What condition on \(k\) is necessary for the equation \(x^2+4x+k=0\) to have no real roots?

Correct answer: \(k>4\)

Here, \(a=1\), \(b=4\), and \(c=k\). Therefore, the discriminant is \(D=b^2-4ac=16-4k\). A quadratic equation has no real roots when \(D<0\), so \(16-4k<0\), which gives \(k>4\). The closest distractor, \(k=4\), makes \(D=0\) and gives one repeated real root. Exam tip: Check the sign of the discriminant first to determine the nature of the roots.