Examine whether the following equation is quadratic or not: $7x = 2x^2$.

$A$ pole has to be erected at a point on the boundary of a circular park of diameter $13 \, m$ in such a way that the differences of its distances from two diametrically opposite fixed gates $A$ and $B$ on the boundary is $7 \, m$. Is it possible to do so? If yes,at what distances from the two gates should the pole be erected?

Examine whether the following equation is quadratic or not: $3x^{2} + 5\sqrt{x} + 3 = 0$.

Verify whether the given value of $x$ is a solution of the quadratic equation or not: $(x-2)(x+3)+1=0$; $x=2$.

Find the roots of the following quadratic equation by the method of completing the square: $16 x^{2}-24 x-1=0$

The faster train takes $1$ hour less than the slower train for covering a $400\, km$ distance. If the speed of the slower train is $20\, km/hr$ less than the speed of the faster train,find the speed of the slower train.