Which condition on \(k\) is necessary for the equation \(x^2-12x+k=0\) to have two real and distinct roots?

Correct answer: \(k<36\)

For a quadratic equation \(ax^2+bx+c=0\) to have two real and distinct roots, its discriminant must satisfy \(D=b^2-4ac>0\). Here, \(a=1\), \(b=-12\), and \(c=k\), so \(D=(-12)^2-4(1)(k)=144-4k=4(36-k)\). Therefore, \(D>0\) gives \(k<36\), making option A correct. At \(k=36\), the roots are real but equal, not distinct. Exam tip: use \(D>0\) specifically for two distinct real roots.