If the expression \(x^2+px+q\) can be written as \((x+r)^2\) for some real number \(r\), which relation between \(p\) and \(q\) is necessary and sufficient?

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

Expanding \((x+r)^2\) gives \(x^2+2rx+r^2\), so \(p=2r\) and \(q=r^2\). Hence \(p^2=4r^2=4q\). The other relations do not ensure a perfect square. Exam tip: square the middle coefficient and compare it with four times the constant term.