Lesson 1

Determining the Roots of a Quadratic Equation

 

In Module 3 you learned about quadratic functions. You learned that a quadratic function is a second-degree expression that can be written in the form y = ax2 + bx + c. A quadratic equation has the same form as a quadratic function and can be expressed as ax2 + bx + c = 0, where a ≠ 0 and a, b, and c are real numbers.

 

Why is it a condition of quadratic functions and equations that a ≠ 0?

 

To solve a quadratic equation means to determine the value(s) of the unknown variable that would make the equation true. The solution(s) of a quadratic equation are known as the roots of the equation.

 

You can determine the roots of a quadratic equation by analyzing the graph of the corresponding quadratic function. See how you can do this by completing Try This 2.

 

Try This 2

1. Consider the quadratic equation x2 − 6x = 0. Use any strategy to find two different values of x that would satisfy the equation. Verify your solutions by substituting the values into the equation and checking that the left-hand side equals 0.
   
   
2. The graph shows the graph of the quadratic function y = x2 − 6x. Study the properties of the graph. See if you can spot the solutions you identified in question 1 on the graph.
   
   
   

Save your responses in your course folder.

 

Share 2

 

With a partner, respond to the following questions:

1. What is the relationship between the roots of the quadratic equation and the x-intercepts of the corresponding quadratic function?
   
   
2. Check your hypothesis by graphing and solving the quadratic equation x2 + 2x − 15 = 0.
   
   
3. The x-intercepts of a graph are also known as the zeros of the graph. Give a possible explanation as to why x-intercepts of a graph are called zeros.

Save your responses in your course folder.

 

Self-Check 1

 

Work through Distinguishing Between Roots, x-intercepts, and Zeros. You will need to select Solving Quadratic Equations: Using Graphs and then the blue Tutorial button at the top of the window. Finally, select Distinguishing Between Roots, x-intercepts, and Zeros. After working through the applet, make any necessary revisions to your responses to Try This 2 and Share 2.