Find the discriminant of the following quadratic equation: $5x^{2} + 2x - 1 = 0$.

Find the roots of the quadratic equation by using the quadratic formula:
$-3x^{2} + 5x + 12 = 0$

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.

Verify whether the given value of $x$ is a solution of the quadratic equation or not: $x^{2}+\sqrt{2}x-4=0$; $x=-2\sqrt{2}$.

If the roots of the equation $6x^2 - 13x + m = 0$ are reciprocal of each other,then $m = $ ...........

Solve the following equation using the method of factorization and write its solution set: $\frac{x+1}{x-1} + \frac{x-2}{x+2} = 3$