If the value of the discriminant of the quadratic equation $2x^{2} + 5x - k = 0$ is $81$,then $k = \dots$

Does there exist a quadratic equation whose coefficients are rational but both of its roots are irrational? 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 roots of the following quadratic equation by using the quadratic formula,if they exist: $x^{2}-5x-1=0$

Solve the following equation using the quadratic formula,if the equation has a solution in $R$: $\frac{1}{x+1} + \frac{2}{x+2} = \frac{4}{x+4}; \, x \neq -1, -2, -4$.

The length of a rectangular garden exceeds its breadth by $6 \, m$. If the area of the garden is $216 \, m^2$,find the length and the breadth of the garden.