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The equation x⁴ - (12K + 5)x² + 16K² = 0 (K > 0) has four real solutions in arithmetic progression. If K = a/b where a and b are coprime positive integers, find (a - 3b).

1. 3
2. 4
3. 6
4. 7

Correct answer: 3

Solution

Four AP roots symmetric about 0: -3d, -d, d, 3d. Substituting u = x²: the two values are d² and 9d². By Vieta for u² - (12K+5)u + 16K² = 0: sum = d²+9d² = 10d² = 12K+5, product = d²*9d² = 9d⁴ = 16K². From product: 3d² = 4K => d² = 4K/3. Substituting in sum: 10*(4K/3) = 12K+5 => 40K/3 - 12K = 5 => 4K/3 = 5 => K = 15/4. a=15, b=4. a-3b = 15-12 = 3.

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