sqrt{9x+4}=sqrt{7x+1}+1 eching

$$\sqrt { 9 x + 4 } = \sqrt { 7 x + 1 } + 1$$

x uchun yechish

$$x=5$$

$$x=0$$

Yechim qadamlari

Tenglamaning ikkala taraf kvadratini chiqarish.

$$\left(\sqrt{9x+4}\right)^{2}=\left(\sqrt{7x+1}+1\right)^{2}$$

2 daraja ko‘rsatkichini \sqrt{9x+4} ga hisoblang va 9x+4 ni qiymatni oling.

$$9x+4=\left(\sqrt{7x+1}+1\right)^{2}$$

\left(a+b\right)^{2}=a^{2}+2ab+b^{2} binom teoremasini \left(\sqrt{7x+1}+1\right)^{2} kengaytirilishi uchun ishlating.

$$9x+4=\left(\sqrt{7x+1}\right)^{2}+2\sqrt{7x+1}+1$$

2 daraja ko‘rsatkichini \sqrt{7x+1} ga hisoblang va 7x+1 ni qiymatni oling.

$$9x+4=7x+1+2\sqrt{7x+1}+1$$

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

$$9x+4=7x+2+2\sqrt{7x+1}$$

Tenglamaning ikkala tarafidan 7x+2 ni ayirish.

$$9x+4-\left(7x+2\right)=2\sqrt{7x+1}$$

7x+2 teskarisini topish uchun har birining teskarisini toping.

$$9x+4-7x-2=2\sqrt{7x+1}$$

2x ni olish uchun 9x va -7x ni birlashtirish.

$$2x+4-2=2\sqrt{7x+1}$$

2 olish uchun 4 dan 2 ni ayirish.

$$2x+2=2\sqrt{7x+1}$$

Tenglamaning ikkala taraf kvadratini chiqarish.

$$\left(2x+2\right)^{2}=\left(2\sqrt{7x+1}\right)^{2}$$

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

$$4x^{2}+8x+4=\left(2\sqrt{7x+1}\right)^{2}$$

\left(2\sqrt{7x+1}\right)^{2} ni kengaytirish.

$$4x^{2}+8x+4=2^{2}\left(\sqrt{7x+1}\right)^{2}$$

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

$$4x^{2}+8x+4=4\left(\sqrt{7x+1}\right)^{2}$$

2 daraja ko‘rsatkichini \sqrt{7x+1} ga hisoblang va 7x+1 ni qiymatni oling.

$$4x^{2}+8x+4=4\left(7x+1\right)$$

4 ga 7x+1 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$4x^{2}+8x+4=28x+4$$

Ikkala tarafdan 28x ni ayirish.

$$4x^{2}+8x+4-28x=4$$

-20x ni olish uchun 8x va -28x ni birlashtirish.

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

Ikkala tarafdan 4 ni ayirish.

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

0 olish uchun 4 dan 4 ni ayirish.

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

x omili.

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

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

$$x=0 x=5$$

\sqrt{9x+4}=\sqrt{7x+1}+1 tenglamasida x uchun 0 ni almashtiring.

$$\sqrt{9\times 0+4}=\sqrt{7\times 0+1}+1$$

Qisqartirish. x=0 tenglamani qoniqtiradi.

$$2=2$$

\sqrt{9x+4}=\sqrt{7x+1}+1 tenglamasida x uchun 5 ni almashtiring.

$$\sqrt{9\times 5+4}=\sqrt{7\times 5+1}+1$$

Qisqartirish. x=5 tenglamani qoniqtiradi.

$$7=7$$

\sqrt{9x+4}=\sqrt{7x+1}+1 boʻyicha barcha yechimlar roʻyxati.

$$x=0 x=5$$