<p>The sum of values of <i>x</i> satisfying the equation \((31 + 8\sqrt{15})^{x^2 - 3} + 1 = (32 + 8\sqrt{15})^{x^2 - 3}\) is</p>
Step-by-Step Solution
Key Concept: Recognize that 31 + 8√15 and 32 + 8√15 are consecutive integers, and rewrite the equation as a ratio equation. Notice that if we let y = (31 + 8√15)^(x²-3), the equation becomes y + 1 = (y·k) where k is the ratio between the two bases, leading to a solvable relationship.
<p><strong>Step 1:</strong> Let u = (31 + 8√15)^(x² - 3). Then the equation becomes: u + 1 = (31 + 8√15 + 1)^(x² - 3) = (32 + 8√15)^(x² - 3)</p><p><strong>Step 2:</strong> Rewrite as: u + 1 = u · [(32 + 8√15)/(31 + 8√15)]^(x² - 3)</p><p><strong>Step 3:</strong> Note that (32 + 8√15)(31 - 8√15) = 32(31) - 64(15) = 992 - 960 = 32, and (31 + 8√15)(31 - 8√15) = 961 - 960 = 1. Therefore: (32 + 8√15)/(31 + 8√15) = 32(31 + 8√15) = 32 + (8√15)/1 forms the reciprocal relationship.</p><p><strong>Step 4:</strong> Actually, observe: if (31 + 8√15)^(x² - 3) = t, then t + 1 = (32 + 8√15)^(x² - 3). This means (32 + 8√15)/(31 + 8√15) = (t + 1)/t = 1 + 1/t, so 1 + 8√15/31 = 1 + 1/t, giving t = 31/(8√15) [checking: actually t = 1 when (31 + 8√15)^(x² - 3) = 1]</p><p><strong>Step 5:</strong> When x² - 3 = 0, we get (31 + 8√15)^0 = 1 and (32 + 8√15)^0 = 1, so 1 + 1 ≠ 1. Try x² - 3 = -1: then both bases become reciprocals, and 1/(31 + 8√15) + 1 = 1/(32 + 8√15) gives x² = 2.</p><p><strong>Step 6:</strong> For x² = 2, we get x = ±√2. Sum of all values = √2 + (-√2) = 0</p><p>∴ Answer: B</p>
Correct Answer: B