Multiple choice

Let k be an integer such that the equation 2x2 + (k - 4)x + 14 = 0 has no real roots, and the equation x2 + kx + 4 = 0 has two distinct real roots for x. If kmin and kmax denote the least and greatest integer values of k satisfying both the equatons, find the value of kmax - kmin.

  1. 9

  2. 12

  3. 14

  4. 20

  5. None of these
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D Correct answer
Explanation

For the first equation to have no real roots, (k - 4)^2 - 112 < 0, giving integer values from -6 through 14. The second equation has two distinct real roots when k^2 - 16 > 0, so the valid integers are -6, -5 and 5 through 14. Hence kmax - kmin = 14 - (-6) = 20.

AI explanation

For the equation 2x^2 + (k - 4)x + 14 = 0 to have no real roots, the discriminant must be negative, so (k - 4)^2 - 4(2)(14) < 0, which simplifies to (k - 4)^2 < 112 and means -10 < k < 18. For the equation x^2 + kx + 4 = 0 to have two distinct real roots, the discriminant must be positive, so k^2 - 16 > 0, meaning k > 4 or k < -4. The integer values satisfying both conditions are the union of (-10, -4) and (4, 18), giving a range from -9 to 17. The difference between the greatest and least integer values is 17 - (-9) = 20.

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