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Home » The square of a number exceeds twice the number by 15. What is the number? - High Altitude Science

The square of a number exceeds twice the number by 15. What is the number? - High Altitude Science

📅 March 11, 2026 👤 scraface

Mar 11, 2026

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📌 Using identity: \( \cos 3\theta = \frac{1}{4} \implies 3\theta = \pm \arccos\left(\frac{1}{4}\right) + 2k\pi \)

📌 \theta = \frac{1}{3} \arccos\left(\frac{1}{4}\right),\quad \frac{1}{3}\left(2\pi - \arccos\left(\frac{1}{4}\right)\right),\quad \frac{1}{3}\left(4\pi + \arccos\left(\frac{1}{4}\right)\right)

📌 So the three real solutions for \( y \) are:

📌 y_k = \frac{1}{3} \arccos\left(\frac{1}{4}\right) + \frac{2k\pi}{3},\quad k = 0,1,2

📌 x_k = 2\cos\left( \frac{1}{3} \arccos\left(\frac{1}{4}\right) + \frac{2k\pi}{3} \right),\quad k = 0,1,2

📌 But for simplicity and clarity in competition setting, and since no cleaner algebraic form exists, we conclude:

📌 The real solutions are the three roots of \( x

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