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Home » A_{\textshaded} = 100 - 25\pi - High Altitude Science

A_{\textshaded} = 100 - 25\pi - High Altitude Science

📅 March 10, 2026 👤 scraface

Mar 10, 2026

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📌 b_4 = \frac{904518 - 97336}{1594323} = \frac{807182}{1594323}

📌 Thus, $ b_4 = \boxed{\dfrac{807182}{1594323}} $.

📌 Question: A science policy analyst studying energy output models a system where total power $ P(x) = x\sqrt{x} - 5x + 6\sqrt{x} $ is produced based on input $ x \geq 0 $. Find the sum of all real values of $ x $ for which $ P(x) = 0 $, given all roots are non-negative real numbers.

📌 Solution: Let $ u = \sqrt{x} $, so $ x = u^2 $, and $ P(x) = u^2 \cdot u - 5u^2 + 6u = u^3 - 5u^2 + 6u $.

📌 Set $ u^3 - 5u^2 + 6u = 0 $. Factor:

📌 u(u^2 - 5u + 6) = 0 \Rightarrow u(u - 2)(u - 3) = 0

📌 So $ u = 0, 2, $ or $ 3 $. Since $ x = u^2 $, the corresponding $ x $-values are:

📌 $ u = 0 \Rightarrow x = 0 $

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