Check whether the following is a quadratic equation:
$(2x - 1)(x - 3) = (x + 5)(x - 1)$

Two water taps together can fill a tank in $9 \frac{3}{8}$ hours. The tap of larger diameter takes $10 \text{ hours}$ less than the smaller one to fill the tank separately. Find the time in which each tap can separately fill the tank.

Find the nature of the roots of the following quadratic equation. If real roots exist,find them:
$3x^2 - 4\sqrt{3}x + 4 = 0$

Find the roots of the following quadratic equation by factorisation:
$2x^{2} + x - 6 = 0$

$A$ motor boat whose speed is $18\, km/h$ in still water takes $1\, hour$ more to go $24\, km$ upstream than to return downstream to the same spot. Find the speed of the stream. (in $ km/h$)

Find the roots of the quadratic equation by applying the quadratic formula:
$2x^{2} + x + 4 = 0$