}$$

Ikkala tarafdan \frac{9}{4} ni ayirish.

$$-4x<\frac{3}{4}-\frac{9}{4}$$

-\frac{3}{2} olish uchun \frac{3}{4} dan \frac{9}{4} ni ayirish.

$$-4x<-\frac{3}{2}$$

Ikki tarafini -4 ga bo‘ling. -4 manfiy boʻlgani uchun tengsizlikning yo‘nalishi o‘zgaradi.

$$x>\frac{-\frac{3}{2}}{-4}$$

\frac{-\frac{3}{2}}{-4} ni yagona kasrga aylantiring.

$$x>\frac{-3}{2\left(-4\right)}$$

-8 hosil qilish uchun 2 va -4 ni ko'paytirish.

$$x>\frac{-3}{-8}$$

Ikkala surat va maxrajdan manfiy belgini olib tashlash bilan \frac{-3}{-8} kasrini \frac{3}{8} ga soddalashtirish mumkin.

$$x>\frac{3}{8}$$