36+sqrt{r^2-36}=r^2 eching

$$36 + \sqrt { r ^ { 2 } - 36 } = r ^ { 2 }$$

r uchun yechish

$$r=\sqrt{37}\approx 6,08276253$$

$$r=-\sqrt{37}\approx -6,08276253$$

$$r=-6$$

$$r=6$$

Yechim qadamlari

Tenglamaning ikkala tarafidan 36 ni ayirish.

$$\sqrt{r^{2}-36}=r^{2}-36$$

Tenglamaning ikkala taraf kvadratini chiqarish.

$$\left(\sqrt{r^{2}-36}\right)^{2}=\left(r^{2}-36\right)^{2}$$

2 daraja ko‘rsatkichini \sqrt{r^{2}-36} ga hisoblang va r^{2}-36 ni qiymatni oling.

$$r^{2}-36=\left(r^{2}-36\right)^{2}$$

\left(a-b\right)^{2}=a^{2}-2ab+b^{2} binom teoremasini \left(r^{2}-36\right)^{2} kengaytirilishi uchun ishlating.

$$r^{2}-36=\left(r^{2}\right)^{2}-72r^{2}+1296$$

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

$$r^{2}-36=r^{4}-72r^{2}+1296$$

Ikkala tarafdan r^{4} ni ayirish.

$$r^{2}-36-r^{4}=-72r^{2}+1296$$

72r^{2} ni ikki tarafga qo’shing.

$$r^{2}-36-r^{4}+72r^{2}=1296$$

73r^{2} ni olish uchun r^{2} va 72r^{2} ni birlashtirish.

$$73r^{2}-36-r^{4}=1296$$

Ikkala tarafdan 1296 ni ayirish.

$$73r^{2}-36-r^{4}-1296=0$$

-1332 olish uchun -36 dan 1296 ni ayirish.

$$73r^{2}-1332-r^{4}=0$$

r^{2} uchun t ni almashtiring.

$$-t^{2}+73t-1332=0$$

ax^{2}+bx+c=0 shaklidagi barcha tenglamalarni bu formula bilan yechish mumkin: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. Kvadrat tenglamada a uchun -1 ni, b uchun 73 ni va c uchun -1332 ni ayiring.

$$t=\frac{-73±\sqrt{73^{2}-4\left(-1\right)\left(-1332\right)}}{-2}$$

Hisoblarni amalga oshiring.

$$t=\frac{-73±1}{-2}$$

t=\frac{-73±1}{-2} tenglamasini ± plus va ± minus boʻlgan holatida ishlang.

$$t=36 t=37$$

r=t^{2} boʻlganda, yechimlar har bir t uchun r=±\sqrt{t} hisoblanishi orqali olinadi.

$$r=6 r=-6 r=\sqrt{37} r=-\sqrt{37}$$

36+\sqrt{r^{2}-36}=r^{2} tenglamasida r uchun 6 ni almashtiring.

$$36+\sqrt{6^{2}-36}=6^{2}$$

Qisqartirish. r=6 tenglamani qoniqtiradi.

$$36=36$$

36+\sqrt{r^{2}-36}=r^{2} tenglamasida r uchun -6 ni almashtiring.

$$36+\sqrt{\left(-6\right)^{2}-36}=\left(-6\right)^{2}$$

Qisqartirish. r=-6 tenglamani qoniqtiradi.

$$36=36$$

36+\sqrt{r^{2}-36}=r^{2} tenglamasida r uchun \sqrt{37} ni almashtiring.

$$36+\sqrt{\left(\sqrt{37}\right)^{2}-36}=\left(\sqrt{37}\right)^{2}$$

Qisqartirish. r=\sqrt{37} tenglamani qoniqtiradi.

$$37=37$$

36+\sqrt{r^{2}-36}=r^{2} tenglamasida r uchun -\sqrt{37} ni almashtiring.

$$36+\sqrt{\left(-\sqrt{37}\right)^{2}-36}=\left(-\sqrt{37}\right)^{2}$$

Qisqartirish. r=-\sqrt{37} tenglamani qoniqtiradi.

$$37=37$$

\sqrt{r^{2}-36}=r^{2}-36 boʻyicha barcha yechimlar roʻyxati.

$$r=6 r=-6 r=\sqrt{37} r=-\sqrt{37}$$