An express train takes $1$ hour less than a passenger train to travel $132 \mathrm{km}$ between stations $A$ and $B$ (without taking into consideration the time they stop at intermediate stations. If the average speed of the express train is $11 \mathrm{km}/\mathrm{hour}$ more than that of the passenger train, find the average speed of the two trains. 

If two pipes function simultaneously, a reservoir will be filled in twelve hours. First pipe fills the reservoir $10$ hours faster than the second pipe. If the second pipe will take $k$ hours to fill the reservoir, then find the value of $k$.

Solve for $x:$

$\frac{1}{x+1}+\frac{2}{x+2}=\frac{4}{x+4}; x\ne -1, -2, -4$

Solve for $x:$

$9x^2-3(a+b)x+ab=0$

Find two positive numbers whose squares have the difference $48$ and the sum of the numbers is $12$.

Solve for $x:$

$a\left(a^2+b^2\right)x^2+b^{2}x-a=0$