MHB Can you solve this double cubic algebraic equation?

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The discussion focuses on solving a double cubic algebraic equation by considering the expression $x+\frac{1}{x}$ as a whole. It identifies four real solutions: $x=\frac{3±\sqrt{5}}{2}$ and $x=\frac{-3±\sqrt{5}}{2}$, along with two complex solutions: $x=±i$. An alternative method suggested involves multiplying both sides by $x^3$ and substituting $y = x^2$, leading to a simpler cubic equation that can be solved using the rational root theorem. The participants express appreciation for each other's ideas and methods. The thread highlights various approaches to solving the equation effectively.
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The key idea is to view $x+\frac{1}{x}$ as a whole. There are four real solutions: $x=\frac{3±\sqrt{5}}{2}$, $x=\frac{-3±\sqrt{5}}{2}$ (and two complex solutions: $x=±i$). Here is the explanation:
 
MathTutoringByDrLiang said:
The key idea is to view $x+\frac{1}{x}$ as a whole. There are four real solutions: $x=\frac{3±\sqrt{5}}{2}$, $x=\frac{-3±\sqrt{5}}{2}$ (and two complex solutions: $x=±i$). Here is the explanation:

Nice idea!

Or you could just multiply both sides by $x^3$ and sub in $y = x^2$. The resulting cubic equation for y is easy to solve using the rational root theorem.

-Dan
 
topsquark said:
Nice idea!

Or you could just multiply both sides by $x^3$ and sub in $y = x^2$. The resulting cubic equation for y is easy to solve using the rational root theorem.

-Dan
Thank you very much for your feedback!

Derek
 

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