Analytic GeometryName______
Notes Discriminant
An Investigation of the Nature of the Root of Quadratic Equations
The standard form for a quadratic equation is where a, b, and c are real numbers and . The discriminant is a value which enables us to describe the nature of the roots of a quadratic equation. The formula for the discriminant is: where a, b and c are the real numbers in the quadratic equation.
Graph each of the equations below on your graphing calculator. Notice how many times the graph crosses the x-axis. (Recall: the point where the graph crosses the x-axis is the x-intercept) Next, determine the discriminant by computing. Complete the chart below.
Quadratic Equation / # times crosses x-axis / # of x-intercepts / DiscriminantUsing the above information can you see a relationship between the discriminant and the number of x-intercepts?
Every time the discriminant > 0, there is/are ______x-intercepts.
Every time the discriminant = 0, there is/are ______x-intercepts.
Every time the discriminant < 0, there is/are ______x-intercepts.
Not only does the discriminant reveal the number of x-intercepts, but in so doing that, it also names the number and type of roots for the quadratic equation.
Every time the discriminant > 0, there are 2 x-intercepts and ______real roots. These real roots are either rational or irrational.
To determine if the real roots are rational: the discriminant is a ______.
To determine if the real roots are irrational: the discriminant is positive but it is not a ______.
Every time the discriminant = 0, there is 1 x-intercept and ______real root(s). It is a ______root.
Every time the discriminant < 0, there are 0 x-intercepts and ______real roots, instead there are ______imaginary conjugate roots.
Let’s put it all together:
Discriminant / # of x-intercepts / Nature of the rootsD > 0 and a perfect square
D > 0 not a perfect square
D = 0
D < 0
Complete the chart:
# / Quadratic Equation / Discriminant / # of x-intercepts / Nature of the roots1 /
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