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Find all real values of x satisfying the equation 2^(x+2) * 5^(6-x) = 10^(x²).

1. x = 1
2. x = 2
3. x = -log(250) (base 10)
4. x = log(4) - 3 (base 10)

Correct answer: x = 2

Solution

Taking log base 10 yields a quadratic x² - x*(2*log2-1) - (6-4*log2) = 0. Its discriminant equals (2*log2-5)², giving solutions x = 2 and x = 2*log2 - 3 = log4 - 3 = -log250. Both x=2 and x=-log250 are valid (options B and C/D are the same value).

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