Multiple choice

What is the greatest integer 'k' for which the roots of the equation x2 - kx + 14 = 0 are imaginary?

  1. 56

  2. 64

  3. 7

  4. 8

  5. 55
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C Correct answer
Explanation

For roots to be imaginary, the discriminant D < 0. D = k^2 - 4(1)(14) = k^2 - 56. k^2 - 56 < 0 => k^2 < 56. The integers k satisfying this are -7 to 7. The greatest integer is 7.

AI explanation

For the quadratic equation x squared - kx + 14 = 0 to have imaginary roots, the discriminant must be strictly less than zero. Using the discriminant formula, we evaluate b squared minus 4ac, which gives k squared minus 4 times 1 times 14, or k squared minus 56. Setting the inequality k squared minus 56 < 0 means k squared must be less than 56, so the value of k must be strictly between the square root of 56 and its negative counterpart. Since the square root of 56 is approximately 7.48, the greatest integer satisfying this condition is 7.

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