If \(a\) is a real number, which statement correctly describes the nature of the roots of the equation \(3x^2-6ax+(3a^2+2)=0\)?

Correct answer: There are no real roots

Here, \(A=3\), \(B=-6a\), and \(C=3a^2+2\). Therefore, the discriminant is \(D=B^2-4AC=(-6a)^2-4(3)(3a^2+2)=36a^2-36a^2-24=-24\). Since \(D<0\), the equation has no real roots for every real value of \(a\). Option D is incorrect because the discriminant remains fixed at \(-24\), independent of \(a\). Exam tip: For a quadratic equation, \(D<0\) indicates that the roots are not real.