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# X^6-X^3=2

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Equation

X^6-X^3=2

Rehan  Jun 19, 2015

#1
+125
+8

$${\mathtt{y}} = {{\mathtt{x}}}^{{\mathtt{3}}}$$

$${{\mathtt{y}}}^{{\mathtt{2}}}{\mathtt{\,-\,}}{\mathtt{y}} = {\mathtt{2}} \Rightarrow \left\{ \begin{array}{l}{\mathtt{y}} = {\mathtt{2}}\\ {\mathtt{y}} = -{\mathtt{1}}\\ \end{array} \right\}$$

$${{\mathtt{x}}}^{{\mathtt{3}}} = {\mathtt{2}} \Rightarrow \left\{ \begin{array}{l}{\mathtt{x}} = {\frac{\left({{\mathtt{2}}}^{\left({\frac{{\mathtt{1}}}{{\mathtt{3}}}}\right)}{\mathtt{\,\times\,}}{\sqrt{{\mathtt{3}}}}{\mathtt{\,\times\,}}{i}{\mathtt{\,-\,}}{{\mathtt{2}}}^{\left({\frac{{\mathtt{1}}}{{\mathtt{3}}}}\right)}\right)}{{\mathtt{2}}}}\\ {\mathtt{x}} = {\mathtt{\,-\,}}{\frac{\left({{\mathtt{2}}}^{\left({\frac{{\mathtt{1}}}{{\mathtt{3}}}}\right)}{\mathtt{\,\times\,}}{\sqrt{{\mathtt{3}}}}{\mathtt{\,\times\,}}{i}{\mathtt{\,\small\textbf+\,}}{{\mathtt{2}}}^{\left({\frac{{\mathtt{1}}}{{\mathtt{3}}}}\right)}\right)}{{\mathtt{2}}}}\\ {\mathtt{x}} = {{\mathtt{2}}}^{\left({\frac{{\mathtt{1}}}{{\mathtt{3}}}}\right)}\\ \end{array} \right\} \Rightarrow \left\{ \begin{array}{l}{\mathtt{x}} = {\mathtt{\,-\,}}{\mathtt{0.629\: \!960\: \!524\: \!947\: \!436\: \!6}}{\mathtt{\,\small\textbf+\,}}{\mathtt{1.091\: \!123\: \!635\: \!972\: \!428\: \!7}}{i}\\ {\mathtt{x}} = {\mathtt{\,-\,}}\left({\mathtt{0.629\: \!960\: \!524\: \!947\: \!436\: \!6}}{\mathtt{\,\small\textbf+\,}}{\mathtt{1.091\: \!123\: \!635\: \!972\: \!428\: \!7}}{i}\right)\\ {\mathtt{x}} = {\mathtt{1.259\: \!921\: \!049\: \!894\: \!873\: \!2}}\\ \end{array} \right\}$$

$${{\mathtt{x}}}^{{\mathtt{3}}} = -{\mathtt{1}} \Rightarrow \left\{ \begin{array}{l}{\mathtt{x}} = {\mathtt{\,-\,}}{\frac{\left({\sqrt{{\mathtt{3}}}}{\mathtt{\,\times\,}}{i}{\mathtt{\,-\,}}{\mathtt{1}}\right)}{{\mathtt{2}}}}\\ {\mathtt{x}} = {\frac{\left({\sqrt{{\mathtt{3}}}}{\mathtt{\,\times\,}}{i}{\mathtt{\,\small\textbf+\,}}{\mathtt{1}}\right)}{{\mathtt{2}}}}\\ {\mathtt{x}} = -{\mathtt{1}}\\ \end{array} \right\} \Rightarrow \left\{ \begin{array}{l}{\mathtt{x}} = {\mathtt{\,-\,}}\left({\mathtt{\,-\,}}{\frac{{\mathtt{1}}}{{\mathtt{2}}}}{\mathtt{\,\small\textbf+\,}}{\mathtt{0.866\: \!025\: \!403\: \!785}}{i}\right)\\ {\mathtt{x}} = {\frac{{\mathtt{1}}}{{\mathtt{2}}}}{\mathtt{\,\small\textbf+\,}}{\mathtt{0.866\: \!025\: \!403\: \!785}}{i}\\ {\mathtt{x}} = -{\mathtt{1}}\\ \end{array} \right\}$$

