The x Coefficient of a Quadratic Function

By

Alex Moore

For this investigation we will consider the function f(x)=(x^2)+bx+n where b and n are allowed to vary.  We start with the case n=1.  Consider the equation (x^2)+bx+1=0.  Here is the graph of this equation in the x,b plane, that is, the graph of the relationship between x and b.

What is the significance of this graph?  For each fix values of b we have at most two corresponding values for x.  For example, if b=3 then we have the new graph

and we see the that the intersection of these graphs occur at the corresponding x values.  What are these x values?  These are the two roots of our quadratic function! To see what these x values are explicitly we use the quadratic formula:

What if b=1? Then we obtain the graph shown below in which no values of x correspond to b=1.

Why are there no roots to our function?  This is because our discriminant is (b^2)-4=1-4=-3 and -3 has no real roots.   To find what values of b correspond to roots we calculate

(b^2)-4>=0

b^2>=4

-2<=b<=2.

Therefore, for values of b smaller than -2 and larger than 2 we have 2 real roots and for b=2 and b=-2 we have one real root.  For values of b between 2 and -2 we have no real roots.

What would happen to the number of roots to our function if we vary n? For example, what if n=-1?  Then for any value of b the function will have two real roots!

Looking back at the discriminant, if n>0 then for b>(4n)^(1/2) and b<-(4n)^(1/2) we have two real roots, one real root if equality holds, and no real roots if  -(4n)^(1/2)<b<(4n)^(1/2).  If n<0, however, we have that (b^2)-4n>0 for all b and so there will always be two real roots.  What happens if n=0? We get the following graph:

It appears as though we get only one real root but this is not the case.  If n=0 we have 2 real roots!  If n=0 then we have (x^2)+bx=x(x+b)=0 and so x=0 or x=-b.  If b=0 then we have 0 as a repeated root.  Clearly the number of roots of our function f(x)=(x^2)+bx+n=0 vary with n and b and there relationship is most seen by the discriminant, that is, the quantity (b^2)-4n.  Below is the graph of values for n=2,1,0,-1, and -2.