If m is a real number, when will the roots of \(x^2-2(m-4)x+m^2-16=0\) be real and distinct?

Correct answer: \(m<4\)

Here, a=1, b=−2(m−4), and c=m²−16. Therefore, the discriminant is \(D=b^2-4ac=32(4-m)\). Real and distinct roots require \(D>0\), so \(32(4-m)>0\), which gives \(m<4\). Note that when \(m=4\), \(D=0\) and the roots are equal; hence \(m\le4\) is not the correct condition. Exam tip: determine the sign of the discriminant first when analysing the nature of quadratic roots.