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

Represent the following situation in the form of a quadratic equation:
The area of a rectangular plot is $528 \ m^{2}$. The length of the plot (in metres) is one more than twice its breadth. We need to find the length and breadth of the plot.

$A$ 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 $km/h$).

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 quadratic equation by applying the quadratic formula: $4 x^{2}+4 \sqrt{3} x+3=0$.

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