So the real solutions are:

• $${\mathtt{x}} = {{\mathtt{2}}}^{\left({\frac{{\mathtt{1}}}{{\mathtt{3}}}}\right)}$$
• $${\mathtt{x}} = -{\mathtt{1}}$$
syllogist  Jun 19, 2015
Sort:

#1
+125
+8

$${\mathtt{y}} = {{\mathtt{x}}}^{{\mathtt{3}}}$$

$${{\mathtt{y}}}^{{\mathtt{2}}}{\mathtt{\,-\,}}{\mathtt{y}} = {\mathtt{2}} \Rightarrow \left\{ \begin{array}{l}{\mathtt{y}} = {\mathtt{2}}\\ {\mathtt{y}} = -{\mathtt{1}}\\ \end{array} \right\}$$

$${{\mathtt{x}}}^{{\mathtt{3}}} = {\mathtt{2}} \Rightarrow \left\{ \begin{array}{l}{\mathtt{x}} = {\frac{\left({{\mathtt{2}}}^{\left({\frac{{\mathtt{1}}}{{\mathtt{3}}}}\right)}{\mathtt{\,\times\,}}{\sqrt{{\mathtt{3}}}}{\mathtt{\,\times\,}}{i}{\mathtt{\,-\,}}{{\mathtt{2}}}^{\left({\frac{{\mathtt{1}}}{{\mathtt{3}}}}\right)}\right)}{{\mathtt{2}}}}\\ {\mathtt{x}} = {\mathtt{\,-\,}}{\frac{\left({{\mathtt{2}}}^{\left({\frac{{\mathtt{1}}}{{\mathtt{3}}}}\right)}{\mathtt{\,\times\,}}{\sqrt{{\mathtt{3}}}}{\mathtt{\,\times\,}}{i}{\mathtt{\,\small\textbf+\,}}{{\mathtt{2}}}^{\left({\frac{{\mathtt{1}}}{{\mathtt{3}}}}\right)}\right)}{{\mathtt{2}}}}\\ {\mathtt{x}} = {{\mathtt{2}}}^{\left({\frac{{\mathtt{1}}}{{\mathtt{3}}}}\right)}\\ \end{array} \right\} \Rightarrow \left\{ \begin{array}{l}{\mathtt{x}} = {\mathtt{\,-\,}}{\mathtt{0.629\: \!960\: \!524\: \!947\: \!436\: \!6}}{\mathtt{\,\small\textbf+\,}}{\mathtt{1.091\: \!123\: \!635\: \!972\: \!428\: \!7}}{i}\\ {\mathtt{x}} = {\mathtt{\,-\,}}\left({\mathtt{0.629\: \!960\: \!524\: \!947\: \!436\: \!6}}{\mathtt{\,\small\textbf+\,}}{\mathtt{1.091\: \!123\: \!635\: \!972\: \!428\: \!7}}{i}\right)\\ {\mathtt{x}} = {\mathtt{1.259\: \!921\: \!049\: \!894\: \!873\: \!2}}\\ \end{array} \right\}$$

