If the roots of the equation \(x^2+2kx+k^2-4k+8=0\) are real, what is the correct condition on \(k\)?

Correct answer: \(k\ge 2\)

For a quadratic equation to have real roots, its discriminant must satisfy \(D\ge0\). Here, \(a=1\), \(b=2k\), and \(c=k^2-4k+8\). Thus, \(D=b^2-4ac=4k^2-4(k^2-4k+8)=16(k-2)\). Therefore, \(16(k-2)\ge0\), giving \(k\ge2\). At \(k=2\), the roots are equal, so equality must be included. Exam tip: use \(D\ge0\) for real roots, whereas distinct real roots require \(D>0\).