Answer ALL the questions

1. Factor out the coefficient of $x^2$ from the $x$ terms

$3x^2 - 8x - 3 = 3\left(x^2 - \frac{8}{3}x\right) - 3$

2. Complete the square for the expression inside the bracket

$= 3\left[ \left(x - \frac{4}{3}\right)^2 - \left(\frac{4}{3}\right)^2 \right] - 3$
$= 3\left[ \left(x - \frac{4}{3}\right)^2 - \frac{16}{9} \right] - 3$

3. Expand and simplify to reach the final form

$= 3\left(x - \frac{4}{3}\right)^2 - 3\left(\frac{16}{9}\right) - 3$
$= 3\left(x - \frac{4}{3}\right)^2 - \frac{16}{3} - \frac{9}{3}$
$= 3\left(x - \frac{4}{3}\right)^2 - \frac{25}{3}$

1. Determine the completed square form from part (a)

From part (a),
$3x^2 - 8x - 3 = 3\left(x^2 - \frac{8}{3}x\right) - 3$
$= 3\left(x - \frac{4}{3}\right)^2 - 3\left(\frac{16}{9}\right) - 3$
$= 3\left(x - \frac{4}{3}\right)^2 - \frac{25}{3}$

2. Identify the turning point and the corresponding $x$-value

The minimum value occurs when $x - \frac{4}{3} = 0$, which gives $x = \frac{4}{3}$.
The minimum value is $-\frac{25}{3}$.
Therefore, the turning point is $\left(\frac{4}{3}, -\frac{25}{3}\right)$.