sqrt{x+14}-sqrt{2x+5}=1 eching

$$\sqrt { x + 14 } - \sqrt { 2 x + 5 } = 1$$

x uchun yechish

$$x=2$$

Yechim qadamlari

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

$$\sqrt{x+14}=1+\sqrt{2x+5}$$

Tenglamaning ikkala taraf kvadratini chiqarish.

$$\left(\sqrt{x+14}\right)^{2}=\left(1+\sqrt{2x+5}\right)^{2}$$

2 daraja ko‘rsatkichini \sqrt{x+14} ga hisoblang va x+14 ni qiymatni oling.

$$x+14=\left(1+\sqrt{2x+5}\right)^{2}$$

\left(a+b\right)^{2}=a^{2}+2ab+b^{2} binom teoremasini \left(1+\sqrt{2x+5}\right)^{2} kengaytirilishi uchun ishlating.

$$x+14=1+2\sqrt{2x+5}+\left(\sqrt{2x+5}\right)^{2}$$

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

$$x+14=1+2\sqrt{2x+5}+2x+5$$

6 olish uchun 1 va 5'ni qo'shing.

$$x+14=6+2\sqrt{2x+5}+2x$$

Tenglamaning ikkala tarafidan 6+2x ni ayirish.

$$x+14-\left(6+2x\right)=2\sqrt{2x+5}$$

6+2x teskarisini topish uchun har birining teskarisini toping.

$$x+14-6-2x=2\sqrt{2x+5}$$

8 olish uchun 14 dan 6 ni ayirish.

$$x+8-2x=2\sqrt{2x+5}$$

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

$$-x+8=2\sqrt{2x+5}$$

Tenglamaning ikkala taraf kvadratini chiqarish.

$$\left(-x+8\right)^{2}=\left(2\sqrt{2x+5}\right)^{2}$$

\left(a+b\right)^{2}=a^{2}+2ab+b^{2} binom teoremasini \left(-x+8\right)^{2} kengaytirilishi uchun ishlating.

$$x^{2}-16x+64=\left(2\sqrt{2x+5}\right)^{2}$$

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

$$x^{2}-16x+64=2^{2}\left(\sqrt{2x+5}\right)^{2}$$

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

$$x^{2}-16x+64=4\left(\sqrt{2x+5}\right)^{2}$$

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

$$x^{2}-16x+64=4\left(2x+5\right)$$

4 ga 2x+5 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$x^{2}-16x+64=8x+20$$

Ikkala tarafdan 8x ni ayirish.

$$x^{2}-16x+64-8x=20$$

-24x ni olish uchun -16x va -8x ni birlashtirish.

$$x^{2}-24x+64=20$$

Ikkala tarafdan 20 ni ayirish.

$$x^{2}-24x+64-20=0$$

44 olish uchun 64 dan 20 ni ayirish.

$$x^{2}-24x+44=0$$

Bu tenglamani yechish uchun x^{2}+\left(a+b\right)x+ab=\left(x+a\right)\left(x+b\right) formulasi yordamida x^{2}-24x+44 ni faktorlang. a va b ni topish uchun yechiladigan tizimni sozlang.

$$a+b=-24 ab=44$$

ab musbat boʻlganda, a va b da bir xil belgi bor. a+b manfiy boʻlganda, a va b ikkisi ham manfiy. 44-mahsulotni beruvchi bunday butun juftliklarni roʻyxat qiling.

$$-1,-44 -2,-22 -4,-11$$

Har bir juftlik yigʻindisini hisoblang.

$$-1-44=-45 -2-22=-24 -4-11=-15$$

Yechim – -24 yigʻindisini beruvchi juftlik.

$$a=-22 b=-2$$

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

$$\left(x-22\right)\left(x-2\right)$$

Tenglamani yechish uchun x-22=0 va x-2=0 ni yeching.

$$x=22 x=2$$

\sqrt{x+14}-\sqrt{2x+5}=1 tenglamasida x uchun 22 ni almashtiring.

$$\sqrt{22+14}-\sqrt{2\times 22+5}=1$$

Qisqartirish. x=22 qiymati bu tenglamani qoniqtirmaydi, chunki oʻng va chap tarafdagi belgilar bir-biriga qarama-qarshi.

$$-1=1$$

\sqrt{x+14}-\sqrt{2x+5}=1 tenglamasida x uchun 2 ni almashtiring.

$$\sqrt{2+14}-\sqrt{2\times 2+5}=1$$

Qisqartirish. x=2 tenglamani qoniqtiradi.

$$1=1$$

\sqrt{x+14}=\sqrt{2x+5}+1 tenglamasi noyob yechimga ega.

$$x=2$$