Given a question like

Find the coordinates of the minimum point on the curve $y=3x^2+2x+9$.

students are often taught to solve this by completing the square.

The class I am currently teaching this to find completing the square difficult, and I think they would find it much easier to use the quadratic formula to find the roots and then average them to find the $x$ coordinate of the minimum point.

Why is finding minima/maxima of quadratic functions not usually taught like this? What are the advantages of solving these problems by completing the square?