sqrt{4+2x-x^2}=x-2 eching

$$\sqrt{ 4+2x- { x }^{ 2 } } =x-2$$

x uchun yechish

$$x=3$$

Yechim qadamlari

Tenglamaning ikkala taraf kvadratini chiqarish.

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

2 daraja ko‘rsatkichini \sqrt{4+2x-x^{2}} ga hisoblang va 4+2x-x^{2} ni qiymatni oling.

$$4+2x-x^{2}=\left(x-2\right)^{2}$$

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

$$4+2x-x^{2}=x^{2}-4x+4$$

Ikkala tarafdan x^{2} ni ayirish.

$$4+2x-x^{2}-x^{2}=-4x+4$$

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

$$4+2x-2x^{2}=-4x+4$$

4x ni ikki tarafga qo’shing.

$$4+2x-2x^{2}+4x=4$$

6x ni olish uchun 2x va 4x ni birlashtirish.

$$4+6x-2x^{2}=4$$

Ikkala tarafdan 4 ni ayirish.

$$4+6x-2x^{2}-4=0$$

0 olish uchun 4 dan 4 ni ayirish.

$$6x-2x^{2}=0$$

x omili.

$$x\left(6-2x\right)=0$$

Tenglamani yechish uchun x=0 va 6-2x=0 ni yeching.

$$x=0 x=3$$

\sqrt{4+2x-x^{2}}=x-2 tenglamasida x uchun 0 ni almashtiring.

$$\sqrt{4+2\times 0-0^{2}}=0-2$$

Qisqartirish. x=0 qiymati bu tenglamani qoniqtirmaydi, chunki oʻng va chap tarafdagi belgilar bir-biriga qarama-qarshi.

$$2=-2$$

\sqrt{4+2x-x^{2}}=x-2 tenglamasida x uchun 3 ni almashtiring.

$$\sqrt{4+2\times 3-3^{2}}=3-2$$

Qisqartirish. x=3 tenglamani qoniqtiradi.

$$1=1$$

\sqrt{4+2x-x^{2}}=x-2 tenglamasi noyob yechimga ega.

$$x=3$$