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Home » To find the roots, solve \( x^3 - 6x^2 + 11x - 6 = 0 \). - Moon Smoking

To find the roots, solve \( x^3 - 6x^2 + 11x - 6 = 0 \). - Moon Smoking

📅 May 15, 2026 👤 scraface

May 15, 2026

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📌 Use formulas:

📌 $ \sum_{k=1}^{n} k = \frac{n(n+1)}{2} $

📌 $ \sum_{k=1}^{n} k^3 = \left( \frac{n(n+1)}{2} \right)^2 $

📌 a_n = \frac{1}{n} \cdot \frac{n(n+1)}{2} - \frac{1}{3n^3} \cdot \left( \frac{n(n+1)}{2} \right)^2

📌 = \frac{n+1}{2} - \frac{1}{3n^3} \cdot \frac{n^2(n+1)^2}{4}

📌 Simplify the second term:

📌 \frac{1}{3n^3} \cdot \frac{n^2(n+1)^2}{4} = \frac{(n+1)^Question: A historian studying ancient navigation techniques discovers a triangular chart used by early mariners, with side lengths of $13$, $14$, and $15$ units. What is the length of the shortest altitude in this triangle?

📌 Solution: To find the shortest altitude, we first compute the area using Heron’s formula. Let the triangle have sides $a = 13$, $b = 14$, and $c = 15$. The semi-perimeter is:

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