If \alpha and \beta are the roots of the equation \log_{10}[998 + \sqrt{x^2 - 18x + 76}] = 3, then what is (\alpha - \beta)^2 equal to?

Explanation

From the definition of logarithms, 998 + \sqrt{x^2 - 18x + 76} = 10^3 = 1000. Thus, \sqrt{x^2 - 18x + 76} = 2. Squaring yields x^2 - 18x + 76 = 4, or x^2 - 18x + 72 = 0. The roots are 12 and 6. (\alpha - \beta)^2 = (12 - 6)^2 = 36.

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