1. SCIENTIFIC CALCULATOR
TOPIC : QUADRATIC EQUATIONS (ADDITIONAL MATHEMATICS F4)
SUBTOPIC : Discriminant of a Quadratic Equation
Learning objective : Understand and use the condition for quadratic equation which is:
 Two different root
 Two equal root
 No root
Learning outcome : Student should be able to state and determine the type of root of quadratic equation.
Introduction
In mathematics, a quadratic equation is a univariate polynomial equation of the second degree. A
general quadratic equation can be written in the form ax 2 bx c 0 where x represents
a variable or an unknown, and a, b, and c are constants with a ≠ 0.
(If a = 0, the equation is a linear equation.)
The constants a, b, and c are called respectively, the quadratic coefficient, the linear coefficient
and the constant term or free term. Quadratic equations can be solved by :
a) factoring (if quadratic equation cannot be factored, then this method will not work)
b) completing the square which is : a( x h) 2 k
b b2 4ac
c) quadratic formulawhich is : x
2a
From the quadratic formula, b 2 4ac is known as the discriminant. The discriminant gives
additional information on the nature of the roots whether there are any repeated roots. It also
gives information on whether the roots are real or complex, and rational or irrational. More
formally, it gives information on whether the roots are in the field over which the polynomial is
defined, or are in an extension field, and hence whether the polynomial factors over the field of
coefficients. This is most easily stated for quadratic and cubic polynomials; for polynomials of
degree 4 or higher this is more difficult to state. The nature of the roots are given as below:

2. Value of the discriminant Type and number of roots Example of graph
Positive Discriminant Two Real Roots
b² − 4ac > 0 If the discriminant is a perfect
square the roots are rational.
Otherwise, they are irrational.
Discriminant is Zero One Real Root
b² − 4ac = 0
Negative Discriminant No Real Roots
b² − 4ac < 0 Two Imaginary Solutions
As a conclusion, by using the following module, it will help student to understand more about
discriminant, quadratic equation and develop their own knowledge by using scientific calculator.
 In order to do this worksheet, student should already know about:
 Quadratic equation general form ( ax 2 bx c 0 )
 Determine the value of a, b, and c
Example :
x 2 5x 24 0
a=1 b= 5 c = -24

3.  Factorize quadratic equations
Example :
x 2 5x 24 0
( x 3)(x 8) 0 factorize
 From the result of worksheet, student should found that :
a) If b 2 4ac 0 , then the quadratic equation has 2 different roots.
b) If b 2 4ac 0 , then the quadratic equation has 2 equal roots.
c) If b 2 4ac 0 , then the quadratic equation has no roots.
Mapping
Form Form 1 Form 2 Form 3 Form 4 Form 5 Form 6
Prior knowledge
Algebraic expression
Graph of function
Quadratic expression & equations
Quadratic equations ( Add math)
Activity
a) How to use calculator?
 Use the key MODE to enter the EQN mode
 Press 3 times and press number 1
 Then, press to display the quadratic/ cubic equation screen.
Example :
Degree ?
2 3

4.  Use this screen to specify 2 (quadratic) and 3 (cubic) as the degree of equation.
 Input the value of the coefficients until reach for the final coefficient which is until C for
quadratic equation.
Example:
Use and keys to move between coefficients and make changes
Coefficient a? Direction to
name 0. view other
elements
Element
value
 Calculation start and one of the solution will appears.
 Then, press = keys to view other solution. Use and to scroll
all solution. Next, press AC key to returns to the coefficient input screen.
Example :
Direction to view
Variable x1= otherelements
name 0.
Solution

5. EXAMPLE :
6x 2 x 2 0 x1=
R I
(Degree?) 2 0.25
a? 6 =
SHIFT R I
b? 1 =
c? -2 =
x2=
R I
x1 0. 5 0.75
i
x2 0.7

6. b) Worksheet
Complete the following table by using scientific calculator.
Equation a b c b2 4ac Type of roots x1 x2
x2 2x 3
3x 2 6 x
x2 4x 4
x2 2x 1
x2 3
2 x 2 3x 8
c) Worksheet ( Answer)
Equation a b c b2 4ac Type of roots x1 x2
Positive discriminant
x2 2x 3
1 -2 -3 16 b 2 4ac 0 3 -1
3x 2 6 x Positive discriminant
3 6 0 36 b 2 4ac 0 0 -2
x2 4x 4 Discriminant is zero
1 -4 4 0 b 2 4ac 0 2 2
Discriminant is zero
x2 2x 1
1 -2 1 0 b 2 4ac 0 1 1
Negative discriminant
x2 3 1 .7 i 1 .7 i
1 0 3 -12 b 2 4ac 0
2 x 2 3x 8 Negative discriminant 0.75 1.9i 0.75 1.9i
2 3 8 -55
b 2 4ac 0

7. d) Conclusion
Unequal real roots Equal real roots No real roots
b 2 4ac 0 b 2 4ac 0 b 2 4ac 0
Exercise
Complete the following table without using scientific calculator.
Equation a b c b2 4ac Type of roots Conclusion
x2 x 2
20 x 2 36 x 9
x 2 8 x 16
4 x 2 12 x 9
9 x 2 81
25x 2 50

8. Exercise Answer
Complete the following table without using scientific calculator.
Equation a b c b2 4ac Type of roots Conclusion
Positive discriminant
x2 x 2
1 1 -2 9 b 2 4ac 0
Unequal
2 Positive discriminant real roots
20 x 36 x 9
20 -36 9 576 b 2 4ac 0
Discriminant is zero
x 2 8 x 16
1 -8 16 0 b 2 4ac 0
Equal real
2 Discriminant is zero roots
4 x 12 x 9
4 -12 9 0 b 2 4ac 0
9 x 2 81 Negative discriminant
-9 0 -81 -2916 b 2 4ac 0
No real
Negative discriminant roots
25x 2 50
25 0 -50 -5000 b 2 4ac 0

9. References
OoiSooHuat, et al. , Additional Mathematics Form 4, Selangor: NurNiagaSdnBhd, 2005.
Zaini B Musa, et al. ,Additional Mathematics Form 4, Selangor: Cerdik Publication SdnBhd,
2005.
Huraiansukatanpelajaranmatematiktingkatan 1,2dan 3
Huraiansukatanpelajaranmatematiktambahantingkatan 4 dan 5
Huraiansukatanpelajaranmatematik T tingkatan 6
The discriminant in quadratic equation. Retrieved from TheMathWarehousePage:
http://www.mathwarehouse.com/quadratic/discriminant-in-quadratic-equation.php
Discriminant.Retrieved from Wikipedia Page:
http://en.wikipedia.org/wiki/Discriminant#Quadratic_formula
Quadratic euqations.Retrieved from Wikipedia Page :
http://en.wikipedia.org/wiki/Quadratic_equation