**Quadratic Equations**

**by**

**Hieu Huy Nguyen**

Our goal is to investigate the behavior of graphs on the xb Plane.

The equation we are considering is: x^2 + bx + 1 = 0 This equation was graphed on the xb Plane and displayed below:

The graph indicates that there are no real roots when -2 < b < 2, exact one root when b = 2, -2, and two roots when -2 > b > 2. This can be better visualized if we just take any value of b and create a line parallel to the x-axis.

For -2 < b < 2, the graph below illustrations a given line on b-axis that has no real roots

For b = 2, -2, the graph below illustrations a given line on b-axis that has one real root

For -2 > b < 2, the graph below illustrations a given line on b-axis that has two real roots

We would be able to better understand all points where the function crosses the line of b twice, once, or none. Mathematically, this means that there are certain values of b in the quadratic equation in which there are no real solutions for x.

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