Problem

$f(x) = 2x^2 - 3 - \frac{1}{2x} + \frac{5}{3x - 5} \quad x \neq \frac{1}{2}, \frac{5}{3}$

The table below shows values of $f(x)$ for some values of $x$, with values of $f(x)$ given to $4$ decimal places where appropriate.

$x$	$1$	$2$	$3$	$4$	$5$
$f(x)$	$0.5$		$-0.2885$		$0.5834$

(a) Complete the table giving the values to $4$ decimal places.

The equation $f(x) = 0$ has exactly one positive root, $\alpha$.

Using the values in the completed table and explaining your reasoning,

(b) determine an interval of width one that contains $\alpha$.

(c) Hence use interval bisection twice to obtain an interval of width $0.25$ that contains $\alpha$.

Given also that the equation $f(x) = 0$ has a negative root, $\beta$, in the interval $[-1, -0.5]$

(d) use linear interpolation once on this interval to find an approximation for $\beta$.

Give your answer to $3$ significant figures.