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

For a quadratic equation,if $\ldots \ldots \ldots \ldots$ then both the roots of an equation are distinct and real.

Verify whether the given value of $x$ is a solution of the quadratic equation or not: $\frac{1}{3-2x} + \frac{1}{5+2x} = \frac{1}{2}$; $x = -\frac{1}{2}$.

Solve the following equation using the quadratic formula,if the equation has a solution in $R$: $5x^{2} - 3x - 2 = 0$.

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

Examine whether the following equation is quadratic or not: $x^{3}-4x^{2}-x+1=(x-2)^{3}$