Find the discriminant of the following quadratic equation and hence determine the nature of the roots of the equation: $4x^{2} + 12x + 9 = 0$.

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

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

In the centre of a rectangular lawn of dimensions $50\, m \times 40\, m$,a rectangular pond has to be constructed so that the area of the grass surrounding the pond would be $1184\, m^{2}$ [see $Fig.$]. Find the length and breadth of the pond. (in $m$)

If $\frac{1}{2}$ is a root of the equation $x^{2}+kx-\frac{5}{4}=0$,then the value of $k$ is

Solve the following equation using the method of factorization: $x - \frac{20}{x} = 1$