If $\alpha $ and $\beta$ are the roots of the equation $(\log_{2}{x})^{2}+4(\log_{2}{x})-1=0$ then the value of $\log_{\beta}{\alpha}+\log_{\alpha}{\beta}$ equal to

This problem leads to a contradiction because the premise is mathematically impossible. For the equation y^2 + 4y - 1 = 0 to have real solutions for the roots alpha and beta, its discriminant must be non-negative. Calculating the discriminant gives 16 - 4(1)(-1) = 20, which yields positive roots y = -2 + sqrt(5) and y = -2 - sqrt(5). Because alpha and beta equal 2 raised to these powers, one root is positive and the other is negative, making it impossible to take real logarithms of both. Therefore, the expression cannot be evaluated as stated, but if we ignore the domain error and compute log(beta, alpha) + log(alpha, beta) using the sum and product of the roots, the result is 18.

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