If a quadratic equation has 2 identical rational roots, how many times will its graph intersect the #x#-axis?

It will just touch the x-axis once so hence two identical rational x-intercepts, hence just the one x-intercept essentially.
The discriminant in this case will be #0#.

Assumption: This is a quadratic in #x#

If the two roots are #ul("identical")# it has to mean that they have the same value.

If they have the same value then the same value is the same as 1 value

If there is only 1 value then it 'intersects the axis just once!

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However; the question categorically states that there are 2 roots.

So if it has 2 roots the graph must cross the x-axis in two different places. If this is so then how can the roots be identical?

This is probably an area of linguistic debate.

The graph of the parabola does not intersect the #x#-axis in the sense of crossing it. It does intersect the #x#-axis in the sense of meeting it.

It touches the #x#-axis at one point.

Note that this means that there is exactly one point of intersection: The graph of the parabola is a set of points that has a non-empty intersection with the set of points that comprise the #x#-axis. This intersection of sets contains one element.