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A = -4.876 , B = -215,617 , C = 28184.227 , D = 461443.354 , E = -14.205

The solutions to the quartic equation:

\[
-4.876 Z_{b1}^4 - 215.617 Z_{b1}^3 + 28184.227 Z_{b1}^2 + 461443.354 Z_{b1} - 14.205 = 0
\]

are approximately:

1. \( Z_{b1} \approx -94.703 \)
2. \( Z_{b1} \approx -15.211 \)
3. \( Z_{b1} \approx 0.00003078 \)
4. \( Z_{b1} \approx 65.694 \)

These are real solutions, and the imaginary parts are negligible (close to zero).

Would you like more details on how these solutions were derived, or any specific step explained? Here are some related questions for further exploration:

1. What does each root represent in terms of the original context?
2. Can this quartic equation have any physical or real-world applications?
3. How does the nature of the coefficients (large/small) affect the behavior of the solutions?
4. What methods exist for solving quartic equations analytically?
5. How does numerical precision influence small solutions like \( Z_{b1} = 0.00003078 \)?

**Tip**: Always verify solutions graphically when dealing with higher-degree polynomials, to visualize their behavior and ensure accuracy.

A = -4.876 , B = -215.617 , C = 28184.227 , D = 461443.354 , E = -14.205

Learn how to solve the quartic equation -4.876Z^4 - 215.617Z^3 + 28184.227Z^2 + 461443.354Z - 14.205 = 0. This detailed solution provides numerical approximations to the roots and explains the key mathematical concepts involved. Suitable for advanced high school students (Grades 10-12).

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