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0
578
6
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 Nov 16, 2014

Best Answer 

 #4
avatar+118723 
+8

How much of this do you need to do by hand?

 

Well here is a relevant graph

https://www.desmos.com/calculator/mtrfxxuwrs

 

I can see from the graph that y=x

$${{\mathtt{x}}}^{{\mathtt{6}}}{\mathtt{\,-\,}}{{\mathtt{x}}}^{{\mathtt{4}}} = {\mathtt{18}} \Rightarrow \left\{ \begin{array}{l}{\mathtt{x}} = {\mathtt{\,-\,}}{\sqrt{{\mathtt{3}}}}\\
{\mathtt{x}} = {\sqrt{{\mathtt{3}}}}\\
{\mathtt{x}} = {\mathtt{\,-\,}}{\sqrt{{\sqrt{{\mathtt{5}}}}{\mathtt{\,\times\,}}{i}{\mathtt{\,-\,}}{\mathtt{1}}}}\\
{\mathtt{x}} = {\sqrt{{\sqrt{{\mathtt{5}}}}{\mathtt{\,\times\,}}{i}{\mathtt{\,-\,}}{\mathtt{1}}}}\\
{\mathtt{x}} = {\mathtt{\,-\,}}{\sqrt{{\mathtt{\,-\,}}{\sqrt{{\mathtt{5}}}}{\mathtt{\,\times\,}}{i}{\mathtt{\,-\,}}{\mathtt{1}}}}\\
{\mathtt{x}} = {\sqrt{{\mathtt{\,-\,}}{\sqrt{{\mathtt{5}}}}{\mathtt{\,\times\,}}{i}{\mathtt{\,-\,}}{\mathtt{1}}}}\\
\end{array} \right\}$$

 

so the real solutions are 

$$(-\sqrt3,-\sqrt3),\quad (\sqrt3,-\sqrt3),\quad (-\sqrt3,\sqrt3),\quad and \quad (\sqrt3,\sqrt3)$$

 

NOW how much of this did you really need to do by hand?   

 Nov 16, 2014
 #1
avatar+118723 
0

Just to be sure,

are these 

$$\\x^6=y^4+18\\
and\\
y^6=x^4+18$$
       ???

 Nov 16, 2014
 #2
avatar
0

Yes it is. It's an equations system...

 Nov 16, 2014
 #3
avatar+6 
+5

so this is as far as I got... And btw the question is Find all pairs of real numbers I don't know if i translate that good because English isn't my native language...

 Nov 16, 2014
 #4
avatar+118723 
+8
Best Answer

How much of this do you need to do by hand?

 

Well here is a relevant graph

https://www.desmos.com/calculator/mtrfxxuwrs

 

I can see from the graph that y=x

$${{\mathtt{x}}}^{{\mathtt{6}}}{\mathtt{\,-\,}}{{\mathtt{x}}}^{{\mathtt{4}}} = {\mathtt{18}} \Rightarrow \left\{ \begin{array}{l}{\mathtt{x}} = {\mathtt{\,-\,}}{\sqrt{{\mathtt{3}}}}\\
{\mathtt{x}} = {\sqrt{{\mathtt{3}}}}\\
{\mathtt{x}} = {\mathtt{\,-\,}}{\sqrt{{\sqrt{{\mathtt{5}}}}{\mathtt{\,\times\,}}{i}{\mathtt{\,-\,}}{\mathtt{1}}}}\\
{\mathtt{x}} = {\sqrt{{\sqrt{{\mathtt{5}}}}{\mathtt{\,\times\,}}{i}{\mathtt{\,-\,}}{\mathtt{1}}}}\\
{\mathtt{x}} = {\mathtt{\,-\,}}{\sqrt{{\mathtt{\,-\,}}{\sqrt{{\mathtt{5}}}}{\mathtt{\,\times\,}}{i}{\mathtt{\,-\,}}{\mathtt{1}}}}\\
{\mathtt{x}} = {\sqrt{{\mathtt{\,-\,}}{\sqrt{{\mathtt{5}}}}{\mathtt{\,\times\,}}{i}{\mathtt{\,-\,}}{\mathtt{1}}}}\\
\end{array} \right\}$$

 

so the real solutions are 

$$(-\sqrt3,-\sqrt3),\quad (\sqrt3,-\sqrt3),\quad (-\sqrt3,\sqrt3),\quad and \quad (\sqrt3,\sqrt3)$$

 

NOW how much of this did you really need to do by hand?   

Melody Nov 16, 2014
 #5
avatar+6 
0

Like this only one... But there is other 9 hardcore problems...

 Nov 16, 2014
 #6
avatar+118723 
0

Welcome to the forum Mataxa   

 

I have given you the final answer that you need but do you need more working.

 

I did almost all of it with online calculators. 

 Nov 16, 2014

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