Page 11 - Focus SPM KSSM Tg 4.5 - Add Maths

Additional Mathematics SPM Chapter 2 Quadratic Functions

SPM Highlights 7

One of the roots of the quadratic equation Determine the range of values of x that satisfy each
x + (n – 5)x – 2n = 0, where n is a constant, is the of the following quadratic inequalities by using graph
2
2
negative of the other root of the equation. Find the
product of the roots. sketching method, number line method or table
method.
Solution: (a) x + x – 6  0
2
Assume that the roots for the equation 2
x + (n – 5)x – 2n = 0 are α and – α. (b) 6 + x – x > 0
2
2
2
SOR = α + (– α) = –(n – 5) (c) 2x – 5x + 2 . 0
0 = –n + 5
n = 5 ............ Solution
2
POR = –2n 2 (a) x + x – 6  0
= –2(5) 2 Graph sketching method
= –50 When f(x) = 0, x + x – 6 = 0
2
(x – 2)(x + 3) = 0
Form 4
SPM Highlights x – 2 = 0 or x + 3 = 0
x = –3
x = 2
2
The quadratic equation px – 4x + q = 0, where p and q The coefficient of x , a = 1 . 0 ⇒ the shape of the
2
are constants, has roots α and 3α. Express p in terms graph is .
of q.
The graph sketch:
Solution:
f(x)
px – 4x + q = 0
2
4
q
x – x + = 0
2
p p –3 2 x

SOR = α + 3α = – – 4 
p
4
4 α =
p Therefore, the range of values of x that satisfy the
α = 1 .............. 2
p quadratic inequality x + x – 6  0 is –3  x  2.

POR = (α)(3α) = q 2
p (b) 6 + x – x > 0
q
3 α = .............b Number line method
2
p Factorise the quadratic equation,
Substitute  into b. 6 + x – x = 0
2
1
3   2 = q (2 + x)(3 – x) = 0
p
p
3 = q (2 + x) = 0 or (3 – x) = 0
p 2 p x = –2 x = 3
3 p = p q
2
3 p – p q = 0 (2 + x) ≥ 0 when x ≥ –2
2
p(3 – pq) = 0 – + +
p = 0 or 3 – pq = 0
( unacceptable) pq = 3 + –2 + 3 –
3
p =
q (3 – x) ≥ 0 when x ≤ 3 +
(+) × (+) = (+)
C Solving quadratic inequalities
Therefore, the range of values of x that satisfy
1. The range of values of x that satisfy a quadratic the quadratic inequality 6 + x – x > 0 is
2
inequality can be determined by using –2 < x < 3.
• graph sketching method SPM Tips
• number line method
• table method Use the graph sketching method to check the
INFO answer.
34