By Jaehong Shin
Graphs in the xb plane
Consider again the equation
We had better substitute y for b because graphing
calculator can recognize only x,y as variables not b. So y will
be used from now on for equation ,
If we take any particular value of y, say y = 3,2,1,-1,-2,-3 and overlay this equation on the graph, we can get several lines parallel to the x-axis as follows.
As we can see above, each value of y we select, we get a horizontal line. It is clear on a single graph that we get two negative real roots of the original equation when y > 2, one negative real root when y = 2, no real roots for -2 < y < 2, one positive real root when y = -2, and two positive real roots when y < -2.
Now, consider the case when c = - 1
As you see, we can get one negative root and one positive root whatever the value y is.
Further, we can see that we can get two real roots or one real root or no real root depending on the value of a such that y=a when c>0, one real root when c = 0, two real roots c < 0 with y=a regardless of the value of a.
