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It has now become a rather standard exercise, with availble technology, to construct graphs to consider the equation

and to overlay several graphs of

for different values of a, b, or c as the other two are held constant. From these graphs discussion of the patterns for the roots of

can be followed.

For example, if we set

for b = -3, -2, -1, 0, 1, 2, 3, and overlay the graphs, the following picture is obtained.
 
 

We can discuss the "movement" of a parabola as b is changed. The parabola always passes through the same point on the y-axis ( the point (0,1) with this equation). For b < -2 the parabola will intersect the x-axis in two points with positive x -values (i.e. the original equation will have two real roots, both positive). For b = -2, the parabola is tangent to the x-axis and so the original equation has one real and positive root at the point of tangency. For -2 < b < 2, the parabola does not intersect the x-axis -- the original equation has no real roots. Similarly for b = 2 the parabola is tangent to the x-axis (one
real negative root) and for b > 2, the parabola intersets the x-axis twice to show two negative real roots for each b.

Show that the locus is the parabola, y = -x^2+1 (black curve)
 
 

Now, I try to examine the roots of a quadratic equation through the graph.

First, the possible way is to graph in the xa plane instead of in the xy plane. This means that we will substitute y for a into a quadratic equation and the graph.
Next, instead of graphing in the xy plane, I will graph in the xb plane. This also means I will substitute y for b into a quadratic equation and the graph.
Finally, instead of graphing in the xy plane, I will graph in the xc plane. This  means I will substitute y for c into a quadratic equation and the graph.

Let's graph this equation in the xa plane. Before doing it, I need to graph the equation

 

Let's take any particular value of a, and notice the equation y = a on the graph.
The intersection points between the line parallel to X-axis (y=a) and yx^2+x+1=0 are the roots of the a quadratic equation.

   yx^2+x+1=0(blue)
   a = 2, the equation y = 2 (purple)
   a = 0.25, the equation y = 0.25(red)
   a = -1, the equation y = -1 (green)

For a = 2, there is real root of the equation.
For a = 0.25, the equation has one negative real root.
For a = -1, there are one positive and negative real roots.

Generalize :

 when a < 0 , there is real root of the equation.
 when 0 < a < 0.25 , the equation has one negative real root.
 When a > 0.25 , there are one positive and negative real roots.

Let's graph this equation in the xb plane. Before doing it, I need to graph the equation.

Let's take any particular value of b, and notice the equation y = b on the graph.
The intersection points between the line parallel to X-axis (y=a) and yx^2+x+1=0 are the roots of the a quadratic equation.

   x^2+yx+1=0(blue)
   b = 3 and -3, the equation y = 3 , -3 (purple)
   b = 2 and -2, the equation y = 2 , -2(red)
   b = 1 and -1, the equation y = 1, -1(green)
 
 

For b = -1, 1, there is real root of the equation.
For b = -2, 2  the equation has one positive or one negative real root.
For b = -3, 3  there are two positive roots or two negative real roots.

Generalize :

 when b <-2  , there is two positive real roots.
 when b = -2 , the equation has one negative real root.
 when -2 < b < 2 , the equation has no real roots.
 when b = 2 , there are one negative real root.
 when b > 2 , there are two negative real roots.

Let's graph this equation in the xc plane. Before doing it, I need to graph the equation.

Let's take any particular value of c, and notice the equation y = c on the graph.
The intersection points between the line parallel to X-axis (y=c) and yx^2+x+1=0 are the roots of the a quadratic equation.

   x^2+5x+y = 0(blue)
   c = -2, the equation y = -2 (purple)
   c = 0 , the equation y = 0 (x-axis)
   c = 2,   the equation y = 2(red)
   c = 6.25, the equation y = 6.25(green)
 
 

For c = -2, there are one negative root and one positive root.
For c =  0, the equation 0 and one negative real root.
For c =  2, the equation has two negative real roots.
For c = 6.25,  there is no real root.

Generalize :

 when c < 0 , there one negative root and one positive root.
 when c = 0 , tthe equation 0  and one negative real root.
 when 0 < c < 6.25 ,the equation has two negative real roots.
 when c > 6.25, there is no real root.