sqrt{2x-1}-1=sqrt{x-1} eching

$$\sqrt{ 2x-1 } -1= \sqrt{ x-1 }$$

x uchun yechish

$$x=5$$

$$x=1$$

Yechim qadamlari

Tenglamaning ikkala taraf kvadratini chiqarish.

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

\left(a-b\right)^{2}=a^{2}-2ab+b^{2} binom teoremasini \left(\sqrt{2x-1}-1\right)^{2} kengaytirilishi uchun ishlating.

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

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

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

0 olish uchun -1 va 1'ni qo'shing.

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

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

$$2x-2\sqrt{2x-1}=x-1$$

Tenglamaning ikkala tarafidan 2x ni ayirish.

$$-2\sqrt{2x-1}=x-1-2x$$

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

$$-2\sqrt{2x-1}=-x-1$$

Tenglamaning ikkala taraf kvadratini chiqarish.

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

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

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

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

$$4\left(\sqrt{2x-1}\right)^{2}=\left(-x-1\right)^{2}$$

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

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

4 ga 2x-1 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$8x-4=\left(-x-1\right)^{2}$$

\left(a-b\right)^{2}=a^{2}-2ab+b^{2} binom teoremasini \left(-x-1\right)^{2} kengaytirilishi uchun ishlating.

$$8x-4=x^{2}+2x+1$$

Ikkala tarafdan x^{2} ni ayirish.

$$8x-4-x^{2}=2x+1$$

Ikkala tarafdan 2x ni ayirish.

$$8x-4-x^{2}-2x=1$$

6x ni olish uchun 8x va -2x ni birlashtirish.

$$6x-4-x^{2}=1$$

Ikkala tarafdan 1 ni ayirish.

$$6x-4-x^{2}-1=0$$

-5 olish uchun -4 dan 1 ni ayirish.

$$6x-5-x^{2}=0$$

Polinomni standart shaklga keltirish uchun uni qayta tartiblang. Shartlarni eng yuqoridan eng pastki qiymat ko'rsatgichiga joylashtirish.

$$-x^{2}+6x-5=0$$

Tenglamani yechish uchun guruhlash orqali chap qoʻl tomonni faktorlang. Avvalo, chap qoʻl tomon -x^{2}+ax+bx-5 sifatida qayta yozilishi kerak. a va b ni topish uchun yechiladigan tizimni sozlang.

$$a+b=6 ab=-\left(-5\right)=5$$

ab musbat boʻlganda, a va b da bir xil belgi bor. a+b musbat boʻlganda, a va b ikkisi ham musbat. Faqat bundan juftlik tizim yechimidir.

$$a=5 b=1$$

-x^{2}+6x-5 ni \left(-x^{2}+5x\right)+\left(x-5\right) sifatida qaytadan yozish.

$$\left(-x^{2}+5x\right)+\left(x-5\right)$$

-x^{2}+5x ichida -x ni ajrating.

$$-x\left(x-5\right)+x-5$$

Distributiv funktsiyasidan foydalangan holda x-5 umumiy terminini chiqaring.

$$\left(x-5\right)\left(-x+1\right)$$

Tenglamani yechish uchun x-5=0 va -x+1=0 ni yeching.

$$x=5 x=1$$

\sqrt{2x-1}-1=\sqrt{x-1} tenglamasida x uchun 5 ni almashtiring.

$$\sqrt{2\times 5-1}-1=\sqrt{5-1}$$

Qisqartirish. x=5 tenglamani qoniqtiradi.

$$2=2$$

\sqrt{2x-1}-1=\sqrt{x-1} tenglamasida x uchun 1 ni almashtiring.

$$\sqrt{2\times 1-1}-1=\sqrt{1-1}$$

Qisqartirish. x=1 tenglamani qoniqtiradi.

$$0=0$$

\sqrt{2x-1}-1=\sqrt{x-1} boʻyicha barcha yechimlar roʻyxati.

$$x=5 x=1$$