Question

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

Answer 1

Once.

Explanation:

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`.

Answer 2

Thank you George and Trevor. This is an interesting debate.
At least it will make the readers of this question think about the importance of words and their interpretation.

Explanation:

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?

Answer 3

It will touch the `x`-axis at exactly one point. Whether that should be called "intersecting" may be a matter of debate.

Explanation:

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.