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

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.

Solve by using the quadratic formula. The area of a rectangular plot is $528 \ m^2$. The length of the plot (in metres) is one more than twice its breadth. We need to find the length and breadth of the plot.

Verify whether the given value of $x$ is a solution of the quadratic equation or not: $4 \sqrt{3} x^{2} + 5 x - 2 \sqrt{3} = 0$; $x = \frac{\sqrt{3}}{4}$.

The sum of the reciprocals of Rehman's ages,(in $years$) $3$ years ago and $5$ years from now is $\frac{1}{3}$. Find his present age. (in $years$)

Find the roots of the quadratic equation by using the quadratic formula:
$5x^{2} + 13x + 8 = 0$