Question

How do you determine whether there are two, one or no real solutions given the graph of a quadratics function does not have an x-intercept?

Answer 1

If a graph of a quadratic, `f(x)`, does not have an x-intercept
then `f(x)=0` has no Real solutions.

Explanation:

The x-axis is composed of all points for which `f(x)` (or, if you prefer, `y`) is equal to `0`

If the graph of `f(x)` does not have an x-intercept
then it has no (Real) points for which `f(x)=0`

Answer 2

See explanation

Explanation:

No x-intercept means that it does not cross the x-axis. Thus two solutions is definitely ruled out.

However, if you DO NOT INCLUDE a point of coincidence (Vertex coincides with the x-axis) in the phrase "does not have an x-intercept". Then there could be a single value solution if you equate the quadratic to 0. Some people say that it still has two in such a case but they are both the same value. I do not like this way of thinking!

On the other hand, if you DO INCLUDE a point of coincidence in the phrase, then the plot does not cross the x-axis nor does any point on the curve coincide with it. In such an interpretation there is NO SOLUTION THAT IS REAL.