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

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

Find the discriminant of the quadratic equation $2x^{2}-4x+3=0$,and hence find the nature of its roots.

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 the real roots exist,find them:
$2x^2 - 6x + 3 = 0$

The difference of squares of two numbers is $180$. The square of the smaller number is $8$ times the larger number. Find the two numbers.