We will first begin by looking at the equation . We can look at the graph below of . And we can also hold one variable constant and see how the graph varies as we change the others.
We can use the graphs to determine the roots of the equation , as two of the variables are held constant. For example in the graph below we will hold a and c constant at a=c=1 and graph several values of b. The purple graph shows b=(-3), the red graph shows b=(-2), the blue graph shows b=(-1), the green graph shows b=0, the light blue graph shows b=1, the yellow graph shows b=2, and the grey graph shows b=3.
We can discuss the "movement" of a parabola as b is changed. The parabola always passes through the same point on the y-axis ( the point (0,1) with this equation). For b < -2 the parabola will intersect the x-axis in two points with positive x values (i.e. the original equation will have two real roots, both positive). For b = -2, the parabola is tangent to the x-axis and so the original equation has one real and positive root at the point of tangency. For -2 < b < 2, the parabola does not intersect the x-axis -- the original equation has no real roots. Similarly for b = 2 the parabola is tangent to the x-axis (one real negative root) and for b > 2, the parabola intersects the x-axis twice to show two negative real roots for each b.
We can also look at the vertices of the above graphs, and notice that the graph of goes through all of the vertices.
We know this is true because is the locus of the equation when a=c=1. We can see by the algebra steps why this is true.
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