Problem

Figure 2 shows a sketch of part of the curve with parametric equations

$$x=2t^2-6t,\qquad y=t^3-4t,\qquad t\in\mathbb{R}$$

The curve cuts the $x$-axis at the origin and at the points $A$ and $B$, as shown in Figure 2.

(a) Find the coordinates of $A$ and show that $B$ has coordinates $(20,0)$.

(b) Show that the equation of the tangent to the curve at $B$ is

$$7y+4x-80=0$$

The tangent to the curve at $B$ cuts the curve again at the point $P$.

(c) Find, using algebra, the $x$ coordinate of $P$.