Nonnegative for every input

7 written marks

Let \(q_k(x)=x^2-2kx+k+6\), where \(k\) is real.

1. Write \(q_k(x)\) in completed-square form.

   [2]

2. Find all \(k\) for which \(q_k(x)\ge0\) for every real \(x\).

   [3]

3. For which values of \(k\) from part (b) is equality \(q_k(x)=0\) possible? Give the corresponding \(x\).

   [2]

Complete-answer checking, worked solutions and written marking guidance.