Quadratic Equations
by Oktay Mercimek
In this writeup I will examine relationship between roots of Quadratic equation and c as in .
Let's say b=0, then .
If we graph this equation in the xc plane,
Adding parallel lines to xaxis will help to investigate for special c values.
To see what happens when c=1, We can add c=1 line to graph.
red line is c=1
So for b=0 and c=1 , we don't have any root.
Let make c=2 and see what happens.
red line is c=2
The line intersects the curve at two points, one of them is between 2 and 1 , and other is between 1 and 2.
So for c=2 we have two roots, one is negative and other is positive.
Let's say b=4, then the equation becomes .
In this case graph of this equation in the xc plane
Let draw a few lines on this graph.
red line is c=5
if b=4 and c=5 then real equation is .
Since we don't see intersection between line and curve, this equation doesn't have a root.
Now let's try c=5
red line is c=3
c=3 line intersects the curve in two positive points. So we have two positive roots. One of them is around 1 and other is around 3
For c=1
red line is c=1
so for c=1, we have two roots. One is negative and other one is negative.
Let's arrange all information in a table to see relationship between c and roots.
b=0 
c=1 
No root 
c=2 
One negative and one positive root 

b=4 
c=5 
No root 
c=3 
Two positive roots 

c=1 
One negative and one positive root 
As you see in the table,
When we have a positive c then we have no root or two root of same sign.
When we have a negative c then we have a negative and a positive root.
It is also significant in graphs that curves always passing through point (0,0), so it means if c=0 then we always have x=0 as a root.