Multiple choice

Let A be a real number. Then the roots of the equation x2 - 4x - log2A = 0 are real and distinct if and only if

  1. A < 1/8

  2. A < 1/16

  3. A > 1/16

  4. A > 1/8
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C Correct answer
Explanation

For the quadratic x^2 - 4x - log2(A) = 0 to have real and distinct roots, the discriminant D must be > 0. D = (-4)^2 - 4(1)(-log2(A)) = 16 + 4log2(A). Setting 16 + 4log2(A) > 0 gives log2(A) > -4, so A > 2^-4, which is A > 1/16.

AI explanation

For the roots of a quadratic equation to be real and distinct, the discriminant must be greater than zero. Calculating the discriminant gives 16 + 4log2A > 0, which means 4 > -log2A and log2A > -2. Converting from logarithmic to exponential form gives A > 1/16.

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