Which condition on \(k\) ensures that the equation \(x^2-2(k-2)x+k^2=0\) has no real roots?

Correct answer: \(k>1\)

For the given quadratic equation, \(a=1\), \(b=-2(k-2)\), and \(c=k^2\). Therefore, the discriminant is \(D=b^2-4ac=4(k-2)^2-4k^2=16(1-k)\). No real roots require \(D<0\), so \(16(1-k)<0\), which gives \(k>1\). At \(k=1\), however, \(D=0\), giving two equal real roots. Exam tip: for a quadratic equation, \(D<0\) indicates that the roots are non-real.