If the following quadratic equation has two equal and real roots,then find the value of $k$: $x^{2} + 2kx + 4 = 0$.

Are the following statements 'True' or 'False'? Justify your answers.
The only value of $k$ for which the quadratic polynomial $k x^{2}+x+k$ has equal zeros is $\frac{1}{2}$.

Find whether the following equation has real roots. If real roots exist,find them.
$8 x^{2}+2 x-3=0$

If both the roots of $25x^{2} - x(m - 2) - 1 = 0$ are opposite,then $m = \ldots$

Find the discriminant of the following quadratic equation: $4x^{2} - 12x + 9 = 0$.

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