sqrt{x-5}=sqrt{3x+7}-4 eching

$$\sqrt { x - 5 } = \sqrt { 3 x + 7 } - 4$$

x uchun yechish

$$x=14$$

$$x=6$$

Yechim qadamlari

Tenglamaning ikkala taraf kvadratini chiqarish.

$$\left(\sqrt{x-5}\right)^{2}=\left(\sqrt{3x+7}-4\right)^{2}$$

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

$$x-5=\left(\sqrt{3x+7}-4\right)^{2}$$

\left(a-b\right)^{2}=a^{2}-2ab+b^{2} binom teoremasini \left(\sqrt{3x+7}-4\right)^{2} kengaytirilishi uchun ishlating.

$$x-5=\left(\sqrt{3x+7}\right)^{2}-8\sqrt{3x+7}+16$$

2 daraja ko‘rsatkichini \sqrt{3x+7} ga hisoblang va 3x+7 ni qiymatni oling.

$$x-5=3x+7-8\sqrt{3x+7}+16$$

23 olish uchun 7 va 16'ni qo'shing.

$$x-5=3x+23-8\sqrt{3x+7}$$

Tenglamaning ikkala tarafidan 3x+23 ni ayirish.

$$x-5-\left(3x+23\right)=-8\sqrt{3x+7}$$

3x+23 teskarisini topish uchun har birining teskarisini toping.

$$x-5-3x-23=-8\sqrt{3x+7}$$

-2x ni olish uchun x va -3x ni birlashtirish.

$$-2x-5-23=-8\sqrt{3x+7}$$

-28 olish uchun -5 dan 23 ni ayirish.

$$-2x-28=-8\sqrt{3x+7}$$

Tenglamaning ikkala taraf kvadratini chiqarish.

$$\left(-2x-28\right)^{2}=\left(-8\sqrt{3x+7}\right)^{2}$$

\left(a-b\right)^{2}=a^{2}-2ab+b^{2} binom teoremasini \left(-2x-28\right)^{2} kengaytirilishi uchun ishlating.

$$4x^{2}+112x+784=\left(-8\sqrt{3x+7}\right)^{2}$$

\left(-8\sqrt{3x+7}\right)^{2} ni kengaytirish.

$$4x^{2}+112x+784=\left(-8\right)^{2}\left(\sqrt{3x+7}\right)^{2}$$

2 daraja ko‘rsatkichini -8 ga hisoblang va 64 ni qiymatni oling.

$$4x^{2}+112x+784=64\left(\sqrt{3x+7}\right)^{2}$$

2 daraja ko‘rsatkichini \sqrt{3x+7} ga hisoblang va 3x+7 ni qiymatni oling.

$$4x^{2}+112x+784=64\left(3x+7\right)$$

64 ga 3x+7 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$4x^{2}+112x+784=192x+448$$

Ikkala tarafdan 192x ni ayirish.

$$4x^{2}+112x+784-192x=448$$

-80x ni olish uchun 112x va -192x ni birlashtirish.

$$4x^{2}-80x+784=448$$

Ikkala tarafdan 448 ni ayirish.

$$4x^{2}-80x+784-448=0$$

336 olish uchun 784 dan 448 ni ayirish.

$$4x^{2}-80x+336=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 4 ni a, -80 ni b va 336 ni c bilan almashtiring.

$$x=\frac{-\left(-80\right)±\sqrt{\left(-80\right)^{2}-4\times 4\times 336}}{2\times 4}$$

-80 kvadratini chiqarish.

$$x=\frac{-\left(-80\right)±\sqrt{6400-4\times 4\times 336}}{2\times 4}$$

-4 ni 4 marotabaga ko'paytirish.

$$x=\frac{-\left(-80\right)±\sqrt{6400-16\times 336}}{2\times 4}$$

-16 ni 336 marotabaga ko'paytirish.

$$x=\frac{-\left(-80\right)±\sqrt{6400-5376}}{2\times 4}$$

6400 ni -5376 ga qo'shish.

$$x=\frac{-\left(-80\right)±\sqrt{1024}}{2\times 4}$$

1024 ning kvadrat ildizini chiqarish.

$$x=\frac{-\left(-80\right)±32}{2\times 4}$$

-80 ning teskarisi 80 ga teng.

$$x=\frac{80±32}{2\times 4}$$

2 ni 4 marotabaga ko'paytirish.

$$x=\frac{80±32}{8}$$

x=\frac{80±32}{8} tenglamasini yeching, bunda ± musbat. 80 ni 32 ga qo'shish.

$$x=\frac{112}{8}$$

112 ni 8 ga bo'lish.

$$x=14$$

x=\frac{80±32}{8} tenglamasini yeching, bunda ± manfiy. 80 dan 32 ni ayirish.

$$x=\frac{48}{8}$$

48 ni 8 ga bo'lish.

$$x=6$$

Tenglama yechildi.

$$x=14 x=6$$

\sqrt{x-5}=\sqrt{3x+7}-4 tenglamasida x uchun 14 ni almashtiring.

$$\sqrt{14-5}=\sqrt{3\times 14+7}-4$$

Qisqartirish. x=14 tenglamani qoniqtiradi.

$$3=3$$

\sqrt{x-5}=\sqrt{3x+7}-4 tenglamasida x uchun 6 ni almashtiring.

$$\sqrt{6-5}=\sqrt{3\times 6+7}-4$$

Qisqartirish. x=6 tenglamani qoniqtiradi.

$$1=1$$

\sqrt{x-5}=\sqrt{3x+7}-4 boʻyicha barcha yechimlar roʻyxati.

$$x=14 x=6$$