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2006- 2026
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2A^2 + 6Ad + 9d^2 = 16A^2 + 48Ad + 36d^2
\Rightarrow 0 = 14A^2 + 42Ad + 27d^2.
Divide by $ d^2 $: let $ x = \frac{A}{d} $, then:
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 $.