What condition between \(p\) and \(q\) is necessary and sufficient for the polynomial \(x^2+px+q\) to be expressible as the square of a binomial with real coefficients?

Correct answer: \(p^2=4q\)

Let \(x^2+px+q=(x+a)^2\). Comparing coefficients gives \(p=2a\) and \(q=a^2\), hence \(p^2=(2a)^2=4q\). The condition \(p^2=q\) does not match the middle-term coefficient. Exam tip: square the coefficient of the middle term and compare it with four times the constant term.