Réitigh x^6=((6x^3)-5^3)

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

Réitigh do x. (complex solution)

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

Céimeanna ag Baint Úsáid as an bhFoirmle Chearnach

Ríomh cumhacht 5 de 3 agus faigh 125.

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

Bain 6x^{3} ón dá thaobh.

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

Cuir 125 leis an dá thaobh.

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

Cuir t in ionad x^{3}.

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

Is féidir gach cothromóid i bhfoirm ax^{2}+bx+c=0 a réiteach ach an fhoirmle chearnach seo a úsáid: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. Cuir 1 in ionad a, -6 in ionad b agus 125 in ionad c san fhoirmle chearnach.

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

Déan áirimh.

$$t=\frac{6±\sqrt{-464}}{2}$$

Réitigh an chothromóid t=\frac{6±\sqrt{-464}}{2} nuair is ionann ± agus luach deimhneach agus ± agus luach diúltach.

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

x=t^{3} agus sin an fáth go dtagtar ar na réitigh tríd an gcothromóid a réiteach do gach t.

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