Solve the following equation using the method of factorization and write its solution set: $\frac{x+2}{x} + \frac{x}{x+2} = \frac{10}{3}$

Write whether the following statements are true or false. Justify your answers.
$(i)$ If the coefficient of $x^{2}$ and the constant term have the same sign and if the coefficient of $x$ term is zero,then the quadratic equation has no real roots.
$(ii)$ Every quadratic equation has at least two roots.

Examine whether the following equation is quadratic or not: $x^{2}+\frac{1}{x^{2}}=-2$ $(x \neq 0)$

Find the discriminant of the following quadratic equation: $x^{2}-7x+10=0$

Examine whether the following equation is quadratic or not: $x^{3}-4x^{2}-x+1=(x-2)^{3}$

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