If one of the roots of the equation $x^{2}-(1+k)x+\sqrt{2}=0$ is $\sqrt{2},$ then find $k$.

The necessary condition for $ax^2 + bx + c = 0$,where $a, b, c \in R$,to be a quadratic equation is:

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

$A$ 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?

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.

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