We want to look at the familiar quadratic equation

.

For a=1, b=1, c=1, the equation equal to y looks like:

We can graph the equation in the xc plane by replacing the c with y.

We can look at any exact value of c by graphing y=c.

For c=3, we get the following graph.

For c=-3, we get the following:

We can see that the horizontal line graphed may or may not intersect the graph depending on the value of c. The points of intersection between the 2 equations represent the roots of the equation for that value of c. That is, for c=-3 as in the image above, the roots of are the points of intersection above.

We can vary the value of c and see the points of intersection.

How can we figure out what values for c will yield a solution/will intersect the graph?

Using algebra!

We want the equation to be yield a solution. Typically, to solve a quadratic equation, we use the quadratic formula: .

In our case of a=1, b=1, we want to see for what values of c is the quadratic formula solvable.

This equation is solvable when 1-4c is greater than or equal to 0.

The equation has a solution when c is less than or equal to 1/4.

Thus, the graph above will have an intersection when c is less than or equal to 1/4.

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