Which of the following is a quadratic equation in one variable?

State whether the following statements are true or false. Justify your answers.
$(i)$ Every quadratic equation has at most two roots.
$(ii)$ If the coefficient of $x^{2}$ and the constant term of a quadratic equation have opposite signs,then the quadratic equation has real roots.

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

The length of the hypotenuse of an isosceles right-angled triangle is $8 \, cm$. Find the length of the other two sides.

Solve the following equation using the method of factorization: $\frac{x-1}{x-2} + \frac{x-3}{x-4} = 3 \frac{1}{3}$ (where $x \neq 2, 4$).

Obtain the roots of the following quadratic equation using the quadratic formula: $\sqrt{3} x^{2} + 10 x - 8 \sqrt{3} = 0$.