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Middle Years

1.01 Factorising non-monic trinomials

Worksheet
Non-monic quadratic trinomials
1

Complete the following factorisations:

a

8 x^{2} + 11 x + 3 = \left( 8 x + 3\right) \left(x + ⬚\right)

b

2 x^{2} - 19 x + 45=\left( ⬚ - 9\right) \left(⬚ - 5\right)

c

2 x^{2} + 3 x - 20 = \left( 2 x - 5\right) \left(x + ⬚\right)

d
10 x^{2} + 29 x + 21=\left(⬚ + 3\right) \left( 5 x + ⬚ \right)
2

Factorise the following expressions:

a
2x^2 -11x-40
b
3y^2+28y+9
c
12t^2-13t-4
d

81 x^{2} + 72 x + 16

e

64 x^{2} - 48 x + 9

f

3 x^{2} - 25 x + 28

g

8 x^{2} - 19 x + 6

h

8 x^{2} - 21 x - 9

i

12 x^{2} + 7 x - 10

j

56 - 41 x - 6 x^{2}

k

- 6 x^{2} + 5 x + 6

l
-6x^2 + 25x-14
3

Factorise the following expressions by first taking out a common factor:

a
3 x^{2} - 21 x - 54
b

4 x^{2} + 40 x + 100

c

4 x^{2} + 24 x + 32

d
10x^2+5x-30
e
8s^2 + 6s -54
f

4 x^{2} - 4 x - 288

g

2 x^{3} + 16 x^{2} + 30 x

h

- 4 x^{2} + 12 x + 40

i

3 x^{2} - 21 x + 30

j

- 3 x^{2} + 12 x - 12.

4

Find the value of k that will make 16 x^{2} - 24 x + k a perfect square trinomial.

Applications
5

Write down an expression in factorised form for the shaded area in the rectangle:

6

A rectangle has an area of 6 x^{2} + 23 x + 20.

If the length and width are linear factors of 6 x^{2} + 23 x + 20, what are the dimensions of the rectangle?

7

The side length of the following regular pentagon is given by S = 2 x^{2} + 21 x + 49.

a

Write the perimeter of the pentagon in terms of x, as a polynomial in expanded form.

b

Express the perimeter in fully factorised form.

8

Let the length of the rectangle below be L = - 56 y + 11 and the width be W = 5 y^{2}.

a

Write the perimeter of the figure in terms of y in expanded form.

b

Fully factorise the expression for the perimeter.

9

Find an expression for the total area of the rectangle in factorised form:

10

A cube has a surface area of \left(6 x^{2} + 36 x + 54\right) square units, where x > 0.

a

Factorise 6 x^{2} + 36 x + 54 completely.

b

Hence, find an expression for the length of a side of the cube.

11

Quadratic trinomials can be factorised using the identity:

a x^{2} + b x + c = \dfrac{\left( a x + m\right) \left( a x + n\right)}{a}where m + n = b and m n = a c.

Find the values m and n for the quadratic 4 x^{2} - 14 x + 12.

12

Consider the figure below:

a

Write an expression in expanded form for the area of the shaded region.

b

Write the expression for the area as a factorised quadratic.

13

A ball is thrown from the top of a 140\text{ m} tall cliff, with an initial velocity of 50\text{ m/s}. The height of the ball after t seconds is approximated by the quadratic - 10 t^{2} + 50 t + 140. Factorise this quadratic.

14

Tara is practising diving. She springs up off a board 32 feet high, and after t seconds, her height in feet above the water is described by the quadratic:

- 16 t^{2} + 16 t + 32

a

Completely factorise the quadratic.

b

Substitute t = 2 into the factorised quadratic and find the value of the expression.

c

Substitute t = 2 into the original quadratic and find the value of the expression.

d

Hence, state what is happening 2 seconds after Tara jumps off the board.

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