Quadratics
A Discriminating Example

Quadratic Equations

A quadratic equation is a second-order polynomial equation in a single variable “x” with a not equal to 0, because it is a second order polynomial, the fundamental theorem of algebra guarantees that is has two solutions. The solutions may be both real, or both complex. In case you have forgotten I have placed the Fundamental Theorem of Algebra below

Before we start dealing with the parts of the quadratic it will become necessary for us to get the given equation in the standard quadratic form.

or sometimes written

The solutions can be found by completing the square, which is known as the quadratic formula.

When we deal with the quadratic form there are several key parts to look at. These parts can be used to determine the shape of a graph, the number of times the graph crosses the x axis…which in turn can tell us the number of solutions to the equation. The parts can also tell us the maximum and minimum values of the quadratic function. This lesson will deal with some of the parts of the quadratic. 

The Discriminant

When an equation is in the standard quadratic form we can identify parts of the equation. The first “part” we will identify is called the “Discriminant”. The discriminant is a value which characterizes certain properties of a quantity’s roots. Using the standard quadratic form the discriminant is defined by

If we would examine several different quadratics and their graphs we would eventually determine that:

• If the discriminant is 0 the curve will cross the x-axis only once. The solution will be one real root
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• If the discriminant is >0 the curve will cross the x-axis twice. The solution will be two real roots
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• If the discriminant is < 0 the curve will not cross the x-axis. The solution will be no real roots
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Discriminant Examples

As an example we have this table below. This table shows an equation and the value of the discriminant. Then shows the number of real solutions based on the discriminant value.

In the examples below find the discriminant and then tell the number of  solutions for the function equaling Zero. Roll over the Bold Text to see the solution.

In the question below match the equation with the graph that comes closest to the graph of the equation. Use the discriminant