For a quadratic equation,if $\ldots \ldots \ldots \ldots$ then both the roots of an equation are distinct and real.

Find the roots of the following quadratic equation by the method of completing the square: $5x^{2}-4x-10=0$

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.

Solve the following equation using the method of factorization and write its solution set: $\frac{x}{x+1} + \frac{x+1}{x} = \frac{5}{2}$

The necessary condition for $ax^2 + bx + c = 0$,where $a, b, c \in R$,to be a quadratic equation is:

Obtain the roots of the following quadratic equation by using the general formula for the solution: $3x^{2} - 2x + 2 = 0$