If the equation \(x^2-2hx+h^2+8h=0\) has real and distinct roots, which condition on \(h\) is necessary?

Correct answer: \(h<0\)

A quadratic equation has real and distinct roots only when its discriminant satisfies \(D>0\). Here, \(a=1\), \(b=-2h\), and \(c=h^2+8h\), so \(D=b^2-4ac=4h^2-4(h^2+8h)=-32h\). Therefore, \(-32h>0\), which gives \(h<0\). Note that \(h=0\) makes \(D=0\), producing equal roots, while \(h>0\) gives non-real roots. Exam tip: For real and distinct roots, always impose \(D>0\).