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Home » The arithmetic mean is \(\boxed11\). - Silent Sales Machine

The arithmetic mean is \(\boxed11\). - Silent Sales Machine

📅 March 10, 2026 👤 scraface

Mar 10, 2026

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📌 14x^2 + 42x + 27 = 0.

📌 Discriminant: $ 42^2 - 4\cdot14\cdot27 = 1764 - 1512 = 252 = 36 \cdot 7 $.

📌 x = \frac{-42 \pm 6\sqrt{7}}{28} = \frac{-21 \pm 3\sqrt{7}}{14}.

📌 Third term: $ A + 2d $, first: $ A $, so ratio:

📌 \frac{A + 2d}{A} = 1 + 2\frac{d}{A} = 1 + 2\frac{1}{x} = 1 + \frac{2x}{-21 \pm 3\sqrt{7}}.

📌 Take positive root for real chance (positive terms): $ x = \frac{-21 + 3\sqrt{7}}{14} $. Compute numerically: $ \sqrt{7} \approx 2.6458 $, $ 3\sqrt{7} \approx 7.937 $, so $ x \approx \frac{-21 + 7.937}{14} = \frac{-13.063}{14} \approx -0.932 $. Then $ \frac{1}{x} \approx -1.073 $, so $ 1 + 2(-1.073) = -1.146 $. Not meaningful.

📌 But from quadratic: $ 14x^2 + 42x + 27 = 0 $. Product of roots $ \frac{27}{14} $, sum $ -3 $.

📌 Let roots be $ r_1, r_2 $, then $ \frac{A + 2d}{A} = 1 + 2\frac{d}{A} = 1 + 2\frac{1}{x} $. Since product is $ x^2 = 28/9 $, $ x = \pm \sqrt{28}/3 $. Use symbolic. From $ 14x^2 + 42x + 27 = 0 $, divide by 14: $ x^2 + 3x + \frac{27}{14} = 0 $.

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