What is the correct condition on \\(k\\) for the roots of the equation \\(x^2-2(k+1)x+k^2=0\\) to be real and distinct?

Correct answer: \\(k>-\\frac{1}{2}\\)

For the quadratic equation, \\(a=1\\), \\(b=-2(k+1)\\), and \\(c=k^2\\). Therefore, the discriminant is \\(D=b^2-4ac=4(k+1)^2-4k^2=4(2k+1)\\). Distinct real roots require \\(D>0\\), so \\(2k+1>0\\), which gives \\(k>-\\frac{1}{2}\\). At \\(k=-\\frac{1}{2}\\), \\(D=0\\), so the roots are equal rather than distinct. Exam tip: determine the sign of the discriminant before classifying the roots.