8. (a) Factorise completely 9x - 4x³ (b) Sketch the curve C with equation y = 9x - 4x³ Show on your sketch the coordinates at which the curve meets the x-axis - Edexcel - A-Level Maths Pure - Question 10 - 2015 - Paper 1

To sketch the curve given by the equation:

y = 9x - 4x³,

we start by identifying the x-intercepts, which occur where y = 0. Setting the equation to zero:

$0 = 9x - 4x³$

This can be factored to find the x-intercepts:

$0 = x(9 - 4x²)$

The solutions to this equation give:

1. $x = 0$
2. From $9 - 4x² = 0$, we find x = ±rac{3}{2}.

Thus, the curve meets the x-axis at the points (0, 0), (1.5, 0), and (-1.5, 0). The shape of the curve is that of a cubic function, opening downward, with its local maxima and minima determined by derivatives or by analyzing its shape.

In the sketch:

• Mark the x-axis intercepts at (0,0), (1.5,0), and (-1.5,0).
• This curve has a downward peak, as established by analyzing the leading coefficient.

To find the length of segment AB, we first need the coordinates of points A and B. Using the given x-coordinates:

• Point A has an x-coordinate of -2: $y_A = 9(-2) - 4(-2)³ = -18 - 4(-8) = -18 + 32 = 14$ So, A is (-2, 14).

• Point B has an x-coordinate of 1: $y_B = 9(1) - 4(1)³ = 9 - 4 = 5$ Thus, B is (1, 5).

Now, we calculate the length of AB:

$AB = ext{Distance} = ext{sqrt}((x_B - x_A)^2 + (y_B - y_A)^2)$
Substituting the coordinates:

$AB = ext{sqrt}((1 - (-2))^2 + (5 - 14)^2)$ $= ext{sqrt}((3)^2 + (-9)^2)$
$= ext{sqrt}(9 + 81)$
$= ext{sqrt}(90)$
$= 3 ext{sqrt}(10)$

Therefore, we find that the length of AB is $k ext{sqrt}(10)$ where $k = 3$.