sqrt{x+3}+sqrt{x-3}=6 eching

$$\sqrt { x + 3 } + \sqrt { x - 3 } = 6$$

x uchun yechish

$$x = \frac{37}{4} = 9\frac{1}{4} = 9,25$$

Yechim qadamlari

Tenglamaning ikkala tarafidan \sqrt{x-3} ni ayirish.

$$\sqrt{x+3}=6-\sqrt{x-3}$$

Tenglamaning ikkala taraf kvadratini chiqarish.

$$\left(\sqrt{x+3}\right)^{2}=\left(6-\sqrt{x-3}\right)^{2}$$

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

$$x+3=\left(6-\sqrt{x-3}\right)^{2}$$

\left(a-b\right)^{2}=a^{2}-2ab+b^{2} binom teoremasini \left(6-\sqrt{x-3}\right)^{2} kengaytirilishi uchun ishlating.

$$x+3=36-12\sqrt{x-3}+\left(\sqrt{x-3}\right)^{2}$$

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

$$x+3=36-12\sqrt{x-3}+x-3$$

33 olish uchun 36 dan 3 ni ayirish.

$$x+3=33-12\sqrt{x-3}+x$$

12\sqrt{x-3} ni ikki tarafga qo’shing.

$$x+3+12\sqrt{x-3}=33+x$$

Ikkala tarafdan x ni ayirish.

$$x+3+12\sqrt{x-3}-x=33$$

0 ni olish uchun x va -x ni birlashtirish.

$$3+12\sqrt{x-3}=33$$

Ikkala tarafdan 3 ni ayirish.

$$12\sqrt{x-3}=33-3$$

30 olish uchun 33 dan 3 ni ayirish.

$$12\sqrt{x-3}=30$$

Ikki tarafini 12 ga bo‘ling.

$$\sqrt{x-3}=\frac{30}{12}$$

\frac{30}{12} ulushini 6 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

$$\sqrt{x-3}=\frac{5}{2}$$

Tenglamaning ikkala taraf kvadratini chiqarish.

$$x-3=\frac{25}{4}$$

3 ni tenglamaning ikkala tarafiga qo'shish.

$$x-3-\left(-3\right)=\frac{25}{4}-\left(-3\right)$$

O‘zidan -3 ayirilsa 0 qoladi.

$$x=\frac{25}{4}-\left(-3\right)$$

\frac{25}{4} dan -3 ni ayirish.

$$x=\frac{37}{4}$$

\sqrt{x+3}+\sqrt{x-3}=6 tenglamasida x uchun \frac{37}{4} ni almashtiring.

$$\sqrt{\frac{37}{4}+3}+\sqrt{\frac{37}{4}-3}=6$$

Qisqartirish. x=\frac{37}{4} tenglamani qoniqtiradi.

$$6=6$$

\sqrt{x+3}=-\sqrt{x-3}+6 tenglamasi noyob yechimga ega.

$$x=\frac{37}{4}$$