If the value of the discriminant of a quadratic equation $6x^{2}-x-k=0$ is $25$,then $k = \ldots$

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

If both the roots of a quadratic equation $ax^2 + bx + c = 0$ $(a \neq 0)$ are equal,then ..... .

State whether the quadratic equation $x^{2}-3x+4=0$ has two distinct real roots. Justify your answer.

Does there exist a quadratic equation whose coefficients are rational but both of its roots are irrational? Justify your answer.

Find the discriminant of the following quadratic equation and hence determine the nature of the roots of the equation: $x^{2}+5x+5=0$