Which of the following is a necessary and sufficient condition for the expression \(x^2+bx+c\), with real coefficients, to be a perfect square of a binomial?

Correct answer: \(b^2=4c\)

Let \(x^2+bx+c=(x+a)^2\). Since \((x+a)^2=x^2+2ax+a^2\), we get \(b=2a\) and \(c=a^2\). Hence \(b^2=4a^2=4c\). In exams, square the middle coefficient and compare it with \(4c\).