Quadratic and Cubic Equations

by

Hyeshin Choi

Investigation 2.

Graphs in the xb plane with x^2+bx+1=0

Here is the graph of x^2+bx+c=0.

I would consider xy plane instead of xb-plane. Then I will set y=b since we work on finding the solution of x^2+bx+1=0.If I set them as a system of equation of these two, then the intersection of these two equation would be the solutions. For each value of b we select, we get a horizontal line. First I will select b=2 and -2, then the intersect of these two equations is only one point.

And the intersections are two negative real roots of the original equation when b > 2, and two positive real roots when b < -2.

And I see that there is no real roots for -2 < b < 2.

Now I want to try different values for y which is my b. First I want to try y=5, 2, and 1to show how many solutions of the equation in each interval.

There are two solutions when y is bigger than 2, it will be the same when y is less than -2.

There is one solution when y=2 or y=-2.

When y is between 2 and -2 there is no solution.

Now I try c=-1 instead of 1. Then this quadratic draw a graph hyperbola. So when I have -1 for c, every place has two different solutions.