What is the nature of the roots of the equation \(x^2-2(3+\sqrt{5})x+(14+6\sqrt{5})=0\)?

Correct answer: Two real and equal

Here, \(a=1\), \(b=-2(3+\sqrt{5})\), and \(c=14+6\sqrt{5}\). Therefore, the discriminant is \(D=b^2-4ac=4(3+\sqrt{5})^2-4(14+6\sqrt{5})=0\), since \((3+\sqrt{5})^2=14+6\sqrt{5}\). Hence, the roots are real and equal. In fact, the repeated root is \(3+\sqrt{5}\), which is irrational; therefore, option D is incorrect because it says the roots are distinct. Exam tip: For a quadratic equation, \(D=0\) always indicates two real and equal roots.