0=1/2sqrt{x_{0-1}}*(0-x_{0})+sqrt{x_{0}-1} eching

$$0 = \frac { 1 } { 2 \sqrt { x _ { 0 } - 1 } } \cdot ( 0 - x _ { 0 } ) + \sqrt { x _ { 0 } - 1 }$$

x_0 uchun yechish

$$x_{0}=2$$

Yechim qadamlari

Tenglamaning ikkala tarafidan \sqrt{x_{0}-1} ni ayirish.

$$-\sqrt{x_{0}-1}=\frac{1}{2\sqrt{x_{0}-1}}\left(-1\right)x_{0}$$

\frac{1}{2\sqrt{x_{0}-1}}x_{0} ni yagona kasrga aylantiring.

$$-\sqrt{x_{0}-1}=\frac{x_{0}}{2\sqrt{x_{0}-1}}\left(-1\right)$$

-1ni ikki tarafidan bekor qilish.

$$\sqrt{x_{0}-1}=\frac{x_{0}}{2\sqrt{x_{0}-1}}$$

Tenglamaning ikkala taraf kvadratini chiqarish.

$$\left(\sqrt{x_{0}-1}\right)^{2}=\left(\frac{x_{0}}{2\sqrt{x_{0}-1}}\right)^{2}$$

2 daraja ko‘rsatkichini \sqrt{x_{0}-1} ga hisoblang va x_{0}-1 ni qiymatni oling.

$$x_{0}-1=\left(\frac{x_{0}}{2\sqrt{x_{0}-1}}\right)^{2}$$

\frac{x_{0}}{2\sqrt{x_{0}-1}}ni darajaga oshirish uchun, surat va maxrajni darajaga oshirib, keyin bo‘ling.

$$x_{0}-1=\frac{x_{0}^{2}}{\left(2\sqrt{x_{0}-1}\right)^{2}}$$

\left(2\sqrt{x_{0}-1}\right)^{2} ni kengaytirish.

$$x_{0}-1=\frac{x_{0}^{2}}{2^{2}\left(\sqrt{x_{0}-1}\right)^{2}}$$

2 daraja ko‘rsatkichini 2 ga hisoblang va 4 ni qiymatni oling.

$$x_{0}-1=\frac{x_{0}^{2}}{4\left(\sqrt{x_{0}-1}\right)^{2}}$$

2 daraja ko‘rsatkichini \sqrt{x_{0}-1} ga hisoblang va x_{0}-1 ni qiymatni oling.

$$x_{0}-1=\frac{x_{0}^{2}}{4\left(x_{0}-1\right)}$$

Tenglamaning ikkala tarafini 4\left(x_{0}-1\right) ga ko'paytirish.

$$4\left(x_{0}-1\right)x_{0}+4\left(x_{0}-1\right)\left(-1\right)=x_{0}^{2}$$

Shartlarni qayta saralash.

$$4x_{0}\left(x_{0}-1\right)-4\left(x_{0}-1\right)=x_{0}^{2}$$

4x_{0} ga x_{0}-1 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$4x_{0}^{2}-4x_{0}-4\left(x_{0}-1\right)=x_{0}^{2}$$

-4 ga x_{0}-1 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$4x_{0}^{2}-4x_{0}-4x_{0}+4=x_{0}^{2}$$

-8x_{0} ni olish uchun -4x_{0} va -4x_{0} ni birlashtirish.

$$4x_{0}^{2}-8x_{0}+4=x_{0}^{2}$$

Ikkala tarafdan x_{0}^{2} ni ayirish.

$$4x_{0}^{2}-8x_{0}+4-x_{0}^{2}=0$$

3x_{0}^{2} ni olish uchun 4x_{0}^{2} va -x_{0}^{2} ni birlashtirish.

$$3x_{0}^{2}-8x_{0}+4=0$$

Tenglamani yechish uchun guruhlash orqali chap qoʻl tomonni faktorlang. Avvalo, chap qoʻl tomon 3x_{0}^{2}+ax_{0}+bx_{0}+4 sifatida qayta yozilishi kerak. a va b ni topish uchun yechiladigan tizimni sozlang.

$$a+b=-8 ab=3\times 4=12$$

ab musbat boʻlganda, a va b da bir xil belgi bor. a+b manfiy boʻlganda, a va b ikkisi ham manfiy. 12-mahsulotni beruvchi bunday butun juftliklarni roʻyxat qiling.

$$-1,-12 -2,-6 -3,-4$$

Har bir juftlik yigʻindisini hisoblang.

$$-1-12=-13 -2-6=-8 -3-4=-7$$

Yechim – -8 yigʻindisini beruvchi juftlik.

$$a=-6 b=-2$$

3x_{0}^{2}-8x_{0}+4 ni \left(3x_{0}^{2}-6x_{0}\right)+\left(-2x_{0}+4\right) sifatida qaytadan yozish.

$$\left(3x_{0}^{2}-6x_{0}\right)+\left(-2x_{0}+4\right)$$

Birinchi guruhda 3x_{0} ni va ikkinchi guruhda -2 ni faktordan chiqaring.

$$3x_{0}\left(x_{0}-2\right)-2\left(x_{0}-2\right)$$

Distributiv funktsiyasidan foydalangan holda x_{0}-2 umumiy terminini chiqaring.

$$\left(x_{0}-2\right)\left(3x_{0}-2\right)$$

Tenglamani yechish uchun x_{0}-2=0 va 3x_{0}-2=0 ni yeching.

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

0=\frac{1}{2\sqrt{x_{0}-1}}\left(0-x_{0}\right)+\sqrt{x_{0}-1} tenglamasida x_{0} uchun 2 ni almashtiring.

$$0=\frac{1}{2\sqrt{2-1}}\left(0-2\right)+\sqrt{2-1}$$

Qisqartirish. x_{0}=2 tenglamani qoniqtiradi.

$$0=0$$

0=\frac{1}{2\sqrt{x_{0}-1}}\left(0-x_{0}\right)+\sqrt{x_{0}-1} tenglamasida x_{0} uchun \frac{2}{3} ni almashtiring. \sqrt{\frac{2}{3}-1} ifodasi noaniq, chunki ildiz ostidagi qiymat manfiy boʻlishi mumkin emas.

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

\sqrt{x_{0}-1}=\frac{x_{0}}{2\sqrt{x_{0}-1}} tenglamasi noyob yechimga ega.

$$x_{0}=2$$