Problem

A curve which passes through $(0, 3)$ has equation $y = f(x)$. It is given that $f'(x) = 1 - \frac{2}{(x - 1)^3}$.

(a) Find the equation of the curve.

The tangent to the curve at $(0, 3)$ intersects the curve again at one other point, $P$.

(b) Show that the $x$-coordinate of $P$ satisfies the equation $(2x + 1)(x - 1)^2 - 1 = 0$.

(c) Verify that $x = \frac32$ satisfies this equation and hence find the $y$-coordinate of $P$.