Be careful!

In the expression $2x^2 - 5x + K$ ,
$a = 2 ; b = -5; c = K$
We see that $a = 2 > 0$
So, if we take $D$ (discriminant) $> 0$ , then the expression is not positive for $\alpha < \beta$ , where $\alpha$ is the smaller root of the expression and $\beta$ is the greater root of the expression.

If we take $D = 0$ , then at $-\dfrac{b}{2a}$ , value of the expression is $0$ .

Now, if we take $D < 0$ , then value of the expression is always positive.

$\therefore b^2 - 4ac < 0\\ \implies 5^2 - 4×2×K < 0\\ \implies 25 - 8K < 0\\ \implies -8K < -25\\ \implies K > \dfrac{-25}{-8}\\ \implies K > 3.125\\ \therefore x \in \boxed{(3.125 , \infty)}$