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Additional Mathematics SPM Chapter 2 Quadratic Functions

(b) 9x = 6x – 1
2
9x – 6x + 1 = 0
2
Compare with ax + bx + c = 0.
2
a = 9, b = –6, c = 1
5 m 2 2
Ground b – 4ac = (–6) – 4(9)(1)
= 0
Find the time of the stone to reach the ground. [Give Therefore, equation 9x = 6x – 1 has two equal real
2
your answer correct to two significant figures.]
roots.
2
2.2 Types of Roots of Quadratic (c) x + 4x + 5 = 0 2
Compare with ax + bx + c = 0.
Equations a = 1, b = 4, c = 5
2
2
A Relating the types of roots b – 4ac = 4 – 4(1)(5) Form 4
of quadratic equations to the = –4  0
discriminant value Therefore, equation x + 4x + 5 = 0 has no real
2
roots.
1. The types of roots of a quadratic equation
ax + bx + c = 0 depends on the value of the SPM Tips
2
2
expression b – 4ac.
2
2. The expression b – 4ac is called the discriminant For the equation x + 4x + 5 = 0 in (c), the imaginary root
2
can be written as
of the quadratic equation. –b ±  –4 ± 
2
x = b – 4ac = –4
3. The relation between the types of roots and the 2a 2(1)
–1
values of the discriminant are shown in the table = –2 ±  4
below. 2
–1
= –2 ± 
Discriminant, Types of roots = –2 ± i
b – 4ac The imaginary part, that is  is represented by the
2
–1,
(a) Two different real symbol i.
. 0
roots Imaginary roots are widely used in the fields of
electric and electronics.
(b) Two equal real
= 0
roots
(c) No roots (or Try Question 1 in ‘Try This! 2.2’
 0
imaginary roots )
B Solving problems involving types of
9 roots of quadratic equations
Determine the types of roots for each of the following 1. The relation between the types of roots and
quadratic equations. the values of the discriminants for quadratic
(a) 3x – 5x – 6 = 0 equations can be used to
2
(b) 9x = 6x – 1 (a) find the value or values of unknown
2
(c) x + 4x + 5 = 0 variables,
2
Solution (b) find a relation between two variables in the
(a) 3x – 5x – 6 = 0 quadratic equation.
2
Compare with ax + bx + c = 0.
2
a = 3, b = –5, c = –6 10
b – 4ac = (–5) – 4(3)(–6)
2
2
= 97 . 0 Find the range of values of p if the quadratic equation
2
2
Therefore, equation 3x – 5x – 6 = 0 has two 3x + 4 = 8x + 2px , where p is a constant, has two
2
different real roots. different real roots.
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