This write up explores graphs of quadratic equation in different planes.
For different values of b with a =c=1, graphs look as below.
If we take any particular value of b, say b = 3, and overlay this equation on the graph we add a line parallel to the x-axis. If it intersects the curve in the xb plane the intersection points correspond to the roots of the original equation for that value of b. We have the following graph.
For each value of b 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 b > 2, one negative real root when b = 2, no real roots for -2 < b < 2, One positive real root when b = -2, and two positive real roots when b < -2.
Moving along, if c = -1, -2 and so on Š30 and for c approaching 0, the graph looks as follows:
For c = 0, the graph would be that of 2x+b = 0 which is a line as seen below.
The graph of 0 = x2+x+c in the xc plane and 0 = ax2+x+1 in xa plane are as below:
And now lets looks at the cubic equations. The general form is y = ax3+bx2+cx+d. If we plot graphs for different values of b with a =c=1, graphs look as below.
LetÕs consider the equation 0 = x3+bx2+x+1.
In the xb plane the graph look like the following.