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

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 quadratic equation by factorisation:
$\sqrt{2} x^{2}+7 x+5 \sqrt{2}=0$

$A$ train travels $360\, km$ at a uniform speed. If the speed had been $5\, km/h$ more,it would have taken $1\, hour$ less for the same journey. Find the speed of the train (in $km/h$).

Represent the following situation mathematically:
John and Jivanti together have $45$ marbles. Both of them lost $5$ marbles each,and the product of the number of marbles they now have is $124$. We would like to find out how many marbles they had to start with.

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