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log10[c + sqrt(quadratic in x)] = 3: reduce to a quadratic and use its roots

This type of question, asked 2 times: CDS 2 2022, CDS 2 2025. All 2 questions below, oldest first.

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  1. Question 1

    CDS 2 2022

    24
    If log⁡10[995+x2−12x+60]=3\log_{10}\left[995 + \sqrt{x^2 - 12x + 60}\right] = 3, then what is the sum of the roots of the equation ?

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  2. Question 2

    CDS 2 2025

    37
    If α\alpha and β\beta are the roots of the equation log⁡10[998+x2−18x+76]=3\log_{10}\left[998 + \sqrt{x^2 - 18x + 76}\right] = 3 then what is (α−β)2(\alpha - \beta)^2 equal to?

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