Consider the quadratic x^{2} + 11 x + 24. To factorise this quadratic, we need to find two numbers.
What should their product be?
What should their sum be?
Find the values of a and b given that:
- b < a
- a + b = - 11
- a b = 18
Find the values of m and n that make the following equation true:
y^{2} + m y + 60 = \left(y + 10\right) \left(y + n\right)
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.
Factorise the following:
x^{2} + 6 x + 5
x^{2} - 15 x + 56
x^{2} + x - 132
n^{2} - n - 12
- 12 - 13 x - x^{2}
k^{2} - 81
3 x + x z - 39 y - 13 y z
x^{2} + 12 x + 36
7 v - v^{2}
4 - m^{2}
17 r s + w r + 17 s t + w t
36 - 12 u + u^{2}
t^{2} - 14 t + 48
u^{2} m^{2} - 121
t^{2} + 6 t - 16
6 y - y w + w^{2} - 6 w
v^{2} - 3 v + \dfrac{9}{4}
11 + 12 q + q^{2}
t^{2} + 10 t + 15 t + 150
What is the largest possible integer value of k that will allow m^{2} + k m + 24 to be factorised?
Find all positive and negative integer values of k such that x^{2} + k x + 14 is able to be factorised.
Find all positive integer values of c such that x^{2} + 4 x + c can be factored into a product of linear factors that contain integer terms.
Complete the following statements:
3 x^{2} + 10 x - 8=\left( 3 x - 2\right)(x+⬚)
15 x^{2} - 29 x+⬚=(3 x-⬚)\left( 5 x - 3\right)
15 x^{2} - 13 x - 6=(5 x-⬚)(3 x+⬚)
Factorise the following:
3 x^{2} - 21 x - 54
- 5 x^{2} + 10 x + 40
16 m^{2} - 81
81 t^{2} + 72 t + 16
z^{2} + 4 z^{4}
7 x^{2} - 75 x + 50
9 x^{2} - 19 x + 10
6 x^{2} + 13 x + 6
- 10 x^{2} - 7 x + 12
8 - 14 p - 49 p^{2}
10 x^{2} + 23 x + 12
56 - 41 b - 6 b^{2}
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.
Find an expression for the total area of the rectangle in factorised form:
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.
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
Completely factorise the quadratic.
Substitute t = 2 into the factorised quadratic and find the value of the expression.
Substitute t = 2 into the original quadratic and find the value of the expression.
Hence, state what is happening 2 seconds after Tara jumps off the board.
Write down an expression in factorised form for the shaded area in the rectangle:
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?
The side length of the following regular pentagon is given by S = 2 x^{2} + 21 x + 49.
Write the perimeter of the pentagon in terms of x, as a polynomial in expanded form.
Express the perimeter in fully factorised form.
Let the length of the rectangle below be L = - 56 y + 11 and the width be W = 5 y^{2}.
Write the perimeter of the figure in terms of y in expanded form.
Fully factorise the expression for the perimeter.
