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Home » ho^2 = c^2 \sin^2\phi - Decision Point

ho^2 = c^2 \sin^2\phi - Decision Point

📅 May 3, 2026 👤 scraface

May 03, 2026

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📌 5Frage: Berechne die Summe der Wurzeln der Gleichung \( v\sqrt{v} - 8v + 9\sqrt{v} - 18 = 0 \), vorausgesetzt, dass alle Wurzeln reell sind.

📌 Lösung: Sei \( w = \sqrt{v} \), also \( v = w^2 \). Einsetzen in die Gleichung ergibt:

📌 w^2 \cdot w - 8w^2 + 9w - 18 = 0 \Rightarrow w^3 - 8w^2 + 9w - 18 = 0.

📌 Wir suchen die Summe der Wurzeln der ursprünglichen Gleichung in \( v \), also der Summe der Quadrate der reellen positiven Wurzeln \( w \), da \( v = w^2 \). Zunächst bestimmen wir die Summe und das Produkt der Wurzeln \( w_i \) des kubischen Polynoms \( w^3 - 8w^2 + 9w - 18 = 0 \). Nach Vieta’s Formeln:

📌 w_1 + w_2 + w_3 = 8, \quad w_1w_2 + w_2w_3 + w_3w_1 = 9, \quad w_1w_2w_3 = 18.

📌 Wir wollen \( \sum v_i = \sum w_i^2 \) berechnen. Mit der Identität:

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