Where this has one root



amounts to the system of equations with y = left side
and y = right side, or



having ony one point of intersection.

We draw the graph of  and various
horizontal lines that have the equations  for
various values of k:

 

We see that the horizontal lines above the 
relative maximum point intersect the graph in only
one point, as do points below the relative minimum
point.  However the horizontal lines between the relative
maximum and relative minimum points intersect the graph
3 times, and the horizontal line that pass through those
relative extrema intersect the graph twice.

Therefore so that a horizontal y = k crosses the graph
only once, we find the relative extrema:



We find the derivative using the product rule:




  


Set that = 0



Factor out (x-1)







, 
,   
             

So the x-coordinates of the relative extrema are
x = 1 and x=-1

We substitute those into the equation of the graph
to find the y-values of the two relative extrema:

, substituting x=-1





So the relative maximum point is (-1,4)

, substituting x=1





So the relative minimum point is (1,0)

So we must have k > 4 or k < 0 in order
for  to have exactly one root.

Edwin