For equations in the following form:

Here are graphs with varying values of b (-5, 5), with a = 1 and c = 1

All the graphs pass through the point (0,1) and the locus of the vertices was found using (0,1) as the vertex with the equation:

Notice that for the following equations and their corresponding graphs:

When b = 2 and b = -2, there is one negative and one positive real root, respectively. When -2 < b < 2, the equation has no real roots.

Now let’s examine the equations in the xb plane:

Graphing

Gives us the graph below:

Notice the gap between b = -2 and b = 2, this corresponds to the previous statement of when -2 < b < 2, there are no real roots.

Let’s use b = 3 in the graph below:

We see that the horizontal line intersects the graph at two points that correspond to the roots of the original equation in the xy plane; in particular and its two negative real roots.

Now let’s take a look at other values for c in the following equations and their corresponding graphs:

For varying values of b and c = -1

We can see from the graph that each equation has two real roots which correspond to the b values (-2,2) that intersect the graph at two points below:

We can use different values for c:

And the corresponding roots to the original equations are found where the y=b line cross the graph.