sqrt{2x-3}=2+sqrt{x-5} eching

$$\sqrt{ 2x-3 } = 2+ \sqrt{ x-5 }$$

x uchun yechish

$$x=14$$

$$x=6$$

Yechim qadamlari

Tenglamaning ikkala taraf kvadratini chiqarish.

$$\left(\sqrt{2x-3}\right)^{2}=\left(2+\sqrt{x-5}\right)^{2}$$

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

$$2x-3=\left(2+\sqrt{x-5}\right)^{2}$$

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

$$2x-3=4+4\sqrt{x-5}+\left(\sqrt{x-5}\right)^{2}$$

2 daraja ko‘rsatkichini \sqrt{x-5} ga hisoblang va x-5 ni qiymatni oling.

$$2x-3=4+4\sqrt{x-5}+x-5$$

-1 olish uchun 4 dan 5 ni ayirish.

$$2x-3=-1+4\sqrt{x-5}+x$$

Tenglamaning ikkala tarafidan -1+x ni ayirish.

$$2x-3-\left(-1+x\right)=4\sqrt{x-5}$$

-1+x teskarisini topish uchun har birining teskarisini toping.

$$2x-3+1-x=4\sqrt{x-5}$$

-2 olish uchun -3 va 1'ni qo'shing.

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

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

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

Tenglamaning ikkala taraf kvadratini chiqarish.

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

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

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

\left(4\sqrt{x-5}\right)^{2} ni kengaytirish.

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

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

$$x^{2}-4x+4=16\left(\sqrt{x-5}\right)^{2}$$

2 daraja ko‘rsatkichini \sqrt{x-5} ga hisoblang va x-5 ni qiymatni oling.

$$x^{2}-4x+4=16\left(x-5\right)$$

16 ga x-5 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$x^{2}-4x+4=16x-80$$

Ikkala tarafdan 16x ni ayirish.

$$x^{2}-4x+4-16x=-80$$

-20x ni olish uchun -4x va -16x ni birlashtirish.

$$x^{2}-20x+4=-80$$

80 ni ikki tarafga qo’shing.

$$x^{2}-20x+4+80=0$$

84 olish uchun 4 va 80'ni qo'shing.

$$x^{2}-20x+84=0$$

Bu tenglamani yechish uchun x^{2}+\left(a+b\right)x+ab=\left(x+a\right)\left(x+b\right) formulasi yordamida x^{2}-20x+84 ni faktorlang. a va b ni topish uchun yechiladigan tizimni sozlang.

$$a+b=-20 ab=84$$

ab musbat boʻlganda, a va b da bir xil belgi bor. a+b manfiy boʻlganda, a va b ikkisi ham manfiy. 84-mahsulotni beruvchi bunday butun juftliklarni roʻyxat qiling.

$$-1,-84 -2,-42 -3,-28 -4,-21 -6,-14 -7,-12$$

Har bir juftlik yigʻindisini hisoblang.

$$-1-84=-85 -2-42=-44 -3-28=-31 -4-21=-25 -6-14=-20 -7-12=-19$$

Yechim – -20 yigʻindisini beruvchi juftlik.

$$a=-14 b=-6$$

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

$$\left(x-14\right)\left(x-6\right)$$

Tenglamani yechish uchun x-14=0 va x-6=0 ni yeching.

$$x=14 x=6$$

\sqrt{2x-3}=2+\sqrt{x-5} tenglamasida x uchun 14 ni almashtiring.

$$\sqrt{2\times 14-3}=2+\sqrt{14-5}$$

Qisqartirish. x=14 tenglamani qoniqtiradi.

$$5=5$$

\sqrt{2x-3}=2+\sqrt{x-5} tenglamasida x uchun 6 ni almashtiring.

$$\sqrt{2\times 6-3}=2+\sqrt{6-5}$$

Qisqartirish. x=6 tenglamani qoniqtiradi.

$$3=3$$

\sqrt{2x-3}=\sqrt{x-5}+2 boʻyicha barcha yechimlar roʻyxati.

$$x=14 x=6$$