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Q. Sum of all values of $n \in N$ for which the equation $\frac{x-8}{n-10}=\frac{n}{x}$ has no solution, equals

Complex Numbers and Quadratic Equations

Solution:

$x ^2-8 x -\left( n ^2-10 n \right)=0 $
$D =64+4\left( n ^2-10 n \right)< 0$
$n ^2-10 n +16< 0 $
$( n -8)( n -2)< 0 $
$n \in(2,8) \quad \text { also }( n =10 $
$\text { Sum of all values of } n =3+4+5+6+7+10=35$