If one of the roots of the equation $kx^{2} + 2x - 3 = 0$ is $2$,then find $k$.

The roots of $4x^{2} + 2x + \frac{1}{4} = 0$ are $\ldots \ldots \ldots \ldots.$

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

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 discriminant of the following quadratic equation and hence determine the nature of the roots of the equation: $4x^{2} + 11x + 10 = 0$

Examine whether the following equation is quadratic or not: $(3x - 4)^2 - (2x - 3)^2 = 7$.