sqrt{y+1}+sqrt{2y-5}=3 eching

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

y uchun yechish

$$y=3$$

Yechim qadamlari

Tenglamaning ikkala tarafidan \sqrt{2y-5} ni ayirish.

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

Tenglamaning ikkala taraf kvadratini chiqarish.

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

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

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

\left(a-b\right)^{2}=a^{2}-2ab+b^{2} binom teoremasini \left(3-\sqrt{2y-5}\right)^{2} kengaytirilishi uchun ishlating.

$$y+1=9-6\sqrt{2y-5}+\left(\sqrt{2y-5}\right)^{2}$$

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

$$y+1=9-6\sqrt{2y-5}+2y-5$$

4 olish uchun 9 dan 5 ni ayirish.

$$y+1=4-6\sqrt{2y-5}+2y$$

Tenglamaning ikkala tarafidan 4+2y ni ayirish.

$$y+1-\left(4+2y\right)=-6\sqrt{2y-5}$$

4+2y teskarisini topish uchun har birining teskarisini toping.

$$y+1-4-2y=-6\sqrt{2y-5}$$

-3 olish uchun 1 dan 4 ni ayirish.

$$y-3-2y=-6\sqrt{2y-5}$$

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

$$-y-3=-6\sqrt{2y-5}$$

Tenglamaning ikkala taraf kvadratini chiqarish.

$$\left(-y-3\right)^{2}=\left(-6\sqrt{2y-5}\right)^{2}$$

\left(a-b\right)^{2}=a^{2}-2ab+b^{2} binom teoremasini \left(-y-3\right)^{2} kengaytirilishi uchun ishlating.

$$y^{2}+6y+9=\left(-6\sqrt{2y-5}\right)^{2}$$

\left(-6\sqrt{2y-5}\right)^{2} ni kengaytirish.

$$y^{2}+6y+9=\left(-6\right)^{2}\left(\sqrt{2y-5}\right)^{2}$$

2 daraja ko‘rsatkichini -6 ga hisoblang va 36 ni qiymatni oling.

$$y^{2}+6y+9=36\left(\sqrt{2y-5}\right)^{2}$$

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

$$y^{2}+6y+9=36\left(2y-5\right)$$

36 ga 2y-5 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$y^{2}+6y+9=72y-180$$

Ikkala tarafdan 72y ni ayirish.

$$y^{2}+6y+9-72y=-180$$

-66y ni olish uchun 6y va -72y ni birlashtirish.

$$y^{2}-66y+9=-180$$

180 ni ikki tarafga qo’shing.

$$y^{2}-66y+9+180=0$$

189 olish uchun 9 va 180'ni qo'shing.

$$y^{2}-66y+189=0$$

Bu tenglamani yechish uchun y^{2}+\left(a+b\right)y+ab=\left(y+a\right)\left(y+b\right) formulasi yordamida y^{2}-66y+189 ni faktorlang. a va b ni topish uchun yechiladigan tizimni sozlang.

$$a+b=-66 ab=189$$

ab musbat boʻlganda, a va b da bir xil belgi bor. a+b manfiy boʻlganda, a va b ikkisi ham manfiy. 189-mahsulotni beruvchi bunday butun juftliklarni roʻyxat qiling.

$$-1,-189 -3,-63 -7,-27 -9,-21$$

Har bir juftlik yigʻindisini hisoblang.

$$-1-189=-190 -3-63=-66 -7-27=-34 -9-21=-30$$

Yechim – -66 yigʻindisini beruvchi juftlik.

$$a=-63 b=-3$$

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

$$\left(y-63\right)\left(y-3\right)$$

Tenglamani yechish uchun y-63=0 va y-3=0 ni yeching.

$$y=63 y=3$$

\sqrt{y+1}+\sqrt{2y-5}=3 tenglamasida y uchun 63 ni almashtiring.

$$\sqrt{63+1}+\sqrt{2\times 63-5}=3$$

Qisqartirish. y=63 qiymati bu tenglamani qoniqtirmaydi.

$$19=3$$

\sqrt{y+1}+\sqrt{2y-5}=3 tenglamasida y uchun 3 ni almashtiring.

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

Qisqartirish. y=3 tenglamani qoniqtiradi.

$$3=3$$

\sqrt{y+1}=-\sqrt{2y-5}+3 tenglamasi noyob yechimga ega.

$$y=3$$