What condition on \(c\) is necessary for the quadratic equation \(x^2-2(c+1)x+c^2-2c+5=0\) to have real roots?

Correct answer: \(c\ge 1\)

For real roots, the discriminant must satisfy \(D\ge0\). Here \(a=1\), \(b=-2(c+1)\), and the constant term is \(c^2-2c+5\), so \(D=b^2-4ac=16(c-1)\). Thus \(16(c-1)\ge0\), giving \(c\ge1\). At \(c=1\), the equation has equal real roots, so the stricter condition \(c>1\) is incorrect. Exam tip: For a quadratic to have real roots, begin by imposing \(D\ge0\).