Page 8 - Ranger SPM 2022 - Additional Mathematics

Additional Mathematics SPM Chapter 2 Quadratic Functions

2.1 Quadratic Equations and Inequalities

1. The general form of the quadratic (b) Formula method
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equation is ax + bx + c = 0 where a, –b ±
2
b
4
ac

–
2
b and c are constants. x = 2a
2. Method for solving quadratic 3. If α and β are the roots of a quadratic
equations: equation then, x – (α + β)x + αβ = 0, Form 4
2
(a) Method of completing the where α + β is the sum of the roots
square and αβ is the product of the roots.
Make sure the coefficient of x is
2
1 before we start completing the
square.
4. Method to determine the range for a quadratic inequality:
(a) Graph sketching
For a quadratic equation in the form of (x – a)(x – b) = 0, where a  b:
If (x – a)(x – b)  0, then x  a or x > b.
If (x – a)(x – b)  0, then a  x  b.
(b) Number line
Select any integer x  a Select any integer x Select any integer x  b
to test whether the value that satisfies a < x < b to test whether the value
of (x – a)(x – b) is positive to test whether the value of of (x – a)(x – b) is positive
or negative. (x – a)(x – b) is positive or or negative.
negative.
a b x
x  a a  x  b x  b
(c) Table
Range of values of x
x  a a  x  b x  b
(x – a) – + +
(x – b) – – +
(x – a)(x – b) + – +

Example 1 Solution
Solve the following quadratic equations by (a) x – 8x + 4 = 0
2
using completing the square method. x – 8x = –4
2
(a) x – 8x + 4 = 0 2   2   2
2
–8
–8
(b) –2x – 7x + 5 = 0 x – 8x + 2 = –4 + 2
2
x – 8x + (–4) = –4 + (–4) 2
2
2
(x – 4) = 12
2
x – 4 = ±12
x = 0.5359 or x = 7.4641
15

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