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

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

x uchun yechish

$$x=1$$

$$x=-1$$

Yechim qadamlari

Tenglamaning ikkala tarafidan \sqrt{1+x} ni ayirish.

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

Tenglamaning ikkala taraf kvadratini chiqarish.

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

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

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

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

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

\sqrt{2} kvadrati – 2.

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

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

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

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

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

Tenglamaning ikkala tarafidan 3+x ni ayirish.

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

3+x teskarisini topish uchun har birining teskarisini toping.

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

-2 olish uchun 1 dan 3 ni ayirish.

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

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

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

Tenglamaning ikkala taraf kvadratini chiqarish.

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

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

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

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

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

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

$$4+8x+4x^{2}=4\left(\sqrt{2}\right)^{2}\left(\sqrt{1+x}\right)^{2}$$

\sqrt{2} kvadrati – 2.

$$4+8x+4x^{2}=4\times 2\left(\sqrt{1+x}\right)^{2}$$

8 hosil qilish uchun 4 va 2 ni ko'paytirish.

$$4+8x+4x^{2}=8\left(\sqrt{1+x}\right)^{2}$$

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

$$4+8x+4x^{2}=8\left(1+x\right)$$

8 ga 1+x ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$4+8x+4x^{2}=8+8x$$

Ikkala tarafdan 8 ni ayirish.

$$4+8x+4x^{2}-8=8x$$

-4 olish uchun 4 dan 8 ni ayirish.

$$-4+8x+4x^{2}=8x$$

Ikkala tarafdan 8x ni ayirish.

$$-4+8x+4x^{2}-8x=0$$

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

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

Ikki tarafini 4 ga bo‘ling.

$$-1+x^{2}=0$$

Hisoblang: -1+x^{2}. -1+x^{2} ni x^{2}-1^{2} sifatida qaytadan yozish. Kvadratlarning farqini ushbu formula bilan hisoblash mumkin: a^{2}-b^{2}=\left(a-b\right)\left(a+b\right).

$$\left(x-1\right)\left(x+1\right)=0$$

Tenglamani yechish uchun x-1=0 va x+1=0 ni yeching.

$$x=1 x=-1$$

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

$$\sqrt{1-1}+\sqrt{1+1}=\sqrt{2}$$

Qisqartirish. x=1 tenglamani qoniqtiradi.

$$2^{\frac{1}{2}}=2^{\frac{1}{2}}$$

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

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

Qisqartirish. x=-1 tenglamani qoniqtiradi.

$$2^{\frac{1}{2}}=2^{\frac{1}{2}}$$

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

$$x=1 x=-1$$