# 1965 AHSME Problems/Problem 9

## Problem 9

The vertex of the parabola will be a point on the -axis if the value of is:

## Solution

Notice that if the vertex of a parabola is on the x-axis, then the x-coordinate of the vertex must be a solution to the quadratic. Since the quadratic is strictly increasing on either side of the vertex, the solution must have double multiplicity, or the quadratic is a perfect square trinomial. This means that for the vertex of to be on the x-axis, the trinomial must be a perfect square, and have discriminant of zero. So,

Therefore , and our answer is