- #1
DrLiangMath
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Can you solve this double cubic algebraic equation?
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- Thread starter DrLiangMath
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SUMMARY
The discussion focuses on solving the double cubic algebraic equation by analyzing the expression $x+\frac{1}{x}$. It establishes that there are 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 involves multiplying both sides by $x^3$ and substituting $y = x^2$, leading to a cubic equation for $y$ that can be solved using the rational root theorem.
PREREQUISITES- Understanding of algebraic equations and their solutions
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- Knowledge of the rational root theorem
- Basic skills in manipulating algebraic expressions
- Study the rational root theorem in detail
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- Explore the properties of complex numbers and their applications
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Mathematicians, students studying algebra, and anyone interested in solving complex polynomial equations will benefit from this discussion.
- #2
DrLiangMath
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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:
- #3
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MathTutoringByDrLiang 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
- #4
DrLiangMath
- 21
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Thank you very much for your feedback!topsquark said:
Derek
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