If the roots of the equation \(x^2-2(\alpha+2)x+\alpha^2=0\) are real and distinct, what is the correct condition on \(\alpha\)?

Correct answer: \(\alpha>-1\)

Here, \(a=1\), \(b=-2(\alpha+2)\), and \(c=\alpha^2\). For real and distinct roots, the discriminant must satisfy \(D>0\). Thus, \(D=[-2(\alpha+2)]^2-4(1)(\alpha^2)=16(\alpha+1)\). Hence, \(16(\alpha+1)>0\), giving \(\alpha>-1\). When \(\alpha=-1\), \(D=0\), so the roots are real but equal, not distinct. Exam tip: For real and distinct roots, always use the condition \(D>0\).