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

Find the roots of the following quadratic equation by the method of completing the square: $3 y^{2}+7 y-20=0$

$A$ train,travelling at a uniform speed for $360\, km$,would have taken $48\, min$ less to travel the same distance,if its speed were $5\, km/h$ more. Find the original speed of the train (in $km/h$).

Obtain the roots of the following quadratic equation by using the quadratic formula: $9x^{2} + 7x - 2 = 0$.

The value of discriminant $D$ for the equation $(3x - 14)^2 = 0$ is ........

State whether the quadratic equation $\sqrt{2} x^{2}-\frac{3}{\sqrt{2}} x + \frac{1}{\sqrt{2}} = 0$ has two distinct real roots. Justify your answer.