2.2a Solving Quadratic Equations – Factorisation


2.2a Solving Quadratic Equations – Factorisation
1. If a quadratic equation can be factorised into a product of two factors such that

(x – p)(x – q) = 0

Hence
 x – p = 0   or  x – q = 0
       x = p   or        x = q

p and q  are the roots of the equation.

Notes
1.The equation must be written in general form ax2 + bx + c = 0 before factorisation.
2. This method can only be used if the quadratic expression can be factorised completely.


Example 1:
Find the roots of the quadratic equations
(a) 
x (2x − 8) = 0 
(b) 
x2 −16x = 0
(c) 
3x2 − 75x = 0
(d) 
5x2 − 100x = 25x

Solution:
(a) 
x (2x − 8) = 0 
x = 0  or  2x − 8 = 0
                        2x − 8 = 0
                        2x = 8
                        x = 4
x = 0  or  x = 4

(b)
x2 −16x = 0
x (x − 16) = 0 
x = 0  or  x − 16 = 0
x = 0  or  x = 16

(c) 
3x2 − 75x = 0
3x (x − 25) = 0 
3x = 0  or  x − 25 = 0
x = 0  or  x = 25

(d) 
5x2 − 100x = 25x
5x2 − 100x − 25x = 0
5x2 − 125x = 0
x (5x − 125) = 0 
x = 0  or  5x − 125 = 0
                        5x = 125
                        x = 25
x = 0  or  x = 25


Example 2:
Solve the following quadratic equations
(a) 
x2 4x 5 = 0
(b) 1 5x + 2x2 = 4

Solution:
(a) 
x2 4x 5 = 0
(x – 5) (x + 1) = 0
x – 5 = 0  or  x + 1 = 0
x = 5  or  x = –1

(b)
1 5x + 2x2 = 4
2x2 5x + 1 – 4 = 0
2x2 5x – 3 = 0
(2x + 1) (x – 3) = 0
2x + 1= 0  or  x – 3 = 0
2x = –1  or  x = 3
x = –½  or  x = 3

Click on the image below to learn how to factorise a quadratic equation.