Form 4 Maths Chapter 2 14y1f

CHAPTER 2 – QUADRATIC EXPRESSIONS & EQUATIONS  Quadratic Expressions  Quadratic Equations QUADRATIC EXPRESSIONS 1. A quadratic expression is written in the form

ax2 + bx + c , where x is an unknown and a, b, c are constants such that a ≠0 Example i. x2 + 3x – 1 ii. 4x2 – 7x 2.

iii. iv.

8 – x2 10p2 + p – 2

Has only one unknown and the highest power of the unknown is 2.

EXERCISE Level 1: Determine which are the following are quadratic expressions: 3 a. 7 + 2x – x2 c. + 4x – 1 𝑥2

b.

1

5x2 – 𝑥 + 3

d.

2x3 – 5x +

Level 1: Form quadratic expression by multiplying two linear expressions: a. (3x – 2) (x + 4)

b.

(2x + 5) (2x – 5)

c.

(4x – 1)2

PART I

d.

(5 + p) (2 + 3p)

Level 2: A

B x cm F 4 cm

D

2x cm

E

5 cm

C

In the diagram, ABCD is a rectangle. E lies on CD and F lies on BC. Express the area of the shaded region in of x.

Level 2

Diagram shows a cuboid. BG is 2cm shorter than AB and BC is 3cm longer than AB. Express the total surface area of the cuboid in of x.

Level 3

In the diagram, BCE and AED are straight lines. a. Find the length of EC in cm. b. Express the area of △ ABE in of x.

Level 3 The perimeter of a rectangular plot of land (4x – 10) cm. Given its length is (x – 7) cm, express area in of x.

3.

Factorisation of quadratic expressions 

Expansion

- when two linear expressions are multiplied.



Factorisation

- when a quadratic expression is written as a product of two linear expressions.



To factorise quadratic expressions ax2 + c, take out the highest common factor of a and c



To factorise quadratic expressions ax2 + bx, take out the highest common factor of a and b



To factorise quadratic expressions px2 – q, where p and q are perfect squares, write p = a2 and q = b2 and px2 – q = (ax + b) (ax – b).



In the form x2 + bx + c = (x + p) (x + q), b = p + q and c = pq.



In the form ax2 + bx + c = (mx + p) (nx +q), b = mq + np and c = pq.

EXERCISE Level 1: Factorise the following completely. a. 6x2 + 8

g.

6e – 2e2

b.

9 + 6p2

h.

20r2 – 8r

c.

12x2 + 18

i.

25x2 – 4

d.

5 + 5k2

j.

1 – 36p2

e.

2x2 + 10x

k.

4m2 – n2

f.

7p + 21p2

l.

9e2 – 49f2

Level 2: a. x2 + 7x + 10

f.

2x2 + 7x + 6

b.

x2 – 8x + 12

g.

2x2 – 13x + 15

c.

x2 + 5x – 6

h.

6x2 + 5x – 6

d.

x2 – 3x – 18

i.

8 + 2x – x2

e.

16 – 8x + x

Level 2 Factorise a. 8x2 – 12x

c.

(2x – 1) (x + 3) – 9

Level 3: Factorise the following completely a. 2x2 + 16x + 24

d.

-18x2 – 33x + 30

b.

12x2 – 12x + 3

e.

c2 – 25p2

c.

4 – 36k2

f.

(3x + 5)2 – 16

b.

3x (x – 7) + 2(7 – x)