Joshua DuMont

Consider the equation x2+bx+c=0. Let’s look at this in the xb plane as c changes:

The curves seem to be hyperbolas. If we look at a horizontal line b=k, the number of times it intersects the curve gives the number of roots of y= x2+bx+c with the x values at the intersection equal to the roots of y= x2+bx+c.

Solving our equation for b yields: .
We can see that this has a horizontal asymptote at x=0. We can also see the other asymptote by looking at the boundary case when c=0; we get the line b=-x. When c is negative we get hyperbolas with a different orientation.

Analytically, we could look at the number of roots by starting to solve:

ax2+bx+c=0

ax2+bx=-c

x2+x=

x2+x+=

==

To solve for x we take the square root of the right hand side, . If   will not be real and we will get no real roots. If =0 then  returns only 0, and we get one real root. If  returns two real values and we get two real roots.

The dividing line between zero and two real roots happens when:

=0

=

b=

Notice that this is imaginary if a and c are oppositely signed. In that case,  must be a positive number and y=ax2+bx+c must have two real roots. Graphically this case is when the hyperbolas in the xb plane are oriented as below. Notice that any horizontal line must intersect the hyperbola twice.

On a graph in the xb plane, b=  represent the horizontal lines that will intersect a hyperbola only once. For example in the equation 4x2+bx+9=0, b==12. If b is greater than 12 we see two intersections, if b is between -12 and 12 we see no intersections and if b is below -12 we see two intersections.