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how do you know what graph has a discriminant of zero?

Mathematics
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If it is tangent to the graph.
if all you have is a graph; the graph will only touch the x axis in one spot then curve back
If it is tangent, then that means it has a discriminant of zero. By the way, if it intersects at two points, that means it has a discriminant larger than 0, and if it does not intersect, that mans it has a discriminant smaller than 0.

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