$${{\mathtt{x}}}^{{\mathtt{3}}} = -{\mathtt{1}} \Rightarrow \left\{ \begin{array}{l}{\mathtt{x}} = {\mathtt{\,-\,}}{\frac{\left({\sqrt{{\mathtt{3}}}}{\mathtt{\,\times\,}}{i}{\mathtt{\,-\,}}{\mathtt{1}}\right)}{{\mathtt{2}}}}\\ {\mathtt{x}} = {\frac{\left({\sqrt{{\mathtt{3}}}}{\mathtt{\,\times\,}}{i}{\mathtt{\,\small\textbf+\,}}{\mathtt{1}}\right)}{{\mathtt{2}}}}\\ {\mathtt{x}} = -{\mathtt{1}}\\ \end{array} \right\} \Rightarrow \left\{ \begin{array}{l}{\mathtt{x}} = {\mathtt{\,-\,}}\left({\mathtt{\,-\,}}{\frac{{\mathtt{1}}}{{\mathtt{2}}}}{\mathtt{\,\small\textbf+\,}}{\mathtt{0.866\: \!025\: \!403\: \!785}}{i}\right)\\ {\mathtt{x}} = {\frac{{\mathtt{1}}}{{\mathtt{2}}}}{\mathtt{\,\small\textbf+\,}}{\mathtt{0.866\: \!025\: \!403\: \!785}}{i}\\ {\mathtt{x}} = -{\mathtt{1}}\\ \end{array} \right\}$$

So the real solutions are:

• $${\mathtt{x}} = {{\mathtt{2}}}^{\left({\frac{{\mathtt{1}}}{{\mathtt{3}}}}\right)}$$
• $${\mathtt{x}} = -{\mathtt{1}}$$
syllogist  Jun 19, 2015
#2
+18829
+5

Equation  X^6-X^3=2

$$\small{\text{ \begin{array}{rcl} x^6-x^3&=&2 \qquad \mathrm{we ~substitute~and ~set~} z = x^3 \\ \mathrm{then~we~have~} \quad z^2-z &=& 2 \\ z^2-z -2 &=& 0\\ z_{1,2} &=& \dfrac{1\pm \sqrt{1-4\cdot(-2)}}{2} \\ z_{1,2} &=& \dfrac{1\pm \sqrt{9}}{2} \\\\ z_{1,2} &=& \dfrac{1\pm 3 }{2} \\\\ z_1 &=& \frac{1 + 3 }{2} = 2 \\\\ z_2 &=& \frac{1 - 3 }{2} = - 1\\\\\\ x_1 & =& \sqrt[3]{z_1}= \sqrt[3]{2}= 1.25992104989\\\\ \mathbf{x_1} & \mathbf{=}& \mathbf{1.25992104989}\\\\\\ x_2 &=& \sqrt[3]{z_2}= \sqrt[3]{-1}= -1\\\\ \mathbf{x_2} &\mathbf{=}& \mathbf{-1}\\\\ \end{array} }}$$

heureka  Jun 19, 2015
#3
+91454
+5

Syllogist is right :))

$$\\X^6-X^3=2\\\\ Let \;Y=X^3\\\\ Y^2-Y=2\\\\ Completing the square\\\\ Y^2-Y+\frac{1}{4}=2+\frac{1}{4}\\\\ (Y-\frac{1}{2})^2=\frac{9}{4}\\\\ Y-\frac{1}{2}=\pm\frac{3}{2}\\\\ Y-\frac{1}{2}=\frac{3}{2}\qquad OR \qquad Y-\frac{1}{2}=-\frac{3}{2}\\\\ Y=2\qquad \qquad OR \qquad \qquad Y=-1\\\\ X^3=2\qquad \qquad OR \qquad \qquad X^3=-1\\\\ X=\sqrt[3]{2}\qquad \qquad OR \qquad \qquad X=-1\\\\$$

These are just the real solutions :)

Let me think about the complex solutions

I think they are

$$\\x=\sqrt[3]{2}\;\;e^{\frac{2\pi i}{3}}, \qquad x=\sqrt[3]{2}\;\;e^{\frac{-2\pi i}{3}}, \qquad x=e^{\frac{-\pi i}{3}}, \quad and \quad x= e^{\frac{\pi i}{3}}$$

I hope I didn't s***w that one up :/

Melody  Jun 19, 2015

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