2.05 Factorise non-monic quadratic trinomials
Complete the following factorisations:
8 x^{2} + 11 x + 3 = \left( 8 x + 3\right) \left(x + ⬚\right)
2 x^{2} - 19 x + 45=\left( ⬚ - 9\right) \left(⬚ - 5\right)
2 x^{2} + 3 x - 20 = \left( 2 x - 5\right) \left(x + ⬚\right)
Factorise the following expressions:
81 x^{2} + 72 x + 16
64 x^{2} - 48 x + 9
3 x^{2} - 25 x + 28
8 x^{2} - 19 x + 6
8 x^{2} - 21 x - 9
12 x^{2} + 7 x - 10
56 - 41 x - 6 x^{2}
- 6 x^{2} + 5 x + 6
Factorise the following expressions by first taking out a common factor:
4 x^{2} + 40 x + 100
Find the value of k that will make 16 x^{2} - 24 x + k a perfect square trinomial.
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.
Find an expression for the total area of the rectangle in factorised form:
A cube has a surface area of \left(6 x^{2} + 36 x + 54\right) square units, where x > 0.
Factorise 6 x^{2} + 36 x + 54 completely.
Hence, find an expression for the length of a side of the cube.
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.
Consider the figure below:
Write an expression in expanded form for the area of the shaded region.
Write the expression for the area as a factorised quadratic.
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.