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How do you determine whether there are two, one or no real solutions given the graph of a quadratics function is tangent at the x-axis?

Answer 1

There's one real solution.

A quadratic function is a parabola, which consists of a single curve with either a maximum or a minimum ( a 'u' shape or an 'n' shape). The location of the tangent indicates the location of this maximum or minimum.

If the tangent, and therefore the max/min, is on the x-axis this means that it touches the x-axis at this point. But since there is only one max/min point, this means that the graph intersects with the x-axis at only this one point - meaning it only has one root.

When a graph only intersects the x-axis at one point, and therefore only has one root, this indicates that it has one real solution. Basically, the number of intersections it has with the x-axis = the number of roots = the number of solutions

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.