Multiple choice

The set of real values of $a$ for which the equation $x^2=a(x+a)$ has its root greater than $a$ is

  1. $(-2,-\dfrac 12)$
  2. $(-\dfrac 12,-\dfrac 14)$
  3. $(-\infty,0)$

  4. None of the above.
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D Correct answer
AI explanation

The set of real values of a for which the equation x^2 = a(x+a) has its root greater than a cannot be restricted to the given intervals (-2, -1/2), (-1/2, -1/4), or (-infinity, 0). Evaluating the equation at a = 1 gives x^2 - x - 1 = 0, which has roots (1 + sqrt(5))/2 and (1 - sqrt(5))/2. The root (1 + sqrt(5))/2 is approximately 1.618, which is greater than a = 1. Therefore, the correct answer is None of the above.

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