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

If one of the roots of the equation $3x^{2} + 2kx - 3 = 0$ is $-\frac{1}{2}$,then find the value of $k$.

Find the discriminant of the following quadratic equation: $5x^{2} - 4\sqrt{2}x - 1 = 0$.

Divide $16$ into two parts such that twice the square of the larger part exceeds the square of the smaller part by $164$.

If Zeba were younger by $5\, \text{years}$ than what she really is, then the square of her age (in years) would have been $11$ more than five times her actual age. What is her age now? (in $\text{years}$)

Obtain the roots of the following equation using the method of 'completing the square': $5x^{2} - 4x - 10 = 0$.