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The quadratic equation $x^{2}+bx+c=0$ has two roots $4a$ and $3a$, where a is an integer. Which of the following is a possible value of $b^{2}+c$?

  1. $3721$
  2. $549$
  3. $427$
  4. $361$
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Ans is an option (B) 

Sum of the roots $=4a+3a=7a=\frac{-b}{1}$    $\Rightarrow$  $b=-7a \quad \longrightarrow (1)$  

Product of roots  $=4a\times3a=12a^{2}=\frac{c}{1}$   $\Rightarrow$   $c=12a^{2} \quad \longrightarrow (2)$  

Squaring $(1)$ and adding to $(2),$ we get  $b^{2}+c=61a^{2}$

Among all the options, only $549$ satisfies the above equation with $a$ being an integer i.e. $\pm 3$. In all other cases, $a$ is not an integer.

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