Learning Resources
Lesson
The 
in the quadratic formula tells a great deal about the roots of a quadratic equation - in fact, it is given a special name. It is called the discriminant. (Usually represented with the letter D)
The purpose of this investigation is to discover the relationships between the discriminant and the roots of a quadratic equation.
If there is more than one student in your school doing this course, it is a good idea to share the work in this investigation and pool your results to discover the pattern.
You should now complete the investigation as outlined. Also, complete the investigation questions (p.55-56). If you have any problems or difficulties, click "b" below for hints and suggestions before seeking extra help from your teacher.
For this investigation it is a good idea to use a table as below to record your results for Steps A, B and C.
General Form |
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Transformational Form |
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Vertex |
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Number of x-intercepts |
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Types of Roots |
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Discriminant |
Step A
Share the work if possible ! Enter your results in the table.
Step B
Use the characteristics from the transformational form to graph each equation or use technology.
Step C
Use the a, b and c from each function to calculate the value of , the discriminant.
Make note that some times the discriminant is less than zero. When this is placed in the quadratic formula to find the roots of the equation, the square root of a negative number occurs which means that the equation has no real roots and does not cross the x- axis. In this case we represent the roots using complex or imaginary numbers. For these types of numbers we represent 
, where i is considered to be an imaginary number. For example if we have 
as a solution then we write as a complex number in the form 
. (Check the discussion in question 55 of the investigation questions)
As a note, mathematicians first referred to these types of numbers as imaginary numbers because they seemed not to exist. However, imaginary numbers do exist and have many applications in the sciences. Despite their name, they are not really imaginary at all. The name dates back to when they were first introduced, before their existence was really understood. At that time, people were imagining what it would be like to have a number system that contained square roots of negative numbers, hence the name "imaginary".
Step D
Repeat the process twice more to determine if patterns are developing. Again share the work and pool your results.
The discriminant tells you a number of facts as you should have discovered from the investigation and its questions. You should have discovered:
- If
, the graph will have two x-intercepts; two real an unequal roots for the equation
- If
, the graph will have one x-intercept ; two real and equal roots for the equation - sometimes called a double root.
- If
, the graph will have no x-intercepts ; two roots are not real but complex
Another interesting result of knowing the roots of a quadratic equation is the ability to find the equation. If you know the roots of a quadratic equation, the equation can be found by working backwards from the roots to the equation. Consider the following example:
Example
Find a quadratic equation with roots 
and 
.
Solution
Since the roots are and , then we know 
. Use the interactive window below to view how to find the equation.
Activity
Investigation Questions p.55 - 56. Complete 51, 52, 53, 54 and 55
Think About p.55
Check Your Understanding p.56 - 57. Complete 56 to 64 inclusive
When you have completed these questions, ask your on-site teacher to get the solutions for you from the Teacher's Resource Binder and check them against your answers. After you do this, if there is something you had trouble with and still do not understand, contact your on-line teacher for help.
After you do the assigned activities, continue on to the Test Yourself section below for a quick quiz on this lesson.
Test Yourself
For what value of k will 
have (a) two equal real roots; (b) two different real roots; (c) no real roots.
Click here for a suggested solution.