If \(a<0\), \(c>0\), and \(b\) is any real number, what will be the nature of the roots of the quadratic equation \(ax^2+bx+c=0\)?

Correct answer: Two real and distinct roots

The discriminant is \(D=b^2-4ac\). Since \(a<0\) and \(c>0\), we have \(ac<0\), so \(-4ac>0\) and \(D=b^2+(-4ac)>0\). Therefore, the equation has two real and distinct roots. Equal roots require \(D=0\), while no real roots require \(D<0\), neither of which is possible here. Exam tip: Whenever \(ac<0\), the roots are automatically real and distinct.