What is the relationship between the “solution” to a quadratic equation and the graph of a quadratic equation? how many possible real solutions might there be when you solve a quadratic equation and how do these possible solutions affect the position the parabola on a coordinate plane?

The solutions of a quadratic equations are the x value of the x-intercepts of the graph. 

If there are 0 real solutions, the parabola will not touch the x-axis.

If there is 1 real solution, the parabola's vertex will be on the x-axis.

If there are 2 real solutions, the parabola will intersect the x-axis in two distinct positions.

Hope this helps!