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Additional Mathematics SPM Chapter 2 Quadratic Functions

2.1 Quadratic Equations and Solution 2 Express the
(a)
x + 6x – 2 = 0
Inequalities x + 6x = 2 equation in the
form of ax + bx + c
2
2
6
6
A Solving quadratic equations using the x + 6x +   2 = 2 +   2 with a = 1.
2
method of completing the square and 2 2
using the quadratic formula x + 6x + 3 = 2 + 9 Add b 2 on
2
2
 
2a
(x + 3) = 11 both sides of the
2
1. Certain quadratic equations can be solved by x + 3 = ±  11 equation.
factorisation. x = –3 ±  11
For example: x + x – 2 = 0 = –3 +  11 or –3 –  11
2
(x – 1)(x + 2) = 0 = 0.3166 or –6.3166
x – 1 = 0 or x + 2 = 0
x = 1 x = –2 SPM Tips
Form 4
2. Quadratic equations such as x = 16, (x – 2) = 6 Use the calculator to check your answer by
2
2
can be solved easily because the left hand side of substituting the answer into the quadratic equation
the equation is a complete square. involved.
For example: (x – 2) = 6
2
2

x – 2 = ± 6 (b) For the case when the coefficient of x is not 1,

x = 2 ± 6 change the coefficient to 1 by dividing each term in


= 2 + 6 or 2 – 6 the equation by that coefficient before completing
= 4.449 or –0.4495 the square.
[to 4 significant figures] 2x + 4x – 1 = 0
2
2
1
4
2
by the coefficient
3. For quadratic equations that are difficult to x + x – = 0 Divide each term
2
2
2
2
factorise or cannot be factorised easily such as 1 of x .
2
x + 6x – 2 = 0 and 2x + 4x – 1 = 0, these equations x + 2x =
2
2
2
can be solved using the 2 2 1 2 2
2
(a) completing the square method, x + 2x +   = +  
2
2
2

–b ± b – 4ac 1
2
(b) quadratic formula method, . x + 2x + 1 = + 1
2
2a 2
3
(x + 1) = 2
2
Derivation of the 3
Quadratic Formula x + 1 = ±
2
INFO x = –1 ± 1.2247
4. It is easier to use the quadratic formula method = 0.2247 or –2.2247
to solve the quadratic equation ax + bx + c = 0
2
when the value of a, b and c are known. Try Question 1 in ‘Try This! 2.1’
1
Determine the roots for each of the following quadratic 2
equations by using the completing the square.
(a) x – 2 + 6x = 0 Solve each of the following quadratic equations by
2
using the quadratic formula method.
(b) 2x + 4x – 1 = 0 (a) 3x = 2x + 5
2
2
(b) 6 – 2x = 5x 2
30