Réitigh 4=1/6x^6+x^3+2

$$4 = \frac { 1 } { 6 } x ^ { 6 } + x ^ { 3 } + 2$$

Réitigh do x. (complex solution)

$$x\in \sqrt[3]{\sqrt{21}+3}e^{\frac{\pi i}{3}},\sqrt[3]{\sqrt{21}+3}e^{\frac{5\pi i}{3}},-\sqrt[3]{\sqrt{21}+3},\sqrt[3]{\sqrt{21}-3}e^{\frac{4\pi i}{3}},\sqrt[3]{\sqrt{21}-3},\sqrt[3]{\sqrt{21}-3}e^{\frac{2\pi i}{3}}$$

Céimeanna ag Baint Úsáid as an bhFoirmle Chearnach

Athraigh na taobhanna ionas go mbeidh na téarmaí inathraitheacha ar fad ar an taobh clé.

$$\frac{1}{6}x^{6}+x^{3}+2=4$$

Bain 4 ón dá thaobh.

$$\frac{1}{6}x^{6}+x^{3}+2-4=0$$

Dealaigh 4 ó 2 chun -2 a fháil.

$$\frac{1}{6}x^{6}+x^{3}-2=0$$

Cuir t in ionad x^{3}.

$$\frac{1}{6}t^{2}+t-2=0$$

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 \frac{1}{6} in ionad a, 1 in ionad b agus -2 in ionad c san fhoirmle chearnach.

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

Déan áirimh.

$$t=\frac{-1±\frac{1}{3}\sqrt{21}}{\frac{1}{3}}$$

Réitigh an chothromóid t=\frac{-1±\frac{1}{3}\sqrt{21}}{\frac{1}{3}} nuair is ionann ± agus luach deimhneach agus ± agus luach diúltach.

$$t=\sqrt{21}-3 t=-\sqrt{21}-3$$

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

$$x=-\sqrt[3]{\sqrt{21}-3}e^{\frac{\pi i}{3}} x=\sqrt[3]{\sqrt{21}-3}ie^{\frac{\pi i}{6}} x=\sqrt[3]{\sqrt{21}-3} x=-\sqrt[3]{\sqrt{21}+3}ie^{\frac{\pi i}{6}} x=-\sqrt[3]{\sqrt{21}+3} x=\sqrt[3]{\sqrt{21}+3}e^{\frac{\pi i}{3}}$$

Réitigh do x.

$$x=\sqrt[3]{\sqrt{21}-3}\approx 1.165345841$$

$$x=-\sqrt[3]{\sqrt{21}+3}\approx -1.964591458$$

Céimeanna ag Baint Úsáid as an bhFoirmle Chearnach

Athraigh na taobhanna ionas go mbeidh na téarmaí inathraitheacha ar fad ar an taobh clé.

$$\frac{1}{6}x^{6}+x^{3}+2=4$$

Bain 4 ón dá thaobh.

$$\frac{1}{6}x^{6}+x^{3}+2-4=0$$

Dealaigh 4 ó 2 chun -2 a fháil.

$$\frac{1}{6}x^{6}+x^{3}-2=0$$

Cuir t in ionad x^{3}.

$$\frac{1}{6}t^{2}+t-2=0$$

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 \frac{1}{6} in ionad a, 1 in ionad b agus -2 in ionad c san fhoirmle chearnach.

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

Déan áirimh.

$$t=\frac{-1±\frac{1}{3}\sqrt{21}}{\frac{1}{3}}$$

Réitigh an chothromóid t=\frac{-1±\frac{1}{3}\sqrt{21}}{\frac{1}{3}} nuair is ionann ± agus luach deimhneach agus ± agus luach diúltach.

$$t=\sqrt{21}-3 t=-\sqrt{21}-3$$

Más x=t^{3}, is féidir teacht ar na réitigh ach x=\sqrt[3]{t} a mheas i gcomhair gach t.

$$x=\sqrt[3]{\sqrt{21}-3} x=-\sqrt[3]{\sqrt{21}+3}$$