The value of the discriminant $D$ of the quadratic equation $x^{2} + 5x + 1 = 0$ is $\ldots \ldots \ldots \ldots .$

Examine whether the following equation is quadratic or not: $(x+6)(x+5)=0$.

Examine whether the following equation is quadratic or not: $(x+2)^{3} = x(x^{2}-1)$

Find the roots of the following quadratic equation by the method of completing the square: $16 x^{2}-24 x-1=0$

Solve the following equation using the method of factorization and write its solution set: $\frac{x}{x+1} + \frac{x+1}{x} = \frac{41}{20}$

State whether the following statements are true or false. Justify your answers.
$(i)$ Every quadratic equation has exactly one root.
$(ii)$ Every quadratic equation has at least one real root.