sqrt{x+5}+sqrt{2x+8}=1 eching

$$\sqrt { x + 5 } + \sqrt { 2 x + 8 } = 1$$

x uchun yechish

$$x=-4$$

Yechim qadamlari

Tenglamaning ikkala tarafidan \sqrt{2x+8} ni ayirish.

$$\sqrt{x+5}=1-\sqrt{2x+8}$$

Tenglamaning ikkala taraf kvadratini chiqarish.

$$\left(\sqrt{x+5}\right)^{2}=\left(1-\sqrt{2x+8}\right)^{2}$$

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

$$x+5=\left(1-\sqrt{2x+8}\right)^{2}$$

\left(a-b\right)^{2}=a^{2}-2ab+b^{2} binom teoremasini \left(1-\sqrt{2x+8}\right)^{2} kengaytirilishi uchun ishlating.

$$x+5=1-2\sqrt{2x+8}+\left(\sqrt{2x+8}\right)^{2}$$

2 daraja ko‘rsatkichini \sqrt{2x+8} ga hisoblang va 2x+8 ni qiymatni oling.

$$x+5=1-2\sqrt{2x+8}+2x+8$$

9 olish uchun 1 va 8'ni qo'shing.

$$x+5=9-2\sqrt{2x+8}+2x$$

Tenglamaning ikkala tarafidan 9+2x ni ayirish.

$$x+5-\left(9+2x\right)=-2\sqrt{2x+8}$$

9+2x teskarisini topish uchun har birining teskarisini toping.

$$x+5-9-2x=-2\sqrt{2x+8}$$

-4 olish uchun 5 dan 9 ni ayirish.

$$x-4-2x=-2\sqrt{2x+8}$$

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

$$-x-4=-2\sqrt{2x+8}$$

Tenglamaning ikkala taraf kvadratini chiqarish.

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

\left(a-b\right)^{2}=a^{2}-2ab+b^{2} binom teoremasini \left(-x-4\right)^{2} kengaytirilishi uchun ishlating.

$$x^{2}+8x+16=\left(-2\sqrt{2x+8}\right)^{2}$$

\left(-2\sqrt{2x+8}\right)^{2} ni kengaytirish.

$$x^{2}+8x+16=\left(-2\right)^{2}\left(\sqrt{2x+8}\right)^{2}$$

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

$$x^{2}+8x+16=4\left(\sqrt{2x+8}\right)^{2}$$

2 daraja ko‘rsatkichini \sqrt{2x+8} ga hisoblang va 2x+8 ni qiymatni oling.

$$x^{2}+8x+16=4\left(2x+8\right)$$

4 ga 2x+8 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$x^{2}+8x+16=8x+32$$

Ikkala tarafdan 8x ni ayirish.

$$x^{2}+8x+16-8x=32$$

0 ni olish uchun 8x va -8x ni birlashtirish.

$$x^{2}+16=32$$

Ikkala tarafdan 32 ni ayirish.

$$x^{2}+16-32=0$$

-16 olish uchun 16 dan 32 ni ayirish.

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

Hisoblang: x^{2}-16. x^{2}-16 ni x^{2}-4^{2} sifatida qaytadan yozish. Kvadratlarning farqini ushbu formula bilan hisoblash mumkin: a^{2}-b^{2}=\left(a-b\right)\left(a+b\right).

$$\left(x-4\right)\left(x+4\right)=0$$

Tenglamani yechish uchun x-4=0 va x+4=0 ni yeching.

$$x=4 x=-4$$

\sqrt{x+5}+\sqrt{2x+8}=1 tenglamasida x uchun 4 ni almashtiring.

$$\sqrt{4+5}+\sqrt{2\times 4+8}=1$$

Qisqartirish. x=4 qiymati bu tenglamani qoniqtirmaydi.

$$7=1$$

\sqrt{x+5}+\sqrt{2x+8}=1 tenglamasida x uchun -4 ni almashtiring.

$$\sqrt{-4+5}+\sqrt{2\left(-4\right)+8}=1$$

Qisqartirish. x=-4 tenglamani qoniqtiradi.

$$1=1$$

\sqrt{x+5}=-\sqrt{2x+8}+1 tenglamasi noyob yechimga ega.

$$x=-4$$