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

Which of the following polynomials does not have any real zero?

If $\ldots \ldots \ldots$ then real roots of a quadratic equation do not exist.

$x = \dots$ is a solution of the quadratic equation $x^{2} + 7x + 12 = 0$.

Find the discriminant of the following quadratic equation and hence determine the nature of the roots of the equation: $x^{2}-2x-15=0$.

Write whether the following statements are true or false. Justify your answers.
$(i)$ If the coefficient of $x^{2}$ and the constant term have the same sign and if the coefficient of $x$ term is zero,then the quadratic equation has no real roots.
$(ii)$ Every quadratic equation has at least two roots.