Criteria for Solving this equation

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SUMMARY

The equation H^5 - 10*L*H^4 + 25*L^2*H^3 - 25*Q^2/K^2 = 0 is a fifth-order polynomial in H, which cannot be solved using a general formula. Instead, numerical methods, such as iterative procedures in Excel, are recommended for finding solutions when specific values for constants L and K are provided. While special functions like Jacobi theta functions can theoretically express solutions, practical applications favor numerical approaches due to their simplicity.

PREREQUISITES
  • Understanding of polynomial equations, particularly fifth-degree polynomials.
  • Familiarity with numerical methods for solving equations.
  • Basic knowledge of Excel for implementing iterative procedures.
  • Concept of special functions, specifically Jacobi theta functions.
NEXT STEPS
  • Research numerical methods for solving polynomial equations.
  • Learn how to implement iterative procedures in Excel for root-finding.
  • Study the properties and applications of Jacobi theta functions.
  • Explore alternative numerical software tools like MATLAB or Python's NumPy for solving complex equations.
USEFUL FOR

Mathematicians, engineers, and anyone involved in solving complex polynomial equations, particularly those seeking practical solutions using numerical methods.

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Is it possible to solve this equation in terms of H without knowing any numbers? If so what method would I use? If I had numbers I think I could solve it numerically in excel. It's been a few years since I was in school and am fairly (very) rusty:

H^5 -10*L*H^4 +25*L^2*H^3 -25*Q^2/K^2 = 0

L,K = Constant

Any help would be appreciated, Thanks!
 
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You have a fifth order polynomial in H. There is no general formula which can solve this equation. If you have numerical values for your constants, then an iterative procedure (perhaps using Excel) would be the way to calculate solutions.
 
There is no simple formula giving Y as a function of L, H, Q, K because it is a polynomial equation of the fifth degree.
Such formula exists, but involving special functions : the Jacobi theta functions. But, in practice, it is much simpler to use numerical methods to solve the equation.
 

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