What condition on \(a\) is necessary for the equation \(3x^2-2(2a+1)x+(a^2+a+1)=0\) to have real roots?

Correct answer: \(a^2+a-2\ge 0\)

For a quadratic equation \(Ax^2+Bx+C=0\) to have real roots, its discriminant must satisfy \(D=B^2-4AC\ge0\). Here, \(A=3\), \(B=-2(2a+1)\), and \(C=a^2+a+1\), so \(D=4(2a+1)^2-12(a^2+a+1)=4(a^2+a-2)\). Therefore, the required condition is \(a^2+a-2\ge0\). Exam tip: apply the discriminant condition first and solve the resulting inequality only if the question asks for the range of \(a\).