$a, b, c$ are real numbers. The quadratic equation $ax^{2}-bx+c=0$ has equal roots, which is $\beta$, then

1. $\beta =b/a$
2. $\beta^{2} =ac$
3. $\beta^{3} =bc/\left ( 2a^{2} \right )$
4. $\beta^{2} \neq 4ac$

edited

In general, for a quaratic equation  $ax^{2}+bx+c=0$ , sum of roots is defined as :  $\frac{-b}{a}$  and product of roots is defined as :  $\frac{-c}{a}$
$\therefore$  Here both the roots are $\beta$.
So,  according to the given quadratic equation , we have  $2\beta=\frac{b}{a}$ ---(1)  and  $\beta^{2}=\frac{c}{a}$-----(2)
Multiplying equations (1) and (2) and simplifying,  we get  $\beta^{3}=\frac{bc}{2a^{2}}$