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1.02 Factorisation

Worksheet
Factorisation techniques
1

Consider the quadratic x^{2} + 11 x + 24. To factorise this quadratic, we need to find two numbers.

a

What should their product be?

b

What should their sum be?

2

Find the values of a and b given that:

  • b < a
  • a + b = - 11
  • a b = 18
3

Find the values of m and n that make both sides of the following equation equal:

y^{2} + m y + 60 = \left(y + 10\right) \left(y + n\right)

4

One expression for the area of the rectangle below is m^{2} + 14 m + 45. The rectangle is made up of four smaller rectangles. Use the diagram to express the area of the large rectangle in factorised form.

5

Factorise the following:

a

x^{2} + 6 x + 5

b

x^{2} - 15 x + 56

c

x^{2} + x - 132

d

n^{2} - n - 12

e

- 12 - 13 x - x^{2}

f

k^{2} - 81

g

3 x + x z - 39 y - 13 y z

h

x^{2} + 12 x + 36

i

7 v - v^{2}

j

4 - m^{2}

k

17 r s + w r + 17 s t + w t

l

36 - 12 u + u^{2}

m

t^{2} - 14 t + 48

n

u^{2} m^{2} - 121

o

t^{2} + 6 t - 16

p

6 y - y w + w^{2} - 6 w

q

v^{2} - 3 v + \dfrac{9}{4}

r

11 + 12 q + q^{2}

s

t^{2} + 10 t + 15 t + 150

6

What is the largest possible integer value of k that will allow m^{2} + k m + 24 to be factorised?

7

Find all positive and negative integer values of k such that x^{2} + k x + 14 is factorisable.

8

Find all positive integer values of c such that x^{2} + 4 x + c can be factored into the product of linear factors that contain integer terms.

Factorisation for non-monic quadratics
9

Complete the following statements:

a

3 x^{2} + 10 x - 8=\left( 3 x - 2\right)(x+⬚)

b

15 x^{2} - 29 x+⬚=(3 x-⬚)\left( 5 x - 3\right)

c

15 x^{2} - 13 x - 6=(5 x-⬚)(3 x+⬚)

10

Factorise the following:

a

3 x^{2} - 21 x - 54

b

- 5 x^{2} + 10 x + 40

c

16 m^{2} - 81

d

81 t^{2} + 72 t + 16

e

z^{2} + 4 z^{4}

f

7 x^{2} - 75 x + 50

g

9 x^{2} - 19 x + 10

h

6 x^{2} + 13 x + 6

i

- 10 x^{2} - 7 x + 12

j

8 - 14 p - 49 p^{2}

k

10 x^{2} + 23 x + 12

l

56 - 41 b - 6 b^{2}

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.

Use this method to factorise 8x^2 - 17x + 2.

12

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

Applications
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.

15

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

16

A rectangle has an area of 6 x^{2} + 23 x + 20. Find the linear dimensions of the rectangle in terms of x.

17

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

Express the perimeter in terms of x in fully factorised form.

18

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

Express the perimeter in terms of y in fully factorised form.

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Outcomes

1.2.2.3

solve quadratic equations algebraically using factorisation, the quadratic formula (both exact and approximate solutions), and completing the square and using technology

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