In this exploration, we want to look at quadratic equations. We know that these kinds of equations can be solved algebraically or graphically. First we want to start out with the graph
Looking at this graph we can see that it has no real solutions when b is in the open interval (-2,2). Look HERE at the graph. However, when b is greater than or equal to 2, there is at least one negative real root and when b is less than or equal -2, there is at least one positive real root. If we look at different values of c we get the following graphs.
Here we can see that changing the value of c creates hyperbolas. Clearly, we see that the graphs will continously get closer to the y-axis but never cross that axis. From this exploration we can also see that the original equation does not intersect the x-axis therefore, the roots of the quadratic are complex numbers.