If the following quadratic equation has two equal and real roots,then find the value of $k$: $5x^{2} - 2kx + 20 = 0$.

If $1$ is the solution of a quadratic equation $3x^{2}-kx+2=0$,then $k=\ldots$

Solve the following equation using the method of factorization and write its solution set: $\frac{x+1}{x+2} + \frac{1}{x} = \frac{5}{4}$

The sum of a natural number and its reciprocal is $\frac{5}{2}$. Then the number is $\ldots \ldots .$

When there is a decrease of $5\, km/hr$ in the usual uniform speed of a train,it takes $4$ hours more than the usual time for covering the distance of $400\, km$. Find the usual speed of the train.

Find the roots of $6x^{2} - \sqrt{2}x - 2 = 0$ by the factorisation of the corresponding quadratic polynomial.