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Investigating the quadratic equation

on the planes xb and xc

 


 

Consider the equation 

 

 

Here, we have two parabolas that are symmetric, with the value of c = 1

Usually when finding the roots for a quadratic equation, we do algebraically, by solving for x.  The quadratic equation is one of the things the Babylonians contributed to the science of mathematics.  Back then, when the roots of an equation were negative, it implied that the roots did exist.  Well, with time the science has developed so enormously that we can even apply negative roots to our daily mathematics today.

When solving for x through the quadratic formula, we are doing the following:  graph b, as the y-intercept and move it on the graph for the quadratic equation.

Here the value of b = 1.  As you can see, the horizontal is not touching the function at all.  Hence, we can conclude that the equation has no solution at b = 1.  As a matter of fact, there are no solutions for

2 > b > - 2.  See this graph

So, to find the existed roots, we change the value of b to 3, and we get

The equation now has two roots.  The horizontal line intersects the function at two points, and these two points are the roots of the equation at b = 3.  If the value of b were to be negative, the roots would be negative as well.

 

Now, observe what happens to graph as the value of the constant c alters.

This is the original graph and with the graph where c = 0.  What you is that we have a straight line.  What happens here are that the two parabolas shifted that they lie along the same points.  Observe what happens when c = -1.

Now, you see that the parabolas are facing toward the x-axis instead of the y-axis.  That is just observation.  If observe a few more graphs, you will see that the parabolas are getting further and further away from each other as c approaches negative infinity (-) and infinity ().  However, the right observation here is to notice that we have many roots solutions as c approaches negative infinity, and the gap of which we have no solution accrues as c approaches positive infinity.

 

Examples:

 

 

If familiar with translation, this is exactly what is happening here.  Since this a quadratic equation, you can observe that we would have no roots solution for b = 1 when c is a positive integer, similarly when c is a negative integer we have rather a greater bracket of solution.

This is what is happening as the value of c alters.  See graph

 


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