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Home » \(n = 8\), \(a = 5\), \(d = 3\). - Aurero

\(n = 8\), \(a = 5\), \(d = 3\). - Aurero

📅 March 14, 2026 👤 scraface

Mar 14, 2026

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📌 Question: A historian of science analyzing Newtonâs early work considers the equation \( v\sqrt{v} - 5v + 6\sqrt{v} = 0 \), where \( v > 0 \) represents velocity-related magnitude. Find the sum of all real roots.

📌 Solution: We are given the equation:

📌 v\sqrt{v} - 5v + 6\sqrt{v} = 0, \quad v > 0

📌 Note that \( v\sqrt{v} = v^{3/2} \) and \( \sqrt{v} = v^{1/2} \). Let \( u = \sqrt{v} \), so \( v = u^2 \), and \( v^{3/2} = u^3 \). Substitute:

📌 u^3 - 5u^2 + 6u = 0

📌 u(u^2 - 5u + 6) = 0

📌 \Rightarrow u(u - 2)(u - 3) = 0

📌 Solutions: \( u = 0 \), \( u = 2 \), \( u = 3 \)

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