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Home » \( D(1) = p + q + r + s = 3 \) - Portal da AcÃºstica

\( D(1) = p + q + r + s = 3 \) - Portal da AcÃºstica

📅 March 1, 2026 👤 scraface

Mar 01, 2026

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📌 Question: A quantum diagnostic device uses a function $ f(u) = u\sqrt{u} - 5u + 6\sqrt{u} $. Find the sum of all real roots of the equation $ f(u) = 0 $, given that all roots are non-negative.

📌 Solution: Let $ x = \sqrt{u} $, so $ u = x^2 $, and the equation becomes:

📌 x^2 \cdot x - 5x^2 + 6x = 0 \Rightarrow x^3 - 5x^2 + 6x = 0

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

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

📌 $ x = 0 \Rightarrow u = 0 $

📌 $ x = 2 \Rightarrow u = 4 $

📌 $ x = 3 \Rightarrow u = 9 $

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