a^2+4+80=(2+sqrt{80-a^2})^2 eching

$$a ^ { 2 } + 4 + 80 = ( 2 + \sqrt { 80 - a ^ { 2 } } ) ^ { 2 }$$

a uchun yechish

$$a=4$$

$$a=-4$$

Yechim qadamlari

84 olish uchun 4 va 80'ni qo'shing.

$$a^{2}+84=\left(2+\sqrt{80-a^{2}}\right)^{2}$$

\left(a+b\right)^{2}=a^{2}+2ab+b^{2} binom teoremasini \left(2+\sqrt{80-a^{2}}\right)^{2} kengaytirilishi uchun ishlating.

$$a^{2}+84=4+4\sqrt{80-a^{2}}+\left(\sqrt{80-a^{2}}\right)^{2}$$

2 daraja ko‘rsatkichini \sqrt{80-a^{2}} ga hisoblang va 80-a^{2} ni qiymatni oling.

$$a^{2}+84=4+4\sqrt{80-a^{2}}+80-a^{2}$$

84 olish uchun 4 va 80'ni qo'shing.

$$a^{2}+84=84+4\sqrt{80-a^{2}}-a^{2}$$

Ikkala tarafdan 4\sqrt{80-a^{2}} ni ayirish.

$$a^{2}+84-4\sqrt{80-a^{2}}=84-a^{2}$$

a^{2} ni ikki tarafga qo’shing.

$$a^{2}+84-4\sqrt{80-a^{2}}+a^{2}=84$$

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

$$2a^{2}+84-4\sqrt{80-a^{2}}=84$$

Tenglamaning ikkala tarafidan 2a^{2}+84 ni ayirish.

$$-4\sqrt{80-a^{2}}=84-\left(2a^{2}+84\right)$$

2a^{2}+84 teskarisini topish uchun har birining teskarisini toping.

$$-4\sqrt{80-a^{2}}=84-2a^{2}-84$$

0 olish uchun 84 dan 84 ni ayirish.

$$-4\sqrt{80-a^{2}}=-2a^{2}$$

Tenglamaning ikkala taraf kvadratini chiqarish.

$$\left(-4\sqrt{80-a^{2}}\right)^{2}=\left(-2a^{2}\right)^{2}$$

\left(-4\sqrt{80-a^{2}}\right)^{2} ni kengaytirish.

$$\left(-4\right)^{2}\left(\sqrt{80-a^{2}}\right)^{2}=\left(-2a^{2}\right)^{2}$$

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

$$16\left(\sqrt{80-a^{2}}\right)^{2}=\left(-2a^{2}\right)^{2}$$

2 daraja ko‘rsatkichini \sqrt{80-a^{2}} ga hisoblang va 80-a^{2} ni qiymatni oling.

$$16\left(80-a^{2}\right)=\left(-2a^{2}\right)^{2}$$

16 ga 80-a^{2} ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$1280-16a^{2}=\left(-2a^{2}\right)^{2}$$

\left(-2a^{2}\right)^{2} ni kengaytirish.

$$1280-16a^{2}=\left(-2\right)^{2}\left(a^{2}\right)^{2}$$

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

$$1280-16a^{2}=\left(-2\right)^{2}a^{4}$$

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

$$1280-16a^{2}=4a^{4}$$

Ikkala tarafdan 4a^{4} ni ayirish.

$$1280-16a^{2}-4a^{4}=0$$

a^{2} uchun t ni almashtiring.

$$-4t^{2}-16t+1280=0$$

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 1280 ni ayiring.

$$t=\frac{-\left(-16\right)±\sqrt{\left(-16\right)^{2}-4\left(-4\right)\times 1280}}{-4\times 2}$$

Hisoblarni amalga oshiring.

$$t=\frac{16±144}{-8}$$

t=\frac{16±144}{-8} tenglamasini ± plus va ± minus boʻlgan holatida ishlang.

$$t=-20 t=16$$

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

$$a=4 a=-4$$

a^{2}+4+80=\left(2+\sqrt{80-a^{2}}\right)^{2} tenglamasida a uchun 4 ni almashtiring.

$$4^{2}+4+80=\left(2+\sqrt{80-4^{2}}\right)^{2}$$

Qisqartirish. a=4 tenglamani qoniqtiradi.

$$100=100$$

a^{2}+4+80=\left(2+\sqrt{80-a^{2}}\right)^{2} tenglamasida a uchun -4 ni almashtiring.

$$\left(-4\right)^{2}+4+80=\left(2+\sqrt{80-\left(-4\right)^{2}}\right)^{2}$$

Qisqartirish. a=-4 tenglamani qoniqtiradi.

$$100=100$$

-4\sqrt{80-a^{2}}=-2a^{2} boʻyicha barcha yechimlar roʻyxati.

$$a=4 a=-4$$