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# 183k^3 + 61k - 3 = 0 How to solve this?

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183k^3 + 61k - 3 = 0 How to solve this?

Guest Feb 13, 2015

#5
+94203
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Desmos is for graphing not for solving.  So there is no mystery about that :))

Melody  Feb 13, 2015
#1
+94203
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It does not have nice integer solutions so I would use Desmos or Wolfram|Alpha to solve it

http://www.wolframalpha.com/input/?i=183k^3%20%2B%2061k%20-%203%20%3D%200

If I needed to do it myself I would use Newton's method of approximating roots

Melody  Feb 13, 2015
#2
+93038
+5

183k^3 + 61k - 3 = 0

The onsite solver will do this, too......the answer is pretty nasty...!!!

$${\mathtt{183}}{\mathtt{\,\times\,}}{{\mathtt{k}}}^{{\mathtt{3}}}{\mathtt{\,\small\textbf+\,}}{\mathtt{61}}{\mathtt{\,\times\,}}{\mathtt{k}}{\mathtt{\,-\,}}{\mathtt{3}} = {\mathtt{0}} \Rightarrow \left\{ \begin{array}{l}{\mathtt{k}} = {\left({\frac{{\sqrt{{\mathtt{15\,613}}}}}{{\mathtt{3\,294}}}}{\mathtt{\,\small\textbf+\,}}{\frac{{\mathtt{1}}}{{\mathtt{122}}}}\right)}^{\left({\frac{{\mathtt{1}}}{{\mathtt{3}}}}\right)}{\mathtt{\,\times\,}}\left({\mathtt{\,-\,}}{\frac{{\sqrt{{\mathtt{3}}}}{\mathtt{\,\times\,}}{i}}{{\mathtt{2}}}}{\mathtt{\,-\,}}{\frac{{\mathtt{1}}}{{\mathtt{2}}}}\right){\mathtt{\,-\,}}{\frac{\left({\frac{{\sqrt{{\mathtt{3}}}}{\mathtt{\,\times\,}}{i}}{{\mathtt{2}}}}{\mathtt{\,-\,}}{\frac{{\mathtt{1}}}{{\mathtt{2}}}}\right)}{\left({\mathtt{9}}{\mathtt{\,\times\,}}{\left({\frac{{\sqrt{{\mathtt{15\,613}}}}}{{\mathtt{3\,294}}}}{\mathtt{\,\small\textbf+\,}}{\frac{{\mathtt{1}}}{{\mathtt{122}}}}\right)}^{\left({\frac{{\mathtt{1}}}{{\mathtt{3}}}}\right)}\right)}}\\ {\mathtt{k}} = {\left({\frac{{\sqrt{{\mathtt{15\,613}}}}}{{\mathtt{3\,294}}}}{\mathtt{\,\small\textbf+\,}}{\frac{{\mathtt{1}}}{{\mathtt{122}}}}\right)}^{\left({\frac{{\mathtt{1}}}{{\mathtt{3}}}}\right)}{\mathtt{\,\times\,}}\left({\frac{{\sqrt{{\mathtt{3}}}}{\mathtt{\,\times\,}}{i}}{{\mathtt{2}}}}{\mathtt{\,-\,}}{\frac{{\mathtt{1}}}{{\mathtt{2}}}}\right){\mathtt{\,-\,}}{\frac{\left({\mathtt{\,-\,}}{\frac{{\sqrt{{\mathtt{3}}}}{\mathtt{\,\times\,}}{i}}{{\mathtt{2}}}}{\mathtt{\,-\,}}{\frac{{\mathtt{1}}}{{\mathtt{2}}}}\right)}{\left({\mathtt{9}}{\mathtt{\,\times\,}}{\left({\frac{{\sqrt{{\mathtt{15\,613}}}}}{{\mathtt{3\,294}}}}{\mathtt{\,\small\textbf+\,}}{\frac{{\mathtt{1}}}{{\mathtt{122}}}}\right)}^{\left({\frac{{\mathtt{1}}}{{\mathtt{3}}}}\right)}\right)}}\\ {\mathtt{k}} = {\left({\frac{{\sqrt{{\mathtt{15\,613}}}}}{{\mathtt{3\,294}}}}{\mathtt{\,\small\textbf+\,}}{\frac{{\mathtt{1}}}{{\mathtt{122}}}}\right)}^{\left({\frac{{\mathtt{1}}}{{\mathtt{3}}}}\right)}{\mathtt{\,-\,}}{\frac{{\mathtt{1}}}{\left({\mathtt{9}}{\mathtt{\,\times\,}}{\left({\frac{{\sqrt{{\mathtt{15\,613}}}}}{{\mathtt{3\,294}}}}{\mathtt{\,\small\textbf+\,}}{\frac{{\mathtt{1}}}{{\mathtt{122}}}}\right)}^{\left({\frac{{\mathtt{1}}}{{\mathtt{3}}}}\right)}\right)}}\\ \end{array} \right\} \Rightarrow \left\{ \begin{array}{l}{\mathtt{k}} = {\mathtt{\,-\,}}{\mathtt{0.024\: \!415\: \!509\: \!892\: \!177\: \!8}}{\mathtt{\,-\,}}{\mathtt{0.578\: \!896\: \!955\: \!168\: \!551\: \!4}}{i}\\ {\mathtt{k}} = {\mathtt{\,-\,}}{\mathtt{0.024\: \!415\: \!509\: \!892\: \!177\: \!8}}{\mathtt{\,\small\textbf+\,}}{\mathtt{0.578\: \!896\: \!955\: \!168\: \!551\: \!4}}{i}\\ {\mathtt{k}} = {\mathtt{0.048\: \!831\: \!019\: \!784\: \!355\: \!5}}\\ \end{array} \right\}$$

CPhill  Feb 13, 2015
#3
+94203
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Thanks Chris,

I often forget how clever our own site calculator is.

Thank you Mr Massow for making it for us.

Melody  Feb 13, 2015
#4
+93038
+5

Note....Desmos is rather funky about solving these equations....

It will not "solve"  183x^3 + 61x - 3 = 0

It will "solve"   183x^3  =  3 - 61x   if the left and right sides are entered as separate functions .....

See this......https://www.desmos.com/calculator/se02owjof1

Go figure....!!!

CPhill  Feb 13, 2015
#5
+94203
+5