If \(a\ne b\) and \(a,b\) are real numbers, what is the nature of the roots of the equation \(x^2-(a+b)x+ab=0\)?

Correct answer: Real and distinct

Here, \(A=1\), \(B=-(a+b)\), and \(C=ab\). Therefore, the discriminant is \(\Delta=B^2-4AC=(a+b)^2-4ab=(a-b)^2\). Since \(a\ne b\), \(\Delta>0\), so the roots are real and distinct. In fact, the equation factors as \((x-a)(x-b)=0\), giving the roots \(a\) and \(b\). Option B would apply only when \(a=b\), and the roots need not be irrational. Exam tip: For a quadratic equation, \(\Delta>0\) indicates two real and distinct roots.