Check whether the following is a quadratic equation:
$x(x+1)+8=(x+2)(x-2)$

Find the roots of the following quadratic equation,if they exist,by the method of completing the square: $2x^{2} + x - 4 = 0$.

Find the roots of the following equation:
$x + \frac{1}{x} = 3, x \neq 0$

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

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,if they exist,by the method of completing the square: $4x^{2} + 4\sqrt{3}x + 3 = 0$.