Criteria for Solving this equation

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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. Without specific numerical values for the constants L and K, the equation lacks a straightforward analytical solution. Numerical methods, such as iterative procedures in Excel, are recommended for finding solutions when constants are known. While special functions like Jacobi theta functions can theoretically provide solutions, they are complex and impractical for most applications. Therefore, numerical approaches are the most effective way to solve this equation.
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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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