sqrt{x-3}+sqrt{2x-2}=2 eching

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

x uchun yechish

$$x=3$$

Yechim qadamlari

Tenglamaning ikkala tarafidan \sqrt{2x-2} ni ayirish.

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

Tenglamaning ikkala taraf kvadratini chiqarish.

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

2 daraja ko‘rsatkichini \sqrt{x-3} ga hisoblang va x-3 ni qiymatni oling.

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

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

$$x-3=4-4\sqrt{2x-2}+\left(\sqrt{2x-2}\right)^{2}$$

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

$$x-3=4-4\sqrt{2x-2}+2x-2$$

2 olish uchun 4 dan 2 ni ayirish.

$$x-3=2-4\sqrt{2x-2}+2x$$

Tenglamaning ikkala tarafidan 2+2x ni ayirish.

$$x-3-\left(2+2x\right)=-4\sqrt{2x-2}$$

2+2x teskarisini topish uchun har birining teskarisini toping.

$$x-3-2-2x=-4\sqrt{2x-2}$$

-5 olish uchun -3 dan 2 ni ayirish.

$$x-5-2x=-4\sqrt{2x-2}$$

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

$$-x-5=-4\sqrt{2x-2}$$

Tenglamaning ikkala taraf kvadratini chiqarish.

$$\left(-x-5\right)^{2}=\left(-4\sqrt{2x-2}\right)^{2}$$

\left(a-b\right)^{2}=a^{2}-2ab+b^{2} binom teoremasini \left(-x-5\right)^{2} kengaytirilishi uchun ishlating.

$$x^{2}+10x+25=\left(-4\sqrt{2x-2}\right)^{2}$$

\left(-4\sqrt{2x-2}\right)^{2} ni kengaytirish.

$$x^{2}+10x+25=\left(-4\right)^{2}\left(\sqrt{2x-2}\right)^{2}$$

2 daraja ko‘rsatkichini -4 ga hisoblang va 16 ni qiymatni oling.

$$x^{2}+10x+25=16\left(\sqrt{2x-2}\right)^{2}$$

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

$$x^{2}+10x+25=16\left(2x-2\right)$$

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

$$x^{2}+10x+25=32x-32$$

Ikkala tarafdan 32x ni ayirish.

$$x^{2}+10x+25-32x=-32$$

-22x ni olish uchun 10x va -32x ni birlashtirish.

$$x^{2}-22x+25=-32$$

32 ni ikki tarafga qo’shing.

$$x^{2}-22x+25+32=0$$

57 olish uchun 25 va 32'ni qo'shing.

$$x^{2}-22x+57=0$$

Bu tenglamani yechish uchun x^{2}+\left(a+b\right)x+ab=\left(x+a\right)\left(x+b\right) formulasi yordamida x^{2}-22x+57 ni faktorlang. a va b ni topish uchun yechiladigan tizimni sozlang.

$$a+b=-22 ab=57$$

ab musbat boʻlganda, a va b da bir xil belgi bor. a+b manfiy boʻlganda, a va b ikkisi ham manfiy. 57-mahsulotni beruvchi bunday butun juftliklarni roʻyxat qiling.

$$-1,-57 -3,-19$$

Har bir juftlik yigʻindisini hisoblang.

$$-1-57=-58 -3-19=-22$$

Yechim – -22 yigʻindisini beruvchi juftlik.

$$a=-19 b=-3$$

Faktorlangan \left(x+a\right)\left(x+b\right) ifodani olingan qiymatlar bilan qaytadan yozing.

$$\left(x-19\right)\left(x-3\right)$$

Tenglamani yechish uchun x-19=0 va x-3=0 ni yeching.

$$x=19 x=3$$

\sqrt{x-3}+\sqrt{2x-2}=2 tenglamasida x uchun 19 ni almashtiring.

$$\sqrt{19-3}+\sqrt{2\times 19-2}=2$$

Qisqartirish. x=19 qiymati bu tenglamani qoniqtirmaydi.

$$10=2$$

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

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

Qisqartirish. x=3 tenglamani qoniqtiradi.

$$2=2$$

\sqrt{x-3}=-\sqrt{2x-2}+2 tenglamasi noyob yechimga ega.

$$x=3$$