If \(a,b,c\) are real numbers and \(a\ne0\), which condition is necessary and sufficient for \(ax^2+bx+c\) to be the square of a linear expression?

Correct answer: \(a>0,\ b^2=4ac\)

A perfect-square trinomial has zero discriminant, so \(b^2=4ac\). Substituting \(c=\frac{b^2}{4a}\) gives \(a\left(x+\frac{b}{2a}\right)^2\). For this to be a square of a real linear expression, \(a\) must be positive. Exam tip: check the sign of \(a\) along with the discriminant.