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.

(A) $(i)$ True. $A$ quadratic equation is of the form $ax^{2} + bx + c = 0$ where $a \neq 0$. By the Fundamental Theorem of Algebra,a polynomial of degree $n$ has at most $n$ roots. Since a quadratic equation has degree $2$,it can have at most $2$ roots.
$(ii)$ True. For a quadratic equation $ax^{2} + bx + c = 0$,the discriminant is $D = b^{2} - 4ac$. If the coefficient of $x^{2}$ $(a)$ and the constant term $(c)$ have opposite signs,then $ac < 0$. Consequently,$-4ac > 0$. Since $b^{2} \geq 0$,it follows that $D = b^{2} - 4ac > 0$. Because the discriminant is positive,the quadratic equation must have two distinct real roots.