sqrt{4-x}+sqrt{9+x}=5 eching

$$\sqrt { 4 - x } + \sqrt { 9 + x } = 5$$

x uchun yechish

$$x=-5$$

$$x=0$$

Yechim qadamlari

Tenglamaning ikkala tarafidan \sqrt{9+x} ni ayirish.

$$\sqrt{4-x}=5-\sqrt{9+x}$$

Tenglamaning ikkala taraf kvadratini chiqarish.

$$\left(\sqrt{4-x}\right)^{2}=\left(5-\sqrt{9+x}\right)^{2}$$

2 daraja ko‘rsatkichini \sqrt{4-x} ga hisoblang va 4-x ni qiymatni oling.

$$4-x=\left(5-\sqrt{9+x}\right)^{2}$$

\left(a-b\right)^{2}=a^{2}-2ab+b^{2} binom teoremasini \left(5-\sqrt{9+x}\right)^{2} kengaytirilishi uchun ishlating.

$$4-x=25-10\sqrt{9+x}+\left(\sqrt{9+x}\right)^{2}$$

2 daraja ko‘rsatkichini \sqrt{9+x} ga hisoblang va 9+x ni qiymatni oling.

$$4-x=25-10\sqrt{9+x}+9+x$$

34 olish uchun 25 va 9'ni qo'shing.

$$4-x=34-10\sqrt{9+x}+x$$

Tenglamaning ikkala tarafidan 34+x ni ayirish.

$$4-x-\left(34+x\right)=-10\sqrt{9+x}$$

34+x teskarisini topish uchun har birining teskarisini toping.

$$4-x-34-x=-10\sqrt{9+x}$$

-30 olish uchun 4 dan 34 ni ayirish.

$$-30-x-x=-10\sqrt{9+x}$$

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

$$-30-2x=-10\sqrt{9+x}$$

Tenglamaning ikkala taraf kvadratini chiqarish.

$$\left(-30-2x\right)^{2}=\left(-10\sqrt{9+x}\right)^{2}$$

\left(a-b\right)^{2}=a^{2}-2ab+b^{2} binom teoremasini \left(-30-2x\right)^{2} kengaytirilishi uchun ishlating.

$$900+120x+4x^{2}=\left(-10\sqrt{9+x}\right)^{2}$$

\left(-10\sqrt{9+x}\right)^{2} ni kengaytirish.

$$900+120x+4x^{2}=\left(-10\right)^{2}\left(\sqrt{9+x}\right)^{2}$$

2 daraja ko‘rsatkichini -10 ga hisoblang va 100 ni qiymatni oling.

$$900+120x+4x^{2}=100\left(\sqrt{9+x}\right)^{2}$$

2 daraja ko‘rsatkichini \sqrt{9+x} ga hisoblang va 9+x ni qiymatni oling.

$$900+120x+4x^{2}=100\left(9+x\right)$$

100 ga 9+x ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$900+120x+4x^{2}=900+100x$$

Ikkala tarafdan 900 ni ayirish.

$$900+120x+4x^{2}-900=100x$$

0 olish uchun 900 dan 900 ni ayirish.

$$120x+4x^{2}=100x$$

Ikkala tarafdan 100x ni ayirish.

$$120x+4x^{2}-100x=0$$

20x ni olish uchun 120x va -100x ni birlashtirish.

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

x omili.

$$x\left(20+4x\right)=0$$

Tenglamani yechish uchun x=0 va 20+4x=0 ni yeching.

$$x=0 x=-5$$

\sqrt{4-x}+\sqrt{9+x}=5 tenglamasida x uchun 0 ni almashtiring.

$$\sqrt{4-0}+\sqrt{9+0}=5$$

Qisqartirish. x=0 tenglamani qoniqtiradi.

$$5=5$$

\sqrt{4-x}+\sqrt{9+x}=5 tenglamasida x uchun -5 ni almashtiring.

$$\sqrt{4-\left(-5\right)}+\sqrt{9-5}=5$$

Qisqartirish. x=-5 tenglamani qoniqtiradi.

$$5=5$$

\sqrt{4-x}=-\sqrt{x+9}+5 boʻyicha barcha yechimlar roʻyxati.

$$x=0 x=-5$$