Multiple choice

Let p and q are roots of the equation: x2 - kx + 45 = 0. If the difference between the roots is at least 12, what is the range of values k can take?

  1. (- ∞ , 18]

  2. [-18, ∞ ).

  3. (- ∞ , -18] U [18, ∞ ).

  4. (- ∞ , -8) U (8, ∞ ).

  5. None of these
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C Correct answer
Explanation

For x^2 - kx + 45 = 0, the roots p and q satisfy |p - q| = sqrt(D)/a = sqrt(k^2 - 180). Given |p - q| >= 12, then k^2 - 180 >= 144, so k^2 >= 324. This implies k >= 18 or k <= -18.

AI explanation

For the equation x2 - kx + 45 = 0 with roots p and q, the sum of the roots p + q = k and the product of the roots pq = 45. The difference between the roots is given by the formula sqrt((p + q)2 - 4pq), which translates to sqrt(k2 - 180). Since the difference is at least 12, we have the inequality sqrt(k2 - 180) >= 12. Squaring both sides gives k2 - 180 >= 144, so k2 >= 324. Taking the square root of both sides gives the range of values for k as (- infinity, -18] union [18, infinity).

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