Quadratic Equations:

A Different Look

By: Kimberly Young

We often look at the graph of quadratics. We are very interested in seeing how a,b, and c affect the graph and the solutions. The solutions, or roots, are the emphasis of this investigation.

Consider the equation . How do we expect b to affect the roots of this equation? Let's consider this graph on the xb-plane.


We can consider a specific b-value and determine from the above graph if it will have a solution and/or what the specific value(s) of the solution will be. Consider b=4.

Notice that the line b=4 creates a parallel line to the x-axis and the intersections point of the line and the curve correspond to the solutions of the original equation when b=4. Also, from looking at the graph on the xb-plane, we see that the original equation has no solution when -2<b<2.

What happens to the graph on the xb-plane as the value of c changes?


Consider the case when c=-1 rather than +1. This yields the following graph. The blue graph is when c=+1 and the red graph is when c=-1.

When c=-1, we can see that when b is a particular value, the original equation will have two real roots, one positive and one negative root.

This brings up the question, for what values of c will I always have two solutions?

Consider the following animation. The red line is a graph of the c-values when -10<c<10.

From this animation we can see that different c-values affect the number of solutions for all b. If c<0, there will always be two real roots, one positive and one negative. When c=0, there will be only one real root. When c>0, the original equation will either have two real roots, one, or none.


We know from our studies of quadratic equations the discriminante of the quadratic formula determines the number of roots a quadratic equation has. Now the following equation corresponds following discriminant . When the discriminant is negative, there are no real roots. So when there will be no real solution. We can conclude:

-When , the equation has two negative real roots.

-When, the equation has one negative real root.

-When , the equation has no real roots.

-When , the equation has one positive real root.

-When , the equation has two positive real roots.

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