Can you solve this double cubic algebraic equation?

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In summary, the key idea is to view $x+\frac{1}{x}$ as a whole. This leads to four real solutions: $x=\frac{3±\sqrt{5}}{2}$, $x=\frac{-3±\sqrt{5}}{2}$, and two complex solutions: $x=±i$. Another approach suggested is to multiply both sides by $x^3$ and substitute $y = x^2$, resulting in a cubic equation for $y$ that can be solved using the rational root theorem.
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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:
 
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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
 
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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
 

1. How do you solve a double cubic algebraic equation?

To solve a double cubic algebraic equation, you can use the cubic formula or factorization. The cubic formula involves plugging in the coefficients of the equation into a formula, while factorization involves finding common factors and using the zero product property.

2. What is the cubic formula?

The cubic formula is a mathematical formula used to solve cubic equations. It is given by x = (-b ± √(b^2 - 4ac - 3b^3) / 2a, where a, b, and c are the coefficients of the cubic equation.

3. Can all double cubic algebraic equations be solved?

Yes, all double cubic algebraic equations can be solved using either the cubic formula or factorization. However, the solutions may involve complex numbers.

4. Are there any special cases when solving a double cubic algebraic equation?

Yes, there are two special cases when solving a double cubic algebraic equation: when all three roots are real and distinct, and when one root is real and the other two are complex conjugates. In these cases, the cubic formula can be simplified.

5. Can a computer solve a double cubic algebraic equation?

Yes, a computer can solve a double cubic algebraic equation by using numerical methods such as the Newton-Raphson method or the bisection method. These methods involve approximating the roots of the equation using iterative calculations.

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