sqrt{2x-1}-2=sqrt{x-4} eching

$$\sqrt { 2 x - 1 } - 2 = \sqrt { x - 4 }$$

x uchun yechish

$$x=13$$

$$x=5$$

Yechim qadamlari

Tenglamaning ikkala taraf kvadratini chiqarish.

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

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

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

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

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

3 olish uchun -1 va 4'ni qo'shing.

$$2x+3-4\sqrt{2x-1}=\left(\sqrt{x-4}\right)^{2}$$

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

$$2x+3-4\sqrt{2x-1}=x-4$$

Tenglamaning ikkala tarafidan 2x+3 ni ayirish.

$$-4\sqrt{2x-1}=x-4-\left(2x+3\right)$$

2x+3 teskarisini topish uchun har birining teskarisini toping.

$$-4\sqrt{2x-1}=x-4-2x-3$$

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

$$-4\sqrt{2x-1}=-x-4-3$$

-7 olish uchun -4 dan 3 ni ayirish.

$$-4\sqrt{2x-1}=-x-7$$

Tenglamaning ikkala taraf kvadratini chiqarish.

$$\left(-4\sqrt{2x-1}\right)^{2}=\left(-x-7\right)^{2}$$

\left(-4\sqrt{2x-1}\right)^{2} ni kengaytirish.

$$\left(-4\right)^{2}\left(\sqrt{2x-1}\right)^{2}=\left(-x-7\right)^{2}$$

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

$$16\left(\sqrt{2x-1}\right)^{2}=\left(-x-7\right)^{2}$$

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

$$16\left(2x-1\right)=\left(-x-7\right)^{2}$$

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

$$32x-16=\left(-x-7\right)^{2}$$

\left(a-b\right)^{2}=a^{2}-2ab+b^{2} binom teoremasini \left(-x-7\right)^{2} kengaytirilishi uchun ishlating.

$$32x-16=x^{2}+14x+49$$

Ikkala tarafdan x^{2} ni ayirish.

$$32x-16-x^{2}=14x+49$$

Ikkala tarafdan 14x ni ayirish.

$$32x-16-x^{2}-14x=49$$

18x ni olish uchun 32x va -14x ni birlashtirish.

$$18x-16-x^{2}=49$$

Ikkala tarafdan 49 ni ayirish.

$$18x-16-x^{2}-49=0$$

-65 olish uchun -16 dan 49 ni ayirish.

$$18x-65-x^{2}=0$$

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

$$-x^{2}+18x-65=0$$

Tenglamani yechish uchun guruhlash orqali chap qoʻl tomonni faktorlang. Avvalo, chap qoʻl tomon -x^{2}+ax+bx-65 sifatida qayta yozilishi kerak. a va b ni topish uchun yechiladigan tizimni sozlang.

$$a+b=18 ab=-\left(-65\right)=65$$

ab musbat boʻlganda, a va b da bir xil belgi bor. a+b musbat boʻlganda, a va b ikkisi ham musbat. 65-mahsulotni beruvchi bunday butun juftliklarni roʻyxat qiling.

$$1,65 5,13$$

Har bir juftlik yigʻindisini hisoblang.

$$1+65=66 5+13=18$$

Yechim – 18 yigʻindisini beruvchi juftlik.

$$a=13 b=5$$

-x^{2}+18x-65 ni \left(-x^{2}+13x\right)+\left(5x-65\right) sifatida qaytadan yozish.

$$\left(-x^{2}+13x\right)+\left(5x-65\right)$$

Birinchi guruhda -x ni va ikkinchi guruhda 5 ni faktordan chiqaring.

$$-x\left(x-13\right)+5\left(x-13\right)$$

Distributiv funktsiyasidan foydalangan holda x-13 umumiy terminini chiqaring.

$$\left(x-13\right)\left(-x+5\right)$$

Tenglamani yechish uchun x-13=0 va -x+5=0 ni yeching.

$$x=13 x=5$$

\sqrt{2x-1}-2=\sqrt{x-4} tenglamasida x uchun 13 ni almashtiring.

$$\sqrt{2\times 13-1}-2=\sqrt{13-4}$$

Qisqartirish. x=13 tenglamani qoniqtiradi.

$$3=3$$

\sqrt{2x-1}-2=\sqrt{x-4} tenglamasida x uchun 5 ni almashtiring.

$$\sqrt{2\times 5-1}-2=\sqrt{5-4}$$

Qisqartirish. x=5 tenglamani qoniqtiradi.

$$1=1$$

\sqrt{2x-1}-2=\sqrt{x-4} boʻyicha barcha yechimlar roʻyxati.

$$x=13 x=5$$