If \(x^2-(2a+5)x+(a+2)(a+3)=0\), where \(a\) is a real constant, what will be the nature of its roots?

Correct answer: Two real and distinct roots for every real \(a\)

Here, \(A=1\), \(B=-(2a+5)\), and \(C=(a+2)(a+3)\). Therefore, the discriminant is \(D=B^2-4AC=(2a+5)^2-4(a+2)(a+3)=1\). Since \(D=1>0\) for every real \(a\), the roots are real and distinct. In fact, the roots are \(a+2\) and \(a+3\), whose difference is always 1. They are rational only when \(a\) is rational. Exam tip: \(D>0\) guarantees real and distinct roots; rationality also requires checking whether the coefficients are rational.