The symbol used for the discriminant of a quadratic equation is $\ldots \ldots \ldots \ldots .$

If the following quadratic equation has two equal and real roots,then find the value of $k$: $2kx^2 - 8x + k = 0$.

Find the discriminant of the following quadratic equation and hence determine the nature of the roots of the equation: $3x^{2} - 18x + 27 = 0$.

State whether the quadratic equation $2 x^{2}-6 x+\frac{9}{2}=0$ has two distinct real roots. Justify your answer.

The formula for finding the value of $D$ (discriminant) for a quadratic equation $ax^2 + bx + c = 0$ is $\ldots \ldots \ldots \ldots.$

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