CHAPTER 2

Quadratic Expressions and Equations

·  Quadratic expressions in the form ax² + bx + c, where a, b and c are constants, a ≠ 0 and x is an unknown.

·  Quadratic expressions can be formed by multiplying two linear expressions.

 

( x + a ) ( x + b )

 

= x² + bx + ax + ab

 

·  Factorisation of Quadratic Expressions

 

1. ax² + bx                              3.  x² + bx + c

= x ( ax + b)                           = ( x + m ) (x + n )

Example:                               Example :

3x² + 2x                                  x² + 3x + 2

= x ( 3x + 2)                           = ( x + 1 ) (x + 2 )

 

2. (ax)² - b²                            4. ax² + bx + c

= ( ax - b) ( ax + b)                = (  px + m ) ( qx + n )         

Example :                              Example :

(2x)² - 1²                                6x² + 5x + 1

= ( 2x - 1) ( 2x + 1)               =(  3x + 1 ) ( 2x + 1)

 

We can also use the cross method to factorise quadratic expressions in the form of x² + bx + c and ax² + bx + c

 

Try this using the method :

1. x² + 6x + 8

2.  2x² + 7x + 6

 

How about this ? Can you use the same method ?

1. 5t² - 20t  - 60

2. 7y² - 14y – 245

 

·  Roots of Quadratic Equations

 

Finding the roots of a quadratic equation.

 

ax² + bx + c = 0 ( General Form )

(  px + m ) ( qx + n ) = 0 ( Factorisation )

 

px + m = 0             qx + n = 0

         x = - m                x  = - n  

                   p                             q

 

1.Expand each of the following : (a)     ( x + 3 ) ( x + 4 ) (b)      ( x + 5 ) ( x + 6 ) (c)     ( x + 2 ) ( x + 8 ) (d)     ( x - 4 ) ( x - 2 ) (e)     ( x - 6 ) ( x – 10 ) (f)      ( x - 9 ) ( x – 12 ) (g)     ( x + 7 ) ( x – 4 ) (h)     ( x - 11) ( x + 13 )   2.  Factorise : (a)        x ² + 10 x  + 21 (b)       x ² + 3 x  - 18 (c)        x ² - 9 x  + 20 (d)       x ² + 2x  - 80 (e)        x ² + 2 x  + 63 (f)         x ² - 17x + 60 (g)       x ² - 12 x – 13 (h)       x ² - 20 x + 75 (i)         x ² - 13x – 68 (j)         2x ² + 11 x  + 5 (k)        3x ² - 19 x  - 14 (l)         5x ² - 37 x  - 24 (m)      7x ² - 59 x  - 36 (n)       6x ² + 7 x  - 20 (o)       12x ² + 5x  - 72 (p)       18x ² + 63x  + 49 (q)       30x ² - 89 x  + 24 (r)         44x ² - 163 x  + 65 (s)        65x ² - 112x  - 9   3. Expand each of the following : (a)       ( 4x + 3 ) ( x + 5 ) (b)       ( x + 6 ) ( 3x + 7 ) (c)       ( 5x - 4 ) ( x – 3 ) (d)       ( x - 7 ) ( 4x – 9 ) (e)       ( 8x + 3 ) ( x – 6 ) (f)        ( x + 5 ) ( 6x – 5 ) (g)       ( 2x + 7 ) ( 5x + 9 ) (h)       ( 6x -5 ) ( 4x – 7 ) (i)         ( 9x + 4 ) ( 5x – 6 ) (j)         ( 6 + 7x ) ( 4x – 1 ) (k)       ( 9 – 10x ) ( 6x + 11 ) (l)         (5 – 4x ) ( 7- 2x ) (m)      ( x + 3 ) ² (n)       ( x - 4 ) ² (o)       ( 3x + 2 ) ² (p)       ( 5 + 4x  ) ²