sqrt{15+x^2}-sqrt{19-x^2}=2 eching

$$\sqrt { 15 + x ^ { 2 } } - \sqrt { 19 - x ^ { 2 } } = 2$$

x uchun yechish

$$x=\sqrt{10}\approx 3,16227766$$

$$x=-\sqrt{10}\approx -3,16227766$$

Yechim qadamlari

Tenglamaning ikkala tarafidan -\sqrt{19-x^{2}} ni ayirish.

$$\sqrt{15+x^{2}}=2+\sqrt{19-x^{2}}$$

Tenglamaning ikkala taraf kvadratini chiqarish.

$$\left(\sqrt{15+x^{2}}\right)^{2}=\left(2+\sqrt{19-x^{2}}\right)^{2}$$

2 daraja ko‘rsatkichini \sqrt{15+x^{2}} ga hisoblang va 15+x^{2} ni qiymatni oling.

$$15+x^{2}=\left(2+\sqrt{19-x^{2}}\right)^{2}$$

\left(a+b\right)^{2}=a^{2}+2ab+b^{2} binom teoremasini \left(2+\sqrt{19-x^{2}}\right)^{2} kengaytirilishi uchun ishlating.

$$15+x^{2}=4+4\sqrt{19-x^{2}}+\left(\sqrt{19-x^{2}}\right)^{2}$$

2 daraja ko‘rsatkichini \sqrt{19-x^{2}} ga hisoblang va 19-x^{2} ni qiymatni oling.

$$15+x^{2}=4+4\sqrt{19-x^{2}}+19-x^{2}$$

23 olish uchun 4 va 19'ni qo'shing.

$$15+x^{2}=23+4\sqrt{19-x^{2}}-x^{2}$$

Tenglamaning ikkala tarafidan 23-x^{2} ni ayirish.

$$15+x^{2}-\left(23-x^{2}\right)=4\sqrt{19-x^{2}}$$

23-x^{2} teskarisini topish uchun har birining teskarisini toping.

$$15+x^{2}-23+x^{2}=4\sqrt{19-x^{2}}$$

-8 olish uchun 15 dan 23 ni ayirish.

$$-8+x^{2}+x^{2}=4\sqrt{19-x^{2}}$$

2x^{2} ni olish uchun x^{2} va x^{2} ni birlashtirish.

$$-8+2x^{2}=4\sqrt{19-x^{2}}$$

Tenglamaning ikkala taraf kvadratini chiqarish.

$$\left(-8+2x^{2}\right)^{2}=\left(4\sqrt{19-x^{2}}\right)^{2}$$

\left(a+b\right)^{2}=a^{2}+2ab+b^{2} binom teoremasini \left(-8+2x^{2}\right)^{2} kengaytirilishi uchun ishlating.

$$64-32x^{2}+4\left(x^{2}\right)^{2}=\left(4\sqrt{19-x^{2}}\right)^{2}$$

Daraja ko‘rsatkichini boshqa ko‘rsatkichga oshirish uchun, darajalarini ko‘paytiring. 2 va 2 ni ko‘paytirib, 4 ni oling.

$$64-32x^{2}+4x^{4}=\left(4\sqrt{19-x^{2}}\right)^{2}$$

\left(4\sqrt{19-x^{2}}\right)^{2} ni kengaytirish.

$$64-32x^{2}+4x^{4}=4^{2}\left(\sqrt{19-x^{2}}\right)^{2}$$

2 daraja ko‘rsatkichini 4 ga hisoblang va 16 ni qiymatni oling.

$$64-32x^{2}+4x^{4}=16\left(\sqrt{19-x^{2}}\right)^{2}$$

2 daraja ko‘rsatkichini \sqrt{19-x^{2}} ga hisoblang va 19-x^{2} ni qiymatni oling.

$$64-32x^{2}+4x^{4}=16\left(19-x^{2}\right)$$

16 ga 19-x^{2} ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$64-32x^{2}+4x^{4}=304-16x^{2}$$

Ikkala tarafdan 304 ni ayirish.

$$64-32x^{2}+4x^{4}-304=-16x^{2}$$

-240 olish uchun 64 dan 304 ni ayirish.

$$-240-32x^{2}+4x^{4}=-16x^{2}$$

16x^{2} ni ikki tarafga qo’shing.

$$-240-32x^{2}+4x^{4}+16x^{2}=0$$

-16x^{2} ni olish uchun -32x^{2} va 16x^{2} ni birlashtirish.

$$-240-16x^{2}+4x^{4}=0$$

x^{2} uchun t ni almashtiring.

$$4t^{2}-16t-240=0$$

ax^{2}+bx+c=0 shaklidagi barcha tenglamalarni bu formula bilan yechish mumkin: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. Kvadrat tenglamada a uchun 4 ni, b uchun -16 ni va c uchun -240 ni ayiring.

$$t=\frac{-\left(-16\right)±\sqrt{\left(-16\right)^{2}-4\times 4\left(-240\right)}}{2\times 4}$$

Hisoblarni amalga oshiring.

$$t=\frac{16±64}{8}$$

t=\frac{16±64}{8} tenglamasini ± plus va ± minus boʻlgan holatida ishlang.

$$t=10 t=-6$$

x=t^{2} boʻlganda, yechimlar musbat t uchun x=±\sqrt{t} hisoblanishi orqali olinadi.

$$x=\sqrt{10} x=-\sqrt{10}$$

\sqrt{15+x^{2}}-\sqrt{19-x^{2}}=2 tenglamasida x uchun \sqrt{10} ni almashtiring.

$$\sqrt{15+\left(\sqrt{10}\right)^{2}}-\sqrt{19-\left(\sqrt{10}\right)^{2}}=2$$

Qisqartirish. x=\sqrt{10} tenglamani qoniqtiradi.

$$2=2$$

\sqrt{15+x^{2}}-\sqrt{19-x^{2}}=2 tenglamasida x uchun -\sqrt{10} ni almashtiring.

$$\sqrt{15+\left(-\sqrt{10}\right)^{2}}-\sqrt{19-\left(-\sqrt{10}\right)^{2}}=2$$

Qisqartirish. x=-\sqrt{10} tenglamani qoniqtiradi.

$$2=2$$

\sqrt{x^{2}+15}=\sqrt{19-x^{2}}+2 boʻyicha barcha yechimlar roʻyxati.

$$x=\sqrt{10} x=-\sqrt{10}$$