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$a, b, c$ 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. $\beta^{2} \neq 4ac$
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In general, for a quadratic equation  $ax^{2}+bx+c=0$ , sum of roots is given by :  $\frac{-b}{a}$  and product of roots by :  $\frac{-c}{a}$

$\therefore$  Here both the roots are $\beta$.

So,  according to the given quadratic equation, we have  

  • $2\beta=\frac{b}{a}\quad \to (1)$ and  
  • $\beta^{2}=\frac{c}{a}\quad \to (2)$

Multiplying equations $(1)$ and $(2)$ and simplifying, we get  $\beta^{3}=\frac{bc}{2a^{2}}$  
Option C is correct

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