Assume that \(a\) and \(b\) are real numbers. Which statement about the roots of the equation \(x^2+2(a-b)x+(a+b)^2=0\) is correct?

Correct answer: Real roots exist if and only if \(ab\le 0\)

For the quadratic equation, the discriminant is \(D=[2(a-b)]^2-4(a+b)^2\). Hence, \(D=-16ab\). Real roots require \(D\ge 0\), so \(-16ab\ge 0\), which gives \(ab\le 0\). Option D is also incorrect because equal roots require \(D=0\), giving \(ab=0\), not generally \(a=b\). Exam tip: To determine whether quadratic roots are real, first check the sign of the discriminant.