Under which condition can the quadratic trinomial \(x^2+px+q\) be classified as a perfect-square trinomial?

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

For a perfect-square trinomial, \(x^2+px+q\) must be expressible as \((x+r)^2\). On expanding, \((x+r)^2=x^2+2rx+r^2\), so \(p=2r\) and \(q=r^2\). Therefore, \(p^2=(2r)^2=4r^2=4q\). The condition \(p^2=q\) misses the required factor of 4. Exam tip: for \(x^2+px+q\), compare the square of the middle coefficient with \(4q\).