Question

If $5x^2-2kx+1<0$ has exactly one integral solution, then find the sum of all positive integral value of $k$.

Answer 1

Given relation is $5x^2-2kx+1<0$
as the coefficient of $x^2$ is positive,
having exactly one integral solution is possible only if the difference between the roots is less than 2
Let a and b be the roots of the given equation.
$\Rightarrow {(a-b)}^2=\frac{4k^2}{25}-\frac{4}{5}\Rightarrow (a-b)=\sqrt{\frac{4k^2-20}{25}}=\frac{\sqrt{4k^2-20}}{5}<2\Rightarrow k^2<30and4k^2-20>0\Rightarrow k^2>5\Rightarrow 5<k^2<30andkbeingapositiveinteger\Rightarrow k=3,4,5$
substitute it in the given equation, gives the possible values of k to be 4 and 5
Therefore the sum of all positive integral values of k will be $9$.