IT Menu  | GSP | LiveMath

QUADRATIC EQUATIONS IT LESSON 1 - NATURE OF ROOTS
(SEC 3 ADDITIONAL MATHS)

Software: LiveMath (previously known as MathView)
Thinking Skills: Induction and Deduction
At the end of the lesson, the students should be able to:
(1) identify the three types of roots for a quadratic equation,
(2) explain how the discriminant determines the nature of these roots,
(3) explain how the nature of these roots affects the number of points of intersection of the quadratic curve and the x-axis.


A.  INVESTIGATION

If you cannot view the LiveMath document below, click here to troubleshoot.
 
No support for LM Objects 
1. The document on the left (IT3AMEqRoots.the) shows the quadratic equation ax2+bx+c=0 and the graph of the corresponding curve. Change the values of a, b and c for the following quadratic equations and record the discriminants D=b²–4ac and the values of the roots in the table in the Worksheet (see below).
 
2. Determine the types of roots and record your answers in the same table.
 
3. Make a sketch of the graphs of the corresponding quadratic expressions including the x-axis in the same table.
   
No.
Quadratic 
Equation
Discriminant D=b²–4ac
Values of Roots
Types of Roots
1.
x²–4x–6=0
40
5.1623, 
–1.1623
See worksheet
2.

9x6x+1=0

 
 
 
3.
2x²+x+5=0
 
 
 
4.

187xx=0

 
 
 
5.

4+4xx=0

 
 
 
6.

32xx=0

 
 
 
 

B.  INDUCTION
 
1. Observe the values of the discriminants and the types of roots of the equations. Infer by induction how you will identify the type of roots for a quadratic equation without finding their actual values. 


   
2. Explain why the discriminant affects the type of roots for a quadratic equation. 


   
3. What relationship do you observe between the type of roots of a quadratic equation and how the graph of the corresponding quadratic expression cuts the x-axis? 


C.  EXERCISE
 
1. Do NOT use your computer to do the exercise.
2. Calculate the discriminants of the following quadratic equations and hence deduce the types of roots they have.
3. Sketch the graphs of the corresponding quadratic expressions including the x-axis.
   

No.
Quadratic 
Equation
Discriminant 
D=b²–4ac
Types of Roots
Sketch of Graph
1.
2x²+3x–2=0
 
 2 distinct real roots? 
2 equal real roots? 
2 non-real roots?
 
2.
y²–7y+15=0
 
 
 
3.

45xx=0

 
 
 
4.
 9k+6k+1=0
 
 
 
5.
–7x²+7x–2=0
 
 
 
6.

9+12x4x=0

 
 
 


IT Menu  | GSP | LiveMath