What is the nature of the roots of the equation \(3x^2-2\sqrt{21}x+7=0\)?

Correct answer: Two real and equal

For a quadratic equation \(ax^2+bx+c=0\), the discriminant is \(D=b^2-4ac\). Here, \(a=3\), \(b=-2\sqrt{21}\), and \(c=7\), so \(D=(-2\sqrt{21})^2-4(3)(7)=84-84=0\). When \(D=0\), the roots are real and equal. In fact, the repeated root is \(x=\frac{-b}{2a}=\frac{\sqrt{21}}{3}\). Therefore, option A is correct; option D is wrong because it describes the roots as distinct. Exam tip: Remember that \(D=0\) indicates two equal real roots.