x^6=((6x^3)-5^3) eching

$${ x }^{ 6 } =((6 { x }^{ 3 } )- { 5 }^{ 3 } )$$

x uchun yechish (complex solution)

$$x\in \sqrt{5}e^{\frac{-\arctan(\frac{2\sqrt{29}}{3})i+2\pi i}{3}},\sqrt{5}e^{-\frac{\arctan(\frac{2\sqrt{29}}{3})i}{3}},\sqrt{5}e^{\frac{-\arctan(\frac{2\sqrt{29}}{3})i+4\pi i}{3}},\sqrt{5}e^{\frac{\arctan(\frac{2\sqrt{29}}{3})i+4\pi i}{3}},\sqrt{5}e^{\frac{\arctan(\frac{2\sqrt{29}}{3})i}{3}},\sqrt{5}e^{\frac{\arctan(\frac{2\sqrt{29}}{3})i+2\pi i}{3}}$$

Kvadrat tenglamalaridan foydalanish qoidalari

3 daraja ko‘rsatkichini 5 ga hisoblang va 125 ni qiymatni oling.

$$x^{6}=6x^{3}-125$$

Ikkala tarafdan 6x^{3} ni ayirish.

$$x^{6}-6x^{3}=-125$$

125 ni ikki tarafga qo’shing.

$$x^{6}-6x^{3}+125=0$$

x^{3} uchun t ni almashtiring.

$$t^{2}-6t+125=0$$

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 -6 ni va c uchun 125 ni ayiring.

$$t=\frac{-\left(-6\right)±\sqrt{\left(-6\right)^{2}-4\times 1\times 125}}{2}$$

Hisoblarni amalga oshiring.

$$t=\frac{6±\sqrt{-464}}{2}$$

t=\frac{6±\sqrt{-464}}{2} tenglamasini ± plus va ± minus boʻlgan holatida ishlang.

$$t=3+2\sqrt{29}i t=-2\sqrt{29}i+3$$

x=t^{3} boʻlganda, yechimlar har bir t uchun tenglamani yechish orqali olinadi.

$$x=\sqrt{5}e^{\frac{\arctan(\frac{2\sqrt{29}}{3})i+4\pi i}{3}} x=\sqrt{5}e^{\frac{\arctan(\frac{2\sqrt{29}}{3})i+2\pi i}{3}} x=\sqrt{5}e^{\frac{\arctan(\frac{2\sqrt{29}}{3})i}{3}} x=\sqrt{5}e^{-\frac{\arctan(\frac{2\sqrt{29}}{3})i}{3}} x=\sqrt{5}e^{\frac{-\arctan(\frac{2\sqrt{29}}{3})i+4\pi i}{3}} x=\sqrt{5}e^{\frac{-\arctan(\frac{2\sqrt{29}}{3})i+2\pi i}{3}}